Parameterized Kolmogorov–Smirnov Test for Normality

Abstract

The first (main) aim of the article is to define and practically apply the parameterized Kolmogorov–Smirnov (PKS) goodness-of-fit test for normality. The second contribution is to expand the family of empirical distribution functions with four new proposals. The third contribution is to construct a family of alternative distributions that includes both older and newer distributions. The fourth contribution is to calculate the power of the analyzed tests under alternative distributions with parameters chosen to be similar to the normal distribution in various ways. The new proposal is distinguished for left-skewed alternative distributions and symmetric distributions characterizing by negative excess kurtosis. The effectiveness of the PKS test is also illustrated by the analysis of thirty real data sets.

1. Introduction

Goodness-of-fit tests (GoFTs) are applied in many scientific fields, including economics and finance (to analyze market behavior, assess market efficiency, identify deviations, and analyze stochastic processes), demography (regarding the fertility curve distribution), and econometrics (to check if regression errors are normally distributed for proper model evaluation).

The Kolmogorov–Smirnov (KS) test is among the most widely utilized normality testing procedures implemented in statistical software [1,2]. This method is categorized within the class of tests based on empirical distribution functions (ECDFs). The other tests are: the Cramér–von Mises (CM) test [3,4], Lilliefors (LF) test [5], Kuiper (K) test [6], Watson (W) test [7], and Anderson–Darling (AD) test [8].

Let $x_{\left(1\right)},x_{\left(2\right)},\dots ,x_{\left(n\right)}$ be random values observed on a sample of size n drawn from some general population. The values are independent, as is understood in probability theory. Moreover, all $x_{\left(i\right)}$ follow the same probability distribution. As a general rule, analysts do not know what is the actual probability distribution. The only thing they can do is to set up an $H_0$ hypothesis that the population distribution has a form $F\left(x\right)$. For the $H_0$ to be accepted, it must pass the goodness-of-fit test. Very often, analysts assume $H_0:F\left(x\right)=\Phi \left(x\right)$ against $H_1:F\left(x\right)\ne \Phi \left(x\right)$, where $\Phi \left(x\right)$ is the CDF of the Normal distribution. The EDF has a form $F_n\left(x\right)={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\mathit{\theta }\left(x-x_i\right)$, where $\mathit{\theta }\left(u\right)$ is the Heaviside step function.

The $\delta$-corrected KS test [9], which was studied by Khamis [10,11,12], involves redefining the EDF value at each observation and comparing it against the CDF at those same points. Let the EDF at the i-th data point be defined as follows:

$$F_{\delta }\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-\delta }{n-2\delta +1}},0\le \delta \le 1.$$

(1)

In his study, Harter [9] examined the specific values of $\delta =0,0.5,1$.

Bloom [13] introduced the $\alpha ,\beta$ transformation:

$$F_{\alpha ,\beta }\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-\alpha }{n-\alpha -\beta +1}},0\le \alpha ,\beta \le 1$$

(2)

to decrease the mean square error (MSE) of certain statistics. Note that $F_{\delta ,\delta }\left(x\right)=F_{\delta }\left(x\right)$. The transformation (2) was used to create the GoFTs.

Using Bloom’s formula, Sulewski [14] proposed a one-component LF GoFT with the statistic defined as

$$LF\left(\alpha ,\beta \right)=\underset{1\le i\le n}{\underbrace{max}}\left\{\left|F_{\alpha ,\beta }\left(x_{\left(i\right)}\right)-\Phi \left(x_{\left(i\right)}\right)\right|\right\}.$$

(3)

It is well-established that the maximum discrepancy between theoretical and empirical distribution functions can occur at various points within a series. The probability of this discrepancy occurring at a specific positional statistic $\mathrm{r}$ decreases as $\mathrm{r}$ reaches more extreme values. This observation led to the development of the two-component test statistic described in [15]. The first component, consistent with the original LF test, represents the absolute value of the maximum difference between the sample and population distributions, while the second component identifies the position within the ordered sample where this maximum discrepancy occurs.

Sulewski and Stoltmann [16] used Bloom’s formula to create the modified CM (MCM) GoFT with statistic

$$MCM\left(\alpha ,\beta \right)={\displaystyle \frac{1}{12n}}+\sum _{i=1}^n{\left[F_{\alpha ,\beta }\left(x_{\left(i\right)}\right)-\Phi \left(x_{\left(i\right)}\right)\right]}^2.$$

(4)

Simulation studies evaluating the MCM test, alongside the one- and two-component LF tests, were conducted using the following methods of calculating $F_{\alpha ,\beta }\left(x_{\left(i\right)}\right)\left(0\le \alpha ,\beta \le 1\right)$:

• $F_{0,1}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i}{n}}$—occurs in the KS statistic,
• $F_{1,0}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-1}{n}}$—occurs in the KS statistic,
• $F_{0.5,0.5}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.5}{n}}={\displaystyle \frac{2i-1}{2n}}$—appears in the CM statistic, expressed as the sum,
• $F_{0,0}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i}{n+1}}$—the i-th order statistic’s mean for the beta distribution
• $F_{0.3,0.3}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.3}{n+0.4}}$—the beta distribution’s i-th order statistic median,
• $F_{0.375,0.375}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.375}{n+0.25}}$ – the mean of the i-th order statistic of the Gaussian distribution,
• $F_{0.3175,0.3175}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.3175}{n+0.365}}$—founded by Filliben [17],
• $F_{1,1}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-1}{n-1}}$ – founded by Harter [9].

Six of the EDF definitions listed above (except $F_{0,1}$ and $F_{1,0}$) have $\alpha =\beta$.

Recently, numerous studies have focused on goodness-of-fit tests (GoFTs) for normality.

Table A1. Literature devoted to normal GoFTs created in the 21st century [16].

Article	Sample Sizes	Article	Sample Sizes
Bonett and Seier [29]	10, 20,…, 50, 100	Afeez et al. [30]	10, 30, 50, 100, 300, 500,1 000
Aliaga et al. [31]	-	Marange and Qin [32]	15, 30, 50, 80, 100, 150, 200
Bontemps and Meddahi [33]	100, 250, 500, 1000	Sulewski [34] (2019)	10, 12,…, 30, 40, 50
Luceno [35]	100	Tavakoli et al. [36]	5, 6,…, 15, 20, 25, 30, 40,50,…, 100
Yazici and Yolacan [37]	20, 30, 40, 50	Mishra et al. [38]	$n<30$, $n>30$
Gel et al. [39]	20, 50, 100	Kellner and Celisse [40]	50, 75, 100, 200, 300, 400
Coin [41]	20, 50, 200	Wijekularathna et al. [42]	5, 10, 20, 30, 50, 75, 100, 200, 500, 1000, 2000
Brys et al. [43]	100, 1000	Sulewski [14]	10, 14, 20
Gel and Gastwirth [44]	30, 50, 100	Hernandez [24]	5, 10,…, 30
Romao et al. [45]	25, 50, 100	Khatun [46]	10, 20, 25, 30, 40, 50, 100, 200, 300
Razali and Wah [47]	20, 30, 50, 100, 200,…, 500, 1000, 2000	Arnastauskaitė et al. [48]	$2^5$, $2^6$,…, $2^{1}0$
Noughabi and Arghami [49]	10, 20, 30, 50	Bayoud [50]	10, 20,…, 50, 60, 80, 100
Yap and Sim [51]	10, 20, 30, 50, 100, 300, 500, 1000, 2000	Uhm and Yi [52]	10, 20, 30, 100, 200, 300
Chernobai et al. [53]		Sulewski [18]	20, 50, 100
Ahmad and Khan [54]	10, 20,…, 50, 100, 200, 500	Desgagné et al. [55]	20, 50, 100, 200
Mbah and Paothong [56]	10, 20, 30, 50, 100, 200,…, 500, 1000, 2500, 5000	Uyanto [27]	10, 30, 50, 70, 100
Torabi et al. [21]	10, 20	Ma et al. [28]	10, 30, 50
Feuerverger [57]	200	Giles [58]	10, 25, 50, 100, 250, 500, 1000
Nosakhare and Bright [59]	5, 10,…, 50, 100	Borrajo et al. [60]	50, 100, 200, 500
Desgagné and Lafaye de Micheaux [61]	10, 12,…, 20, 50, 100, 200	Terán-García and Pérez-Fernández [62]	25, 900

(see Appendix A) summarizes publications from the 21st century alongside their analyzed sample sizes, with values of $n\le 50$ highlighted in bold. These small sample sizes, which predominate in Table A1, are particularly relevant in fields such as experimental economics, where studies involving only a dozen or so participants per group are frequently published. This is where strong tests can be put to great use. It may be that the results are as follows: in the original paper, the hypothesis was accepted, whereas under a stronger test, the hypothesis is rejected.

The first (main) aim of the article is to define and practically apply the parameterized Kolmogorov–Smirnov goodness-of-fit test for normality. These modifications consist of varying the formula used to compute the empirical distribution function (EDF). The second contribution is to expand the EDF family with four new proposals. The third contribution is to construct a family of alternative distributions, comprising both older and newer distributions, which, owing to their flexibility, span all combinations of skewness and kurtosis signs. Critical values are obtained using ${10}^6$ order statistics for sample sizes $n=10,20$ and at a significance level $0.05$. The fourth contribution is to compute the power of the analyzed tests under alternative distributions, based on ${10}^5$ test-statistic values, with parameters selected to be similar to the normal distribution in various ways. The effectiveness of the analyzed tests is demonstrated using thirty real datasets. The new proposal is distinguished by left-skewed alternative distributions and symmetric distributions characterized by negative excess kurtosis.

The rest of this article is structured as follows. In Section 2, we define the parameterized KS GoFT test for normality. Section 3 discusses the similarity measures between the alternatives and the normal distribution. In Section 4, we introduce the alternative distributions, categorized into nine groups based on their skewness and excess kurtosis. Section 5 presents the power study, followed by real-world data applications in Section 6. Finally, Section 7 offers concluding remarks, while other material is provided in Appendix A.

2. Parameterized Kolmogorov–Smirnov Test for Normality

Before we present the parameterized KS test (the first contribution of the paper), we would like to expand the EDF family (the second contribution of the paper) with four new variants, $F_{0.1,0.1},F_{0.9,0.1},F_{0.9,0.9}$ and $F_{0.1,0.9}$, given by the following:

$$F_{{\displaystyle \frac{1}{10}},{\displaystyle \frac{1}{10}}}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.1}{n+0.8}},$$

$${F_{{\displaystyle \frac{9}{10}},{\displaystyle \frac{1}{10}}}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.9}{n}},F}_{{\displaystyle \frac{9}{10}},{\displaystyle \frac{9}{10}}}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.9}{n-0.8}},F_{{\displaystyle \frac{1}{10}},{\displaystyle \frac{9}{10}}}\left(x_{\left(i\right)}\right)={\displaystyle \frac{i-0.1}{n}},$$

thus, eight of the EDF definitions listed earlier are on the line $\beta =\alpha$ (blue line), and five of them are on the line $\beta =-\alpha +1$ (green line) (see Figure 1). The previously analyzed values are unevenly distributed along the $\beta =\alpha$ line in the interval $\left[0,1\right]$. Four values belong to the interval $\left[0.3,0.5\right]$. The value $0.1$ represents the interval $\left(0,0.3\right)$, and the values $0.9$ represent the interval $\left(0.5,1\right)$. The new representatives of $\beta =-\alpha +1$ line, also located at the corners of the square, are EDFs with $\alpha =0.9, \beta =0.1$ and $\alpha =0.1, \beta =0.9$.

Let $\widehat{x}=\left({\sum }_{i=1}^n{x}_i\right)/n, s^2=\left({\sum }_{i=1}^n{\left(x_i-\widehat{x}\right)}^2\right)/\left(n-1\right), z_{\left(i\right)}=\left(x_{\left(i\right)}-\overline{x}\right)/s$. Let us remember that the KS test statistic is given by the following formula:

$$KS=max\left\{\underset{1\le i\le n}{\underbrace{max}}\left[{\displaystyle \frac{i}{n}}-\Phi \left(z_{\left(i\right)}\right)\right],\underset{1\le i\le n}{\underbrace{max}}\left[\Phi \left(z_{\left(i\right)}\right)-{\displaystyle \frac{i-1}{n}}\right]\right\}.$$

(5)

Our idea is to parametrize the EDF in (5) using Bloom’s formula (2). Parametrized KS (PKS) test statistic is defined as follows:

$$PKS\left(\alpha ,\beta \right)=max\left\{\underset{1\le i\le n}{\underbrace{max}}\left[{\displaystyle \frac{i-\alpha }{n-\alpha -\beta +1}}-\Phi \left(z_{\left(i\right)}\right)\right],\underset{1\le i\le n}{\underbrace{max}}\left[\Phi \left(z_{\left(i\right)}\right)-{\displaystyle \frac{i-\alpha -1}{n-\alpha -\beta +1}}\right]\right\}.$$

(6)

Note that $PKS\left(0,1\right)=KS$. It should be noted that the $PKS\left(0,1\right)$ statistic is equivalent to the $KS$ statistic.

Sulewski and Stoltmann [16] as well as Sulewski [14,18] showed that noteworthy parameterized tests are defined using $F_{\alpha ,\beta }\left(x_{\left(i\right)}\right)\left(\alpha ,\beta \in \left\{0,1\right\}\right)$, so the values of $\left(\alpha ,\beta \right)$ chosen for the simulation study, except for new proposals, are as follows: $\left(0,0\right),\left(1,0\right),\left(1,1\right),\left(0,1\right)$.

3. Similarity Measure

Let us assume that $m_k={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(x_{\left(i\right)}-\overline{x}\right)}^k$ and ${\gamma }_1={\displaystyle \frac{m_3}{s^3}}$, ${\overline{\gamma }}_2={\displaystyle \frac{m_4}{s^4}}-3$. The Malachov inequality is defined as ${\overline{\gamma }}_2\ge {{\gamma }_1}^2-2$ [19]. This formula not only presents the relationship between two characteristics, but also helps to choose the parameter values of the Edgeworth series and Pearson distribution.

A review of the current statistical literature reveals that cases characterized by low skewness ${\gamma }_1$ and excess kurtosis ${\overline{\gamma }}_2$ are not predominant in normality testing. Consequently, it is of particular interest to examine how GoFTs perform when applied to samples coming from alternative distributions that closely resemble the normal distribution.

Consider an alternative distribution with the PDF $f\left(x;\mathit{\theta }\right)$, where $\mathit{\theta }$ denotes the vector of parameters. The similarity measure M between the alternative distribution (A) and the normal distribution is given by [14]

$$M_A\left(\mathit{\theta };\mu ,\sigma \right)={\int }_{-\infty }^{\infty }min\left[f\left(x;\mathit{\theta }\right),\varphi \left(x;\mu ,\sigma \right)\right]dx,$$

(7)

where $\varphi \left(x;\mu ,\sigma \right)$ denotes the PDF of the normal distribution. The measure $M_A\left(\mathit{\theta };\mu ,\sigma \right)$ is bounded within the interval $\left[0,1\right]$, reaching unity only when the PDFs are identical. Calculated as a definite integral, this measure represents the shared area beneath the probability density curves.

Figure 2 shows values of the similarity measure $M_A\left(\mathit{\theta };\mu ,\sigma \right)$, when an alternative distribution is the Student t distribution with v degrees of freedom. Note that if $v\to +\infty$, then obviously $M_t\left(v;0,1\right)\to 1$.

4. Alternative Distributions

As previously noted, a vast body of literature is dedicated to normality testing. These studies employ a wide range of alternative distributions, encompassing both symmetric and asymmetric forms. Notably, the Cauchy and slash distributions serve as examples of symmetric alternatives for which ${\gamma }_1$ and ${\overline{\gamma }}_2$ remain undefined.

Alternative distributions can be classified into four primary groups based on their support and the shape of their density functions (see e.g., [20,21]. These categories comprise symmetric and asymmetric distributions with support on $\left(-\infty ,\infty \right)$, as well as those with support on $\left(0,\infty \right)$ and $\left(0,1\right)$. Alternative distributions are also classified into five groups based on symmetry and tail behavior (asymmetric short/long-tailed, symmetric short/long-tailed, and symmetric close-to-normal), as proposed in [21,22,23]. Sulewski [14,15] divided alternatives into twelve groups, A1–F2, based on their ${\gamma }_1$ and ${\overline{\gamma }}_2$ signs as well as bimodality.

We propose a classification of the alternative distributions into nine distinct groups based on the signs of their skewness ${\gamma }_1$ and excess kurtosis ${\overline{\gamma }}_2$ [16]. These categories, designated as Group 0 and Groups A–H, are formally defined in

Table 1. Classification of alternatives based on the signs (values) of ${\gamma }_1$ and ${\overline{\gamma }}_2$.

Group	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	Group	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$
0	zero	zero
A	positive	positive	E	positive	negative
B	negative	positive	F	negative	negative
C	zero	positive	G	positive	zero
D	zero	negative	H	negative	zero

.

The primary criterion for selecting alternative distributions for Monte Carlo simulations is that the skewness ${\gamma }_1$ and excess kurtosis ${\overline{\gamma }}_2$, calculated from their parameters, must fall within the designated groups. Five distributions defined on an infinite domain satisfy this requirement: the Edgeworth series (ES) and Pearson (P) distributions—both monolithic models with ${\gamma }_1$ and ${\overline{\gamma }}_2$ parameters—as well as the normal mixture (NM). Additionally, this includes the normal distribution with a plasticizing component (NDPC), a mixture of normal and non-normal distributions, and the plasticizing component mixture (PCM), comprising two identical non-normal distributions.

The second criterion for selecting an alternative distribution for Monte Carlo simulation is that the ${\gamma }_1$ and ${\overline{\gamma }}_2$ values, calculated based on the distribution’s parameters, must fall within all analyzed groups with the exception of Group 0.This criterion is fulfilled by the Laplace mixture (LM) distribution belonging to the groups A–H and defined in an infinite domain.

The third criterion for selecting an alternative for Monte Carlo simulation is that the ${\gamma }_1$ and ${\overline{\gamma }}_2$ values, calculated from the distribution’s parameters, must fall within the analyzed groups, with the exception of two specific categories. These alternatives can be very similar to the normal distribution. This criterion is fulfilled by the SB Johnson distribution (except groups 0, C) and the SU Johnson distribution (except groups 0, D).

The fourth criterion for selecting an alternative for Monte Carlo simulation is that the ${\gamma }_1$ and ${\overline{\gamma }}_2$ values, calculated based on the distribution’s parameters, must fall within Groups C and D. These symmetric alternatives can be very similar to the normal distribution. This criterion is satisfied by the extended easily changeable kurtosis (EECK) distribution defined on a finite domain and the exponential power (EP) distribution defined on an infinite domain.

PDFs of selected alternatives and their special cases are presented in Appendix A.

Let $\left({\gamma }_1,{\overline{\gamma }}_2\right)$ be the coordinates of a point described by skewness and excess kurtosis, respectively. For every alternative, values of $\left({\gamma }_1,{\overline{\gamma }}_2\right)$ are calculated for ${10}^4$ randomly determined values of parameters influencing ${\gamma }_1$ and ${\overline{\gamma }}_2$ in the Malakhov area (MA) ${\overline{\gamma }}_2\ge {\gamma }_1^2-2$ [19]. If ${\overline{\gamma }}_2\in \left[-2,14\right]$, then ${\gamma }_1\in \left[-4,4\right]$. The parameter ranges of the alternatives are selected to maximize MA filling (see

Table 2. A numerical range of ${\gamma }_1$ and ${\overline{\gamma }}_2$ for alternatives in the MA $14\ge {\overline{\gamma }}_2\ge {\gamma }_1^2-2$.

Alternative	Parameter Ranges	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$
ES	${\gamma }_1\in \left(-4,4\right),{\overline{\gamma }}_2\in \left(-2,14\right)$	$\left[-3.986,3.972\right]$	$\left[-1.991,13.998\right]$
P	${\gamma }_1\in \left(-4,4\right),{\overline{\gamma }}_2\in \left(-2,14\right)$	$\left[-3.972,3.924\right]$	$\left[-1.984,13.997\right]$
NM	${\mu }_1,{\mu }_2\in \left(-100,100\right)$${\sigma }_1,{\sigma }_2\in \left(0,50\right),\omega \in \left(0,1\right)$	$\left[-3.926,3.779\right]$	$\left[-1.999,13.979\right]$
NDPC	${\mu }_1,{\mu }_2\in \left(-100,100\right),c_2\in \left(1,100\right)$${\sigma }_1,{\sigma }_2\in \left(0,50\right),\omega \in \left(0,1\right)$	$\left[-3.999,3.983\right]$	$\left[-2.000,14.000\right]$
PCM	${\mu }_1,{\mu }_2\in \left(-100,100\right),c_1,c_2\in \left(1,100\right)$${\sigma }_1,{\sigma }_2\in \left(0,50\right),\omega \in \left(0,1\right)$	$\left[-3.982,3.936\right]$	$\left[-1.999,13.993\right]$
LM	${\mu }_1,{\mu }_2\in \left(-100,100\right)$${\sigma }_1,{\sigma }_2\in \left(0,50\right),\omega \in \left(0,1\right)$	$\left[-3.891,3.814\right]$	$\left[-1.987,13.992\right]$
SB	$a\in \left(-5,5\right),b\in \left(0,10\right)$	$\left[-3.728,3.483\right]$	$\left[-1.999,13.964\right]$
SU	$a\in \left(-5,5\right),b\in \left(0,10\right)$	$\left[-2.584,2.586\right]$	$\left[0.041,13.994\right]$
EECK	$a\in \left(-1,100\right),b\in \left(0,50\right)$	0	$\left[-1.977,13.728\right]$
EP	$a\in \left(-50,50\right),b,c\in \left(0,50\right)$	0	$\left[-2,13.592\right]$

). Figure 3, Figure 4, Figure 5 and Figure 6 present sets of points $\left({\gamma }_1,{\overline{\gamma }}_2\right)$ located in the MA for non-symmetric alternatives. Interestingly, the MA for SB and SU are separate; they complement each other.

Additionally, we employ the skewness–kurtosis–square (SKS) measure to evaluate the flexibility of the alternative distributions. For this purpose, circles of diameter $\delta$ are mapped within the MA space, with their center coordinates determined by the respective values of ${\gamma }_1$ and ${\overline{\gamma }}_2$. Subsequently, the fraction of the colored area is determined. Utilizing squares with sides of length $\delta$ serves as a practical alternative to circles, as this approach simplifies the computation of the total colored area. In cases where squares overlap, the area is counted only once to avoid double-counting. The SKS measure is thus defined as follows [15]:

$${SKS}_{\delta }=SS/TT,$$

where $TT$ denotes the total number of squares within the MA region, and $SS$ represents the number of squares containing at least one $\left({\gamma }_1,{\overline{\gamma }}_2\right)$ point. The ${SKS}_{\delta }$ measure is bounded within the interval $\left[0,1\right]$, where the maximum value signifies an ideal dispersal of the $\left({\gamma }_1,{\overline{\gamma }}_2\right)$ coordinates throughout the MA space.

Table 2 shows that the numerical ranges of ${\gamma }_1$ and ${\overline{\gamma }}_2$ for asymmetric alternatives in the MA $14\ge {\overline{\gamma }}_2\ge {\gamma }_1^2-2$, due to the appropriately randomly selected parameter of the alternatives, except the SU, are similar. The range is not the most important. The interior is also important and therefore we also present values of SKS measures obtained for square side $\delta =0.5,0.1,0.075,0.05$ (see

Table 3. The SKS measure for alternatives in the MA $14\ge {\overline{\gamma }}_2\ge {\gamma }_1^2-2$ for the square side $\delta$.

Alternative	${\mathbf{SKS}}_{\mathit{\delta }=0.5}$	${\mathbf{SKS}}_{\mathit{\delta }=0.1}$	${\mathbf{SKS}}_{\mathit{\delta }=0.075}$	${\mathbf{SKS}}_{\mathit{\delta }=0.05}$
ES	0.9764	0.6733	0.4744	0.2507
P	0.9712	0.6791	0.4754	0.2524
NM	0.8874	0.3727	0.2779	0.1701
NDPC	0.9529	0.3845	0.2823	0.1677
PCM	0.9607	0.3644	0.2620	0.1546
LM	0.9005	0.3697	0.2770	0.1698
SB	0.4162	0.1052	0.0736	0.0432
SU	0.3953	0.0945	0.0687	0.0428
EECK	0	0	0	0
EP	0	0	0	0

). Figure 3 and Table 3 confirm what could be expected, that the most flexible distributions are P and ES.

We select specific values for the alternative parameters to achieve the appropriate similarity measure M. This measure is predominantly analyzed using values of $M=0.5,0.75,0.9$.

Appendix A contains

Table A2. Vectors of ES parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups 0, A–H.

Group	$\mathit{\theta }=\left({\mathit{\gamma }}_{1,},{\overline{\mathit{\gamma }}}_2\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
0	$\left(0,0\right)$	0	1	0	0	$M\left(\mathit{\theta };0,1\right)=1$
	(0.4, 3.33)	0	1	0.4	3.33	$M\left(\mathit{\theta };0,1\right)=0.8$
A	(0.3, 2.499)	0	1	0.3	2.499	$M\left(\mathit{\theta };0,1\right)=0.85$
	(0.2, 1.666)	0	1	0.2	1.666	$M\left(\mathit{\theta };0,1\right)=0.9$
	(−0.4, 3.33)	0	1	−0.4	3.33	$M\left(\mathit{\theta };0,1\right)=0.8$
B	(−0.3, 2.499)	0	1	−0.3	2.499	$M\left(\mathit{\theta };0,1\right)=0.85$
	(−0.2, 1.666)	0	1	−0.2	1.666	$M\left(\mathit{\theta };0,1\right)=0.9$
	(0, 3.428)	0	1	0	3.428	$M\left(\mathit{\theta };0,1\right)=0.8$
C	(0, 2.571)	0	1	0	2.571	$M\left(\mathit{\theta };0,1\right)=0.85$
	(0, 1.71)	0	1	0	1.71	$M\left(\mathit{\theta };0,1\right)=0.9$
	(0, −3.428)	0	1	0	−3.428	$M\left(\mathit{\theta };0,1\right)=0.8$
D	(0, −2.571)	0	1	0	−2.571	$M\left(\mathit{\theta };0,1\right)=0.85$
	(0, −1.71)	0	1	0	−1.71	$M\left(\mathit{\theta };0,1\right)=0.9$
	(1.39, −0.067)	0	1	1.39	−0.067	$M\left(\mathit{\theta };0,1\right)=0.825$
E	(1.175, −0.46)	0	1	1.175	−0.46	$M\left(\mathit{\theta };0,1\right)=0.85$
	(0.775, −0.408)	0	1	0.775	−0.408	$M\left(\mathit{\theta };0,1\right)=0.9$
	(−1.39, −0.067)	0	1	−1.39	−0.067	$M\left(\mathit{\theta };0,1\right)=0.825$
F	(−1.175, −0.46)	0	1	−1.175	−0.46	$M\left(\mathit{\theta };0,1\right)=0.85$
	(−0.775, −0.408)	0	1	−0.775	−0.408	$M\left(\mathit{\theta };0,1\right)=0.9$
	(1.391, 0)	0	1	1.391	0	$M\left(\mathit{\theta };0,1\right)=0.825$
G	(1.19, 0)	0	1	1.19	0	$M\left(\mathit{\theta };0,1\right)=0.85$
	(0.795, 0)	0	1	0.795	0	$M\left(\mathit{\theta };0,1\right)=0.9$
	(−1.391, 0)	0	1	−1.391	0	$M\left(\mathit{\theta };0,1\right)=0.825$
H	(−1.19, 0)	0	1	−1.19	0	$M\left(\mathit{\theta };0,1\right)=0.85$
	(−0.795, 0)	0	1	−0.795	0	$M\left(\mathit{\theta };0,1\right)=0.9$

,

Table A3. Vectors of the P parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups 0, A–H.

Group	$\mathit{\theta }=\left({\mathit{\gamma }}_{1,},{\overline{\mathit{\gamma }}}_2\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
0	(0, 0)	0	1	0	0	$M(\mathit{\theta };0,1)=1$
	(2.04, 4.1)	0	1	2.04	4.1	$M(\mathit{\theta };0,1)=0.5$
A	(1.62, 3.845)	0	1	1.62	3.845	$M(\mathit{\theta };0,1)=0.75$
	(0.9, 2)	0	1	0.9	2	$M(\mathit{\theta };0,1)=0.9$
	(−2.04, 4.1)	0	1	−2.04	4.1	$M(\mathit{\theta };0,1)=0.5$
B	(−1.62, 3.845)	0	1	−1.62	3.845	$M(\mathit{\theta };0,1)=0.75$
	(−0.9, 2)	0	1	−0.9	2	$M(\mathit{\theta };0,1)=0.9$
	(0, 11.2)	0	1	0	11.2	$M(\mathit{\theta };0,1)=0.9$
C	(0, 3.65)	0	1	0	3.65	$M(\mathit{\theta };0,1)=0.925$
	(0, 1.521)	0	1	0	1.521	$M(\mathit{\theta };0,1)=0.95$
	(0, −1.695)	0	1	0	−1.695	$M(\mathit{\theta };0,1)=0.5$
D	(0, −1.315)	0	1	0	−1.315	$M(\mathit{\theta };0,1)=0.75$
	(0, −0.89)	0	1	0	−0.89	$M(\mathit{\theta };0,1)=0.9$
	(0.985, −0.5)	0	1	0.985	−0.5	$M(\mathit{\theta };0,1)=0.5$
E	(0.715, −0.475)	0	1	0.715	−0.475	$M(\mathit{\theta };0,1)=0.75$
	(0.515, −0.2)	0	1	0.515	−0.2	$M(\mathit{\theta };0,1)=0.9$
	(−0.985, −0.5)	0	1	−0.985	−0.5	$M(\mathit{\theta };0,1)=0.5$
F	(−0.715, −0.475)	0	1	−0.715	−0.475	$M(\mathit{\theta };0,1)=0.75$
	(−0.515, −0.2)	0	1	−0.515	−0.2	$M(\mathit{\theta };0,1)=0.9$
	(1.164, 0)	0	1	1.164	0	$M(\mathit{\theta };0,1)=0.5$
G	(0.879, 0)	0	1	0.879	0	$M(\mathit{\theta };0,1)=0.75$
	(0.578, 0)	0	1	0.578	0	$M(\mathit{\theta };0,1)=0.9$
	(−1.164, 0)	0	1	−1.164	0	$M(\mathit{\theta };0,1)=0.5$
H	(−0.879, 0)	0	1	−0.879	0	$M(\mathit{\theta };0,1)=0.75$
	(−0.578, 0)	0	1	−0.578	0	$M(\mathit{\theta };0,1)=0.9$

,

Table A4. Vectors of the NM parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups 0, A–H.

Group	$\mathit{\theta }=\left({\mathit{\mu }}_1,{\mathit{\sigma }}_1,{\mathit{\mu }}_2,{\mathit{\sigma }}_2,\mathit{\omega }\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
0	$\left({\mu }_1,{\sigma }_1,{\mu }_2,{\sigma }_2,1\right)$	0	1	0	0	$M\left(\mathit{\theta };{\mu }_1,{\sigma }_1\right)=1$
	$\left({\mu }_1,{\sigma }_1,{\mu }_2,{\sigma }_2,0\right)$	0	1	0	0	$M\left(\mathit{\theta };{\mu }_2,{\sigma }_2\right)=1$
	(0.572, 2.472, 5.614, 3.454, 0.787)	1.646	3.408	0.685	0.755	$M(\mathit{\theta };0,1)=0.5$
A	(−0.215, 1.254, 1.979, 1.99, 0.639)	0.577	1.883	0.645	0.502	$M(\mathit{\theta };0,1)=0.75$
	(0.497, 1.376, −0.268, 0.884, 0.612)	0.2	1.265	0.287	0.249	$M(\mathit{\theta };0,1)=0.9$
	(0.502, 2.019, 1.708, 0.953, 0.36)	1.274	1.544	−0.748	1.502	$M(\mathit{\theta };0,1)=0.5$
B	(0.06, 1.437, 1.004, 0.609, 0.634)	0.406	1.285	−0.5	0.499	$M(\mathit{\theta };0,1)=0.75$
	(0.709, 0.368, −0.072, 1.115, 0.193)	0.079	1.06	−0.301	0.15	$M(\mathit{\theta };0,1)=0.9$
	(0.519, 6.599, 0.519, 1.058, 0.665)	0.519	5.416	0	1.398	$M(\mathit{\theta };0,1)=0.5$
C	(0.137, 0.581, 0.137, 2.391, 0.294)	0.137	2.034	0	1.054	$M(\mathit{\theta };0,1)=0.75$
	(0.1, 0.988, 0.1, 1.543, 0.532)	0.1	1.278	0	0.554	$M(\mathit{\theta };0,1)=0.9$
	(−0.511, 1.353, 4.293, 1.021, 0.551)	1.645	2.681	0	−1.28	$M(\mathit{\theta };0,1)=0.5$
D	(2.707, 0.013, 0.017, 1.125, 0.238)	0.657	1.509	0	−1.001	$M(\mathit{\theta };0,1)=0.75$
	(1.243, 0.621, −0.39, 0.811, 0.347)	0.111	1.09	0	−0.63	$M(\mathit{\theta };0,1)=0.9$
	(−0.475, 2.22, 5.318, 2.427, 0.721)	1.141	3.457	0.5	−0.204	$M(\mathit{\theta };0,1)=0.5$
E	(−0.019, 1.369, 2.979, 1.15, 0.829)	0.494	1.748	0.339	−0.1	$M(\mathit{\theta };0,1)=0.75$
	(2.635, 0.35, −0.015, 1.166, 0.038)	0.086	1.253	0.137	−0.075	$M(\mathit{\theta };0,1)=0.9$
	(−0.692, 0.705, 2.1, 0.679, 0.324)	1.195	1.476	−0.542	−0.852	$M(\mathit{\theta };0,1)=0.5$
F	(−0.055, 1.277, 1.781, 0.443, 0.775)	0.358	1.377	−0.3	−0.5	$M(\mathit{\theta };0,1)=0.75$
	(−0.09, 1.08, −1.581, 0.92, 0.9)	−0.239	1.155	−0.071	−0.042	$M(\mathit{\theta };0,1)=0.9$
	(2.686, 3.099, −0.964, 2.217, 0.471)	0.755	3.232	0.4	0	$M(\mathit{\theta };0,1)=0.5$
G	(−0.56, 1.465, 1.411, 1.45, 0.8)	−0.166	1.661	0.151	0	$M(\mathit{\theta };0,1)=0.75$
	(−0.286, 1.114, 0.984, 1.105, 0.801)	−0.033	1.222	0.101	0	$M(\mathit{\theta };0,1)=0.9$
	(2.425, 1.101, 0.272, 1.693, 0.526)	1.404	1.775	−0.499	0	$M(\mathit{\theta };0,1)=0.5$
H	(0.864, 1.125, −1.339, 1.241, 0.735)	0.28	1.511	−0.386	0	$M(\mathit{\theta };0,1)=0.75$
	(0.429, 1.078, −0.364, 1.228, 0.434)	−0.02	1.23	−0.1	0	$M(\mathit{\theta };0,1)=0.9$

,

Table A5. Vectors of NDPC parameter $\mathit{\theta }$, mean ${\mu }_a$,standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups 0, A–H.

Group	$\mathit{\theta }=\left({\mathit{\mu }}_1,{\mathit{\sigma }}_1,{\mathit{\mu }}_2,{\mathit{\sigma }}_2,{\mathit{c}}_2,\mathit{\omega }\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
0	(${\mu }_1,{\sigma }_1,{\mu }_2,{\sigma }_2,c_2,1$)	0	1	0	0	$M\left(\mathit{\theta };{\mu }_1,{\sigma }_1\right)=1$
	(${\mu }_1,{\sigma }_1,{\mu }_2,{\sigma }_2,1,0$)	0	1	0	0	$M\left(\mathit{\theta };{\mu }_2,{\sigma }_2\right)=1$
	(1.194, 0.601, 2.186, 2.592, 2,0.666)	1.526	1.5	1.002	1.001	$M(\mathit{\theta };0,1)=0.5$
A	(0.265, 0.415, 0.996, 1.541, 1.16, 0.313)	0.767	1.288	0.426	0.152	$M(\mathit{\theta };0,1)=0.75$
	(0.173, 0.358, 0.289, 1.268, 1.132, 0.198)	0.266	1.104	0.056	0.071	$M(\mathit{\theta };0,1)=0.9$
	(−1.321, 1.842, 0.741, 0.459, 2.56, 0.287)	0.15	1.4	−1.764	3.3	$M(\mathit{\theta };0,1)=0.5$
B	(0.539, 0.632, −1.078, 2.061, 1.174, 0.741)	0.12	1.34	−1.499	2.986	$M(\mathit{\theta };0,1)=0.75$
	(−0.966, 1.824, 0.259, 0.889, 1.1, 0.26)	−0.059	1.305	−0.899	1.999	$M(\mathit{\theta };0,1)=0.9$
	(1.308, 0.656, 1.308, 3.261, 2, 0.613)	1.308	1.884	0	0.504	$M(\mathit{\theta };0,1)=0.5$
C	(0.571, 1.023, 0.571, 1.962, 1.15, 0.505)	0.571	1.508	0	0.325	$M(\mathit{\theta };0,1)=0.75$
	(−0.097, 1.332, −0.097, 1.058, 1.1, 0.614)	−0.097	1.223	0	0.101	$M(\mathit{\theta };0,1)=0.9$
	(−0.692, 2.203, −0.692, 2.544, 1.759, 0.25)	−0.692	2.265	0	−1	$M(\mathit{\theta };0,1)=0.5$
D	(0.323, 1.312, 0.605, 1.335, 1.2, 0.01)	0.602	1.266	0	−0.587	$M(\mathit{\theta };0,1)=0.75$
	(0.179, 0.494, 0.179, 1.163, 1.426, 0.443)	0.179	0.862	0	−0.202	$M(\mathit{\theta };0,1)=0.9$
	(0.675, 0.284, 2.122, 1.968, 2.104, 0.374)	1.581	1.565	0.749	−0.849	$M(\mathit{\theta };0,1)=0.5$
E	(0.423, 1.032, 1.058, 2.077, 1.815, 0.494)	0.744	1.544	0.311	−0.667	$M(\mathit{\theta };0,1)=0.75$
	(−0.134, 0.993, 0.671, 1.211, 1.479, 0.583)	0.202	1.115	0.115	−0.4	$M(\mathit{\theta };0,1)=0.9$
	(1.609, 0.59, 0.322, 2.194, 1.609, 0.309)	0.72	1.784	−0.491	−0.728	$M(\mathit{\theta };0,1)=0.5$
F	(0.617, 0.737, 0.129, 1.752, 1.465, 0.332)	0.291	1.395	−0.239	−0.526	$M(\mathit{\theta };0,1)=0.75$
	(−0.046, 1.156, 1.261, 0.799, 1.87, 0.876)	0.116	1.191	−0.1	−0.2	$M(\mathit{\theta };0,1)=0.9$
	(1.88, 2.736, −0.848, 1.122, 6.437, 0.679)	1.005	2.656	0.524	0	$M(\mathit{\theta };0,1)=0.5$
G	(2.419, 1.56, 0.237, 1.384, 1.476, 0.074)	0.398	1.409	0.35	0	$M(\mathit{\theta };0,1)=0.75$
	(0.055, 0.702, 0.474, 1.586, 1.328, 0.473)	0.276	1.191	0.31	0	$M(\mathit{\theta };0,1)=0.9$
	(1.642, 1.247, 0.202, 2.681, 1.428, 0.554)	1	2.018	−0.594	0	$M(\mathit{\theta };0,1)=0.5$
H	(−1.246, 1.326, 0.858, 1.103, 1.242, 0.313)	0.2	1.496	−0.5	0	$M(\mathit{\theta };0,1)=0.75$
	(−0.115, 1.286, 0.306, 1.091, 1.093, 0.465)	0.11	1.189	−0.1	0	$M(\mathit{\theta };0,1)=0.9$

,

Table A6. Vectors of PCM parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups 0, A–H.

Group	$\mathit{\theta }=\left({\mathit{\mu }}_1,{\mathit{\sigma }}_1,{\mathit{c}}_1,{\mathit{\mu }}_2,{\mathit{\sigma }}_2,{\mathit{c}}_2,\mathit{\omega }\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
0	$({\mu }_1,{\sigma }_1,{1,\mu }_2,{\sigma }_2,c_2,1$)	0	1	0	0	$M\left(\mathit{\theta };{\mu }_1,{\sigma }_1\right)=1$
	(${\mu }_1,{\sigma }_1,{c_1,\mu }_2,{\sigma }_2,1,0$)	0	1	0	0	$M\left(\mathit{\theta };{\mu }_2,{\sigma }_2\right)=1$
	(1.415, 1.684, 2.194, 11.252, 5.474, 2.331, 0.9)	2.399	3.622	2.647	7.663	$M(\mathit{\theta };0,1)=0.5$
A	(0.444, 0.899, 1.602, 1.653, 2.506, 1.876, 0.64)	0.879	1.604	0.913	0.412	$M(\mathit{\theta };0,1)=0.75$
	(−0.076, 1.056, 1.1, 0.701, 1.646, 1.095, 0.71)	0.149	1.268	0.374	0.374	$M(\mathit{\theta };0,1)=0.9$
	(1.366, 0.572, 1.11, 0.502, 1.669, 1.253, 0.658)	1.071	1.099	−0.978	1.565	$M(\mathit{\theta };0,1)=0.5$
B	(0.67, 0.425, 1.576, -0.323, 1.696, 1.05, 0.349)	0.024	1.444	−0.569	0.606	$M(\mathit{\theta };0,1)=0.75$
	(−0.204, 2.209, 1.205, 0.133, 1.139, 1.05, 0.076)	0.107	1.224	−0.122	0.457	$M(\mathit{\theta };0,1)=0.9$
	(1.597, 2.518, 1.263, 1.596, 0.856, 1.285, 0.526)	1.597	1.797	0	0.601	$M(\mathit{\theta };0,1)=0.5$
C	(0.012, 0.274, 1.256, 0.012, 2.046, 1.01, 0.183)	0.012	1.846	0	0.598	$M(\mathit{\theta };0,1)=0.75$
	(0.127, 1.089, 1.01, 0.127, 0.183, 1.01, 0.863)	0.127	1.01	0	0.401	$M(\mathit{\theta };0,1)=0.9$
	(1.631, 0.893, 1.05, 1.632, 2.104, 1.554, 0.498)	1.632	1.488	0	−0.268	$M(\mathit{\theta };0,1)=0.5$
D	(0.639, 1.576, 1.167, 0.64, 1.085, 1.199, 0.163)	0.64	1.12	0	−0.251	$M(\mathit{\theta };0,1)=0.75$
	(0.666, 1.123, 4.041, 0.233, 1.069, 1.05, 0.01)	0.237	1.052	0	−0.198	$M(\mathit{\theta };0,1)=0.9$
	(1.472, 0.782, 1.11, 0.236, 0.291, 3.203, 0.692)	1.091	0.861	0.38	−0.8	$M(\mathit{\theta };0,1)=0.5$
E	(−0.196, 0.341, 1.064, 0.613, 0.758, 1.204, 0.153)	0.489	0.734	0.201	−0.7	$M(\mathit{\theta };0,1)=0.75$
	(0.722, 0.703, 1.304, −0.57, 0.598, 1.05, 0.455)	0.018	0.893	0.179	−0.617	$M(\mathit{\theta };0,1)=0.9$
	(0.261, 1.419, 1.909, 3.099, 0.744, 1.567, 0.57)	1.481	1.757	−0.3	−1.107	$M(\mathit{\theta };0,1)=0.5$
F	(0.037, 1.295, 1.076, 1.316, 1.171, 1.654, 0.485)	0.696	1.326	−0.204	−0.4	$M(\mathit{\theta };0,1)=0.75$
	(0.201, 0.121, 1.573, 0.184, 1.177, 1.161, 0.066)	0.185	1.087	−0.003	−0.331	$M(\mathit{\theta };0,1)=0.9$
	(1.088, 0.894, 3.782, 1.969, 2.71, 1.792, 0.55)	1.484	1.793	0.6	0	$M(\mathit{\theta };0,1)=0.5$
G	(1.515, 2.553, 3.55, 0.07, 1.328, 1.619, 0.07)	0.171	1.359	0.501	0	$M(\mathit{\theta };0,1)=0.75$
	(−0.034, 1.072, 1.159, 1.146, 1.51, 1.301, 0.756)	0.254	1.238	0.401	0	$M(\mathit{\theta };0,1)=0.9$
	(0.816, 1.867, 1.24, 1.787, 1.272, 1.05, 0.278)	1.517	1.475	−0.302	0	$M(\mathit{\theta };0,1)=0.5$
H	(−0.364, 1.889, 1.057, 0.29, 1.413, 1.05, 0.527)	−0.055	1.682	−0.154	0	$M(\mathit{\theta };0,1)=0.75$
	(0.286, 0.405, 1.27, -0.263, 1.261, 1.05, 0.112)	−0.202	1.188	−0.128	0	$M(\mathit{\theta };0,1)=0.9$

,

Table A7. Vectors of LM parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups A–H.

Group	$\mathit{\theta }=\left({\mathit{\mu }}_1,{\mathit{\sigma }}_1,{\mathit{\mu }}_2,{\mathit{\sigma }}_2,\mathit{\omega }\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
	(4.521, 7.174, −0.757, 1.959, 0.313)	0.895	6.594	1.172	9.074	$M(\mathit{\theta };0,1)=0.5$
A	(1.169, 1.491, −0.019, 0.849, 0.56)	0.646	1.863	0.4	3.454	$M(\mathit{\theta };0,1)=0.75$
	(0.452, 0.818, −0.947, 0.482, 0.762)	0.119	1.219	0.224	1.644	$M(\mathit{\theta };0,1)=0.9$
	(−0.358, 0.405, −2.549, 2.309, 0.234)	−2.036	3.018	−0.407	3.5	$M(\mathit{\theta };0,1)=0.5$
B	(0.94, 0.335, −0.571, 1.585, 0.122)	−0.387	2.164	−0.202	3.136	$M(\mathit{\theta };0,1)=0.75$
	(−0.736, 0.911, 0.04, 0.878, 0.132)	−0.062	1.275	−0.034	2.773	$M(\mathit{\theta };0,1)=0.9$
	(1.445, 1.571, −2.516, 1.87, 1)	1.445	2.222	0	3	$M(\mathit{\theta };0,1)=0.5$
C	(0.246, 0.844, −0.59, 0.905, 0.043)	−0.554	1.287	0	2.894	$M(\mathit{\theta };0,1)=0.75$
	(0.319, 0.86, −0.21, 0.874, 0.222)	−0.092	1.251	0	2.815	$M(\mathit{\theta };0,1)=0.9$
	(−6.131, 0.945, −0.386, 1.54, 0.366)	−2.487	3.364	0	−0.648	$M(\mathit{\theta };0,1)=0.5$
D	(−4.898, 0.343, −0.415, 1.234, 0.29)	−1.716	2.523	0	−0.597	$M(\mathit{\theta };0,1)=0.6$
	(2.115, 0.07, −-0.512, 0.822, 0.208)	0.034	1.486	0	−0.005	$M(\mathit{\theta };0,1)=0.7$
	(7.186, 1.509, -0.869, 0.58, 0.309)	1.62	3.966	1.005	−0.403	$M(\mathit{\theta };0,1)=0.5$
E	(−1.711, 0.177, 0.773, 0.823, 0.421)	−0.274	1.522	0.5	−0.32	$M(\mathit{\theta };0,1)=0.6$
	(1.023, 0.358, −0.118, 0.348, 0.428)	0.37	0.753	0.15	−0.014	$M(\mathit{\theta };0,1)=0.75$
	(−3.863, 0.348, 1.522, 1.359, 0.248)	0.184	2.872	−0.18	−0.556	$M(\mathit{\theta };0,1)=0.4$
F	(0.006, 0.065, 0.703, 0.189, 0.227)	0.545	0.378	−0.17	−0.286	$M(\mathit{\theta };0,1)=0.45$
	(−0.466, 0.161, 0.08, 0.159, 0.48)	−0.182	0.354	−0.05	−0.2	$M(\mathit{\theta };0,1)=0.5$
	(2.309, 1.022, −1.1, 0.418, 0.391)	0.233	1.949	0.85	0	$M(\mathit{\theta };0,1)=0.5$
G	(−0.208, 1.335, 7.917, 1.899, 0.712)	2.132	4.261	0.839	0	$M(\mathit{\theta };0,1)=0.6$
	(0.679, 0.702, −1.434, 0.642, 0.532)	−0.31	1.422	0.036	0	$M(\mathit{\theta };0,1)=0.75$
	(−9.234, 0.124, 1.581, 2.321, 0.161)	−0.159	4.983	−0.556	0	$M(\mathit{\theta };0,1)=0.4$
H	(−1.322, 0.83, 2.398, 1.181, 0.291)	1.317	2.287	−0.1	0	$M(\mathit{\theta };0,1)=0.45$
	(0.81, 0.479, 2.254, 0.229, 0.736)	1.191	0.878	−0.032	0	$M(\mathit{\theta };0,1)=0.5$

,

Table A8. Vectors of the SB parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups A–B, D–H.

Group	$\mathit{\theta }=\left({\mathit{\mu }}_1,{\mathit{\sigma }}_1,{\mathit{\mu }}_2,{\mathit{\sigma }}_2,\mathit{\omega }\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
	(1.972, 1.819, −0.45, 4)	0.613	0.411	0.649	0.3	$M(\mathit{\theta };0,1)=0.5$
A	(2.482, 2.23, −1.665, 7.423)	0.237	0.618	0.584	0.298	$M(\mathit{\theta };0,1)=0.75$
	(3.092, 2.702, −2.908, 12.271)	0.132	0.832	0.518	0.267	$M(\mathit{\theta };0,1)=0.9$
	(−4.086, 2.097, −5.424, 6.348)	0.074	0.351	−1	1.488	$M(\mathit{\theta };0,1)=0.5$
B	(−2.614, 2.258, −5.722, 7.58)	−0.021	0.611	−0.6	0.341	$M(\mathit{\theta };0,1)=0.75$
	(−1.992, 2.198, −6.446, 8.974)	−0.129	0.823	−0.485	0.099	$M(\mathit{\theta };0,1)=0.9$
	(0, 3.149, −2.116, 4.115)	−0.059	0.319	0	−0.176	$M(\mathit{\theta };0,1)=0.5$
D	(0, 3.958, −4.707, 9.414)	0	0.585	0	−0.117	$M(\mathit{\theta };0,1)=0.75$
	(0, 4.304, −8.154, 15.856)	−0.227	0.909	0	−0.1	$M(\mathit{\theta };0,1)=0.9$
	(0.664, 0.45, −0.027, 4.679)	1.377	1.38	0.856	−0.558	$M(\mathit{\theta };0,1)=0.5$
E	(0.834, 0.754, −0.727, 3.258)	0.26	0.726	0.788	−0.25	$M(\mathit{\theta };0,1)=0.75$
	(0.867, 2.297, −4.627, 10.828)	−0.18	1.095	0.2	−0.227	$M(\mathit{\theta };0,1)=0.9$
	(−0.716, 0.448, −0.622, 1.618)	0.534	0.47	−0.931	−0.4	$M(\mathit{\theta };0,1)=0.5$
F	(−1.044, 1.22, −4.394, 5.493)	−0.665	0.88	−0.603	−0.145	$M(\mathit{\theta };0,1)=0.75$
	(−1.202, 1.515, −4.252, 6.217)	−0.065	0.837	−0.522	−0.1	$M(\mathit{\theta };0,1)=0.9$
	(1.64, 2.044, −3.761, 8.045)	−1.199	0.819	0.452	0	$M(\mathit{\theta };0,1)=0.5$
G	(1.825, 2.345, −1.984, 6.623)	0.145	0.596	0.401	0	$M(\mathit{\theta };0,1)=0.75$
	(2.952, 4.082, −-5.487, 16.27)	−0.135	0.87	0.24	0	$M(\mathit{\theta };0,1)=0.9$
	(−1.357; 1.565; −1.601; 3.202)	0.605	0.41	−0.563	0	$M(\mathit{\theta };0,1)=0.5$
H	(−2.046; 2.695; −5.081; 7.468)	−0.032	0.592	−0.354	0	$M(\mathit{\theta };0,1)=0.75$
	(−2.068; 2.73; −7.098; 10.398)	−0.07	0.814	−0.35	0	$M(\mathit{\theta };0,1)=0.9$

,

Table A9. Vectors of the SU parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups A–C, E–H.

Group	$\mathit{\theta }=\left(\mathit{a},\mathit{b},\mathit{c},\mathit{d}\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
	(−1.246, 2.021, 0.257, 0.731)	0.800	0.501	1.014	2.911	$M(\mathit{\theta };0,1)=0.5$
A	(−0.569, 2.063, −1.301, 2.625)	−0.477	1.499	0.493	1.720	$M(\mathit{\theta };0,1)=0.75$
	(−0.11, 2.762, −0.069, 3.319)	0.072	1.286	0.049	0.648	$M(\mathit{\theta };0,1)=0.9$
	(2.502, 2.889, 2.029, 3.828)	−1.949	2	−0.8	1.455	$M(\mathit{\theta };0,1)=0.5$
B	(2.564, 3.308, 2.137, 1.902)	0.435	0.8	−0.636	0.926	$M(\mathit{\theta };0,1)=0.75$
	(2.296, 5.558,2.36,6.031)	−0.246	1.2	−0.218	0.2	$M(\mathit{\theta };0,1)=0.9$
	(0, 1.821, −1.617, 3.096)	−1.617	1.992	0	2	$M(\mathit{\theta };0,1)=0.5$
C	(0, 1.829, −0.205, 2.967)	−0.205	1.897	0	1.97	$M(\mathit{\theta };0,1)=0.75$
	(0, 3.372, 0.204, 2.935)	0.204	0.91	0	0.403	$M(\mathit{\theta };0,1)=0.9$
	(−22.518, 45.262, −11.095, 19.766)	−0.848	0.492	0.031	−0.007	$M(\mathit{\theta };0,1)=0.5$
E	(1.29, 40.539, −0.294, −17.564)	0.155	0.484	0.491	−0.784	$M(\mathit{\theta };0,1)=0.6$
	(0.244, 21.027, −0.134, −12.383)	0.01	0.59	0.002	−0.007	$M(\mathit{\theta };0,1)=0.75$
	(0.861, 18.997, −1.158, 14.674)	−1.824	0.774	−0.011	−0.005	$M(\mathit{\theta };0,1)=0.3$
F	(0.756, 3.676, 0.166, 0.819)	−0.010	0.236	−0.359	−0.450	$M(\mathit{\theta };0,1)=0.4$
	(13.843, 36.36, 4.174, 11.623)	−0.360	0.343	−0.030	−0.077	$M(\mathit{\theta };0,1)=0.5$
	(−9.342, 11.021, −1.575, 1.981)	0.32	0.25	0.207	0	$M(\mathit{\theta };0,1)=0.4$
G	(−23.944, 18.041, −8.486, 5.409)	1.009	0.606	0.15	0	$M(\mathit{\theta };0,1)=0.5$
	(−9.349, 85.071, −3.763, 35.754)	0.174	0.423	0.004	0	$M(\mathit{\theta };0,1)=0.6$
	(0.738, 49.723, 3.6, 73.029)	2.516	1.469	−0.001	0	$M(\mathit{\theta };0,1)=0.3$
H	(2.547, 7.276, 0.835, 1.65)	0.24	0.243	−0.141	0	$M(\mathit{\theta };0,1)=0.4$
	(4.211, 10.507, 1.959, 3.55)	0.491	0.367	−0.11	0	$M(\mathit{\theta };0,1)=0.5$

,

Table A10. Vectors of the EECK parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M. Groups C-D.

Group	$\mathit{\theta }=\left(\mathit{a},\mathit{b}\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
	(46.018, 1.043)	0	0.032	0	2.256	$M(\mathit{\theta };0,0.096)=0.5$
C	(40.914, 1.366)	0	0.06	0	0.921	$M(\mathit{\theta };0,0.096)=0.75$
	(10.676, 1.184)	0	0.128	0	0.912	$M(\mathit{\theta };0,0.096)=0.9$
	(60.495, 4.846)	0	0.244	0	−0.921	$M(\mathit{\theta };0,0.096)=0.5$
D	(48.76, 2.738)	0	0.15	0	−0.51	$M(\mathit{\theta };0,0.096)=0.75$
	(48.76, 2.211)	0	0.115	0	−0.238	$M(\mathit{\theta };0,0.096)=0.9$

,

Table A11. Vectors of the EP parameter $\mathit{\theta }$, mean ${\mu }_a$, standard deviation ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measure M.

Group	$\mathit{\theta }=\left(\mathit{a},\mathit{b},\mathit{c}\right)$	${\mathit{\mu }}_{\mathit{a}}$	${\mathit{\sigma }}_{\mathit{a}}$	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$	$\mathit{M}\left(\mathit{\theta };\mathit{\mu },\mathit{\sigma }\right)$
	(−0.796, 2.985, 1.609)	−0.796	3.257	0	0.536	$M(\mathit{\theta };0,1)=0.5$
C	(90.611, 1.385, 1.695)	0.611	1.478	0	0.386	$M(\mathit{\theta };0,1)=0.75$
	(90.251, 1.033, 1.785)	0.251	1.079	0	0.253	$M(\mathit{\theta };0,1)=0.9$
	(−0.611, 3.71, 28.792)	−0.611	2.368	0	−1.188	$M(\mathit{\theta };0,1)=0.5$
D	(−0.673, 1.198, 3.828)	−0.673	0.994	0	−0.783	$M(\mathit{\theta };0,1)=0.75$
	(−0.05, 1.272, 3.117)	−0.05	1.11	0	−0.619	$M(\mathit{\theta };0,1)=0.9$

, which provide the alternative parameter vectors $\mathit{\theta }$, along with the corresponding means ${\mu }_a$, standard deviations ${\sigma }_a$, skewness ${\gamma }_1$, excess kurtosis ${\overline{\gamma }}_2$, and similarity measures M for all considered distributions. The skewness and excess kurtosis tend to zero, while the similarity measure tends to unity. Often, the mean tends to zero, and the standard deviation tends to unity, while the similarity measure tends to unity. The PDF formulas and their corresponding curves (see Figure A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10), associated with the alternative parameter vectors $\mathit{\theta }$, are also included in the Appendix.

As illustrated in Figure A1, the ES distribution is unsuitable for simulation studies concerning groups D–H; despite satisfying the normalization condition, the density function exhibits invalid negative values.

Figure A2 shows bathtub shapes. Figure A3, Figure A4, and Figure A6 show unimodal and bimodal shapes. In Figure A5 we can see very interesting multimodal shapes. In Figure A7 dominate unimodal shapes and Figure A8 shows only unimodal shapes. In Figure A9, we observe flat modes, and in Figure A10, very flat modes with table shapes.

5. Power Study

In [24], a sample of the most recent comparisons (since 1990) has been used to rank 55 different normality tests. The overall winner of this analysis is the regression-based Shapiro–Wilk (SW) test of normality.

The parametrized KS (PKS) test with the statistic (6) was compared with the one-component LF test with the statistic (3), Shapiro–Wilk (SW) [25], Shapiro–Francia (SF) [26], AD and CM tests. To study the power of tests, critical values ${cv}_{0.05}$ (the type I error equals $\alpha =0.05$) were calculated using ${m=10}^6$ order statistics. The power of tests (PoTs) was calculated based on ${rep=10}^5$ values of test statistics.

Table 4. Critical Values (CVs) and test sizes (TSs) of the analyzed GoFTs for sample sizes $n=10,20$.

No	GoFT	CV		TS
		$\mathbf{n}=\mathbf{10}$	$\mathbf{n}=\mathbf{20}$	$\mathbf{n}=\mathbf{10}$	$\mathbf{n}=\mathbf{20}$
1	$LF\left(0,0\right)$	0.2010	0.1622	0.049	0.050
2	$LF\left(1,0\right)$	0.2417	0.1784	0.051	0.050
3	$LF\left(1,1\right)$	0.2316	0.1748	0.051	0.049
4	$LF\left(0,1\right)$	0.2419	0.1785	0.049	0.050
5	$LF\left(0.1,0.1\right)$	0.2026	0.163	0.049	0.050
6	$LF\left(0.9,0.1\right)$	0.2325	0.1746	0.051	0.050
7	$LF\left(0.9,0.9\right)$	0.2268	0.173	0.050	0.049
8	$LF\left(0.1,0.9\right)$	0.2327	0.1747	0.050	0.049
9	$PKS\left(0,0\right)$	0.2741	0.1971	0.052	0.050
10	$PKS\left(1,0\right)$	0.3413	0.2268	0.051	0.050
11	$PKS\left(1,1\right)$	0.3211	0.2133	0.051	0.050
12	$PKS\left(0,1\right)$	0.2619	0.192	0.050	0.050
13	$PKS\left(0.1,0.1\right)$	0.2769	0.1981	0.051	0.050
14	$PKS\left(0.9,0.1\right)$	0.3313	0.2218	0.051	0.050
15	$PKS\left(0.9,0.9\right)$	0.3141	0.211	0.051	0.050
16	$PKS\left(0.1,0.9\right)$	0.2635	0.1925	0.050	0.050
17	$CM$	0.1194	0.1232	0.050	0.050
18	$AD$	0.6867	0.7227	0.050	0.050
19	$SF$	0.8424	0.9034	0.050	0.051
20	$SW$	0.8445	0.9044	0.050	0.051

shows critical values (CVs) and test sizes (TSs) for sample sizes $n=10,20$ as representative of values not exceeding 50 (see Table A1 and [21]), [14,15,16,27,28]. The test TS values are approximately 0.05, confirming the validity of the simulation procedures.

The R codes related to the pseudo-random number generators (PRNGs) for all analyzed alternatives, as well as the R codes for calculating the $PKS\left(\alpha ,\beta \right)$ statistics, critical values, and power analysis are provided in Appendix A.

Complete simulation results with power values are presented in a table with 20 columns (20 tests) and 60 rows (10 alternatives, 2 sample sizes, 3 similarity measures). Presenting such large tables is difficult due to the size of the article. Therefore, the conclusions are applied to the full results, and only the most interesting results (tests with the highest power) will be shown in

Table 5. The power of numbered tests (see Table 4) for group A of alternatives (ALTs). The highest values are in bold.

ALT	n	19	20	18	17	3	n	19	20	18	17	3
${ES}_1$	10	0.285	0.254	0.248	0.234	0.220	20	0.570	0.502	0.473	0.419	0.371
${ES}_2$	10	0.201	0.175	0.173	0.163	0.159	20	0.396	0.333	0.311	0.275	0.248
${ES}_3$	10	0.134	0.117	0.115	0.108	0.110	20	0.242	0.200	0.179	0.159	0.149
ALT	n	20	18	19	17	4	n	20	19	18	17	11
$P_1$	10	0.846	0.815	0.804	0.788	0.758	20	0.997	0.993	0.993	0.986	0.976
$P_2$	10	0.846	0.812	0.804	0.783	0.754	20	0.997	0.993	0.993	0.986	0.977
$P_3$	10	0.845	0.814	0.805	0.785	0.757	20	0.997	0.994	0.993	0.987	0.977
ALT	n	4	8	19	20	18	n	4	8	19	20	18
${NM}_1$	10	0.135	0.133	0.111	0.104	0.103	20	0.199	0.193	0.194	0.188	0.171
${NM}_2$	10	0.136	0.133	0.105	0.102	0.100	20	0.200	0.194	0.177	0.177	0.167
${NM}_3$	10	0.082	0.081	0.066	0.064	0.064	20	0.098	0.094	0.085	0.081	0.079
ALT	n	4	8	17	18	19	n	4	8	17	18	3
${NDPC}_1$	10	0.387	0.384	0.340	0.339	0.334	20	0.680	0.673	0.662	0.665	0.621
${NDPC}_2$	10	0.180	0.178	0.133	0.129	0.119	20	0.299	0.292	0.244	0.227	0.222
${NDPC}_3$	10	0.063	0.063	0.057	0.056	0.058	20	0.069	0.069	0.062	0.059	0.064
ALT	n	20	4	8	18	17	n	20	18	19	4	17
${PCM}_1$	10	0.594	0.549	0.550	0.589	0.578	20	0.896	0.895	0.863	0.831	0.871
${PCM}_2$	10	0.227	0.259	0.255	0.219	0.204	20	0.493	0.479	0.465	0.458	0.434
${PCM}_3$	10	0.067	0.079	0.078	0.064	0.062	20	0.094	0.082	0.097	0.098	0.076
ALT	n	19	18	17	20	8	n	19	18	20	17	3
${LM}_1$	10	0.409	0.393	0.393	0.376	0.378	20	0.704	0.691	0.659	0.689	0.638
${LM}_2$	10	0.174	0.152	0.148	0.154	0.143	20	0.316	0.267	0.272	0.250	0.230
${LM}_3$	10	0.100	0.092	0.090	0.094	0.079	20	0.161	0.130	0.143	0.119	0.115
ALT	n	4	8	20	19	18	n	4	8	20	19	18
${SB}_1$	10	0.117	0.115	0.089	0.087	0.084	20	0.172	0.166	0.165	0.152	0.143
${SB}_2$	10	0.107	0.105	0.079	0.079	0.076	20	0.146	0.141	0.132	0.125	0.116
${SB}_3$	10	0.096	0.095	0.071	0.071	0.069	20	0.127	0.122	0.112	0.108	0.100
ALT	n	4	8	19	20	18	n	19	20	4	18	8
${SU}_1$	10	0.150	0.147	0.137	0.129	0.124	20	0.250	0.240	0.225	0.214	0.219
${SU}_2$	10	0.100	0.100	0.100	0.091	0.089	20	0.165	0.146	0.135	0.131	0.132
${SU}_3$	10	0.061	0.061	0.071	0.066	0.065	20	0.096	0.083	0.068	0.076	0.068

,

Table 6. The power of numbered tests (see Table 4) for group B of alternatives (ALTs). The highest values are in bold.

ALT	n	19	20	18	17	11	n	19	20	18	17	11
${ES}_1$	10	0.282	0.249	0.249	0.233	0.225	20	0.572	0.501	0.472	0.418	0.382
${ES}_2$	10	0.202	0.177	0.174	0.163	0.163	20	0.394	0.331	0.306	0.270	0.258
${ES}_3$	10	0.132	0.115	0.117	0.111	0.115	20	0.240	0.198	0.181	0.162	0.164
ALT	n	20	18	19	17	9	n	20	19	18	17	9
$P_1$	10	0.846	0.815	0.804	0.787	0.780	20	0.997	0.994	0.993	0.986	0.970
$P_2$	10	0.845	0.814	0.804	0.786	0.778	20	0.997	0.993	0.993	0.986	0.971
$P_3$	10	0.847	0.813	0.806	0.784	0.777	20	0.997	0.994	0.993	0.987	0.970
ALT	n	15	14	10	11	2	n	14	10	15	11	2
${NM}_1$	10	0.170	0.164	0.164	0.171	0.163	20	0.261	0.261	0.266	0.269	0.253
${NM}_2$	10	0.141	0.143	0.143	0.141	0.142	20	0.223	0.223	0.215	0.214	0.214
${NM}_3$	10	0.102	0.107	0.107	0.101	0.106	20	0.152	0.152	0.141	0.139	0.146
ALT	n	15	11	14	10	2	n	19	20	18	11	15
${NDPC}_1$	10	0.682	0.678	0.689	0.689	0.689	20	0.950	0.949	0.960	0.952	0.952
${NDPC}_2$	10	0.401	0.403	0.390	0.390	0.389	20	0.710	0.689	0.682	0.676	0.672
${NDPC}_3$	10	0.165	0.166	0.157	0.157	0.156	20	0.302	0.282	0.247	0.260	0.257
ALT	n	15	11	14	10	2	n	14	10	11	15	2
${PCM}_1$	10	0.269	0.270	0.258	0.258	0.257	20	0.454	0.454	0.464	0.461	0.444
${PCM}_2$	10	0.237	0.234	0.243	0.243	0.242	20	0.431	0.431	0.416	0.418	0.421
${PCM}_3$	10	0.060	0.061	0.060	0.060	0.060	20	0.068	0.068	0.068	0.068	0.067
ALT	n	19	20	18	17	11	n	19	20	18	17	11
${LM}_1$	10	0.202	0.189	0.186	0.179	0.180	20	0.377	0.340	0.338	0.312	0.293
${LM}_2$	10	0.167	0.148	0.146	0.141	0.141	20	0.305	0.260	0.247	0.227	0.217
${LM}_3$	10	0.162	0.140	0.144	0.142	0.131	20	0.284	0.237	0.239	0.230	0.204
ALT	n	14	10	2	15	6	n	14	10	2	20	13
${SB}_1$	10	0.167	0.167	0.166	0.165	0.163	20	0.274	0.274	0.263	0.293	0.260
${SB}_2$	10	0.111	0.111	0.110	0.108	0.108	20	0.158	0.158	0.151	0.140	0.150
${SB}_3$	10	0.097	0.097	0.096	0.094	0.094	20	0.130	0.130	0.123	0.104	0.123
ALT	n	15	11	14	10	2	n	14	10	11	15	2
${SU}_1$	10	0.130	0.130	0.130	0.130	0.129	20	0.192	0.192	0.189	0.188	0.184
${SU}_2$	10	0.112	0.112	0.112	0.112	0.111	20	0.156	0.156	0.152	0.152	0.149
${SU}_3$	10	0.070	0.070	0.068	0.068	0.068	20	0.078	0.078	0.077	0.077	0.075

,

Table 7. The power of numbered tests (see Table 4) for group C of alternatives (ALTs). The highest values are in bold.

ALT	n	19	20	18	17	3	n	19	20	18	17	3
${ES}_1$	10	0.284	0.249	0.245	0.230	0.220	20	0.571	0.495	0.470	0.413	0.366
${ES}_2$	10	0.201	0.174	0.173	0.161	0.159	20	0.397	0.330	0.308	0.272	0.245
${ES}_3$	10	0.135	0.117	0.116	0.109	0.111	20	0.240	0.196	0.179	0.159	0.149
ALT	n	19	18	20	3	17	n	19	20	18	17	3
$P_1$	10	0.137	0.124	0.121	0.119	0.119	20	0.243	0.209	0.193	0.178	0.169
$P_2$	10	0.137	0.123	0.122	0.119	0.117	20	0.242	0.210	0.190	0.174	0.166
$P_3$	10	0.138	0.120	0.122	0.116	0.115	20	0.240	0.206	0.190	0.174	0.166
ALT	n	3	7	19	17	18	n	17	3	7	18	19
${NM}_1$	10	0.222	0.219	0.204	0.216	0.202	20	0.391	0.387	0.380	0.357	0.325
${NM}_2$	10	0.138	0.135	0.135	0.130	0.127	20	0.214	0.213	0.208	0.203	0.205
${NM}_3$	10	0.064	0.064	0.072	0.063	0.064	20	0.070	0.072	0.070	0.075	0.095
ALT	n	3	17	7	18	19	n	17	3	18	7	11
${NDPC}_1$	10	0.181	0.179	0.178	0.169	0.166	20	0.318	0.302	0.292	0.295	0.279
${NDPC}_2$	10	0.059	0.059	0.059	0.060	0.064	20	0.063	0.064	0.065	0.064	0.062
${NDPC}_3$	10	0.052	0.053	0.052	0.054	0.052	20	0.052	0.051	0.054	0.051	0.050
ALT	n	3	7	19	11	17	n	3	7	11	17	15
${PCM}_1$	10	0.085	0.083	0.093	0.082	0.082	20	0.108	0.105	0.104	0.109	0.101
${PCM}_2$	10	0.119	0.117	0.103	0.107	0.108	20	0.175	0.172	0.161	0.159	0.158
${PCM}_3$	10	0.084	0.082	0.079	0.079	0.077	20	0.106	0.104	0.102	0.097	0.099
ALT	n	19	18	3	17	7	n	19	18	20	17	3
${LM}_1$	10	0.177	0.159	0.159	0.157	0.156	20	0.312	0.271	0.260	0.265	0.253
${LM}_2$	10	0.170	0.152	0.151	0.150	0.148	20	0.300	0.257	0.251	0.249	0.238
${LM}_3$	10	0.164	0.146	0.145	0.142	0.141	20	0.290	0.242	0.240	0.232	0.221
ALT	n	19	20	18	3	7	n	19	20	18	17	3
${SU}_1$	10	0.104	0.093	0.091	0.090	0.088	20	0.169	0.144	0.131	0.120	0.115
${SU}_2$	10	0.106	0.094	0.092	0.091	0.089	20	0.169	0.143	0.130	0.120	0.116
${SU}_3$	10	0.063	0.059	0.058	0.059	0.059	20	0.079	0.071	0.065	0.063	0.063
ALT	n	19	3	7	18	17	n	19	18	20	17	3
${EECK}_1$	10	0.155	0.140	0.137	0.137	0.135	20	0.263	0.223	0.217	0.217	0.210
${EECK}_2$	10	0.091	0.084	0.082	0.082	0.080	20	0.134	0.107	0.111	0.103	0.104
${EECK}_3$	10	0.096	0.088	0.086	0.085	0.084	20	0.141	0.116	0.116	0.113	0.115
ALT	n	19	3	7	18	11	n	19	20	3	18	7
${EP}_1$	10	0.072	0.067	0.066	0.065	0.065	20	0.093	0.079	0.076	0.076	0.074
${EP}_2$	10	0.066	0.063	0.062	0.060	0.060	20	0.079	0.069	0.067	0.066	0.066
${EP}_3$	10	0.060	0.057	0.056	0.057	0.057	20	0.069	0.062	0.060	0.059	0.059

,

Table 8. The power of numbered tests (see Table 4) for group D of alternatives (ALTs). The highest values are in bold.

ALT	n	20	18	17	19	1	n	20	18	19	17	1
$P_1$	10	0.667	0.601	0.528	0.481	0.481	20	0.981	0.949	0.917	0.888	0.802
$P_2$	10	0.666	0.598	0.524	0.483	0.480	20	0.981	0.949	0.919	0.888	0.801
$P_3$	10	0.664	0.597	0.523	0.483	0.477	20	0.980	0.951	0.918	0.890	0.804
ALT	n	9	1	13	5	20	n	18	9	20	13	1
${NM}_1$	10	0.199	0.208	0.189	0.199	0.184	20	0.471	0.416	0.413	0.402	0.426
${NM}_2$	10	0.237	0.218	0.218	0.206	0.213	20	0.432	0.470	0.479	0.448	0.413
${NM}_3$	10	0.055	0.058	0.053	0.056	0.047	20	0.060	0.064	0.054	0.062	0.071
ALT	n	1	9	5	13	17	n	1	17	5	9	18
${NDPC}_1$	10	0.126	0.119	0.120	0.114	0.111	20	0.227	0.237	0.219	0.212	0.231
${NDPC}_2$	10	0.063	0.064	0.062	0.062	0.052	20	0.084	0.070	0.081	0.082	0.066
${NDPC}_3$	10	0.049	0.049	0.049	0.050	0.047	20	0.049	0.046	0.048	0.049	0.045
ALT	n	9	1	13	5	2	n	1	9	5	13	4
${PCM}_1$	10	0.049	0.049	0.049	0.048	0.048	20	0.050	0.049	0.049	0.049	0.049
${PCM}_2$	10	0.061	0.061	0.059	0.060	0.054	20	0.077	0.076	0.075	0.074	0.066
${PCM}_3$	10	0.050	0.050	0.050	0.049	0.048	20	0.053	0.052	0.052	0.051	0.049
ALT	n	1	18	5	17	20	n	18	17	1	5	20
${LM}_1$	10	0.215	0.197	0.208	0.201	0.177	20	0.422	0.432	0.412	0.401	0.341
${LM}_2$	10	0.270	0.270	0.260	0.260	0.250	20	0.572	0.539	0.514	0.502	0.507
${LM}_3$	10	0.213	0.224	0.205	0.206	0.212	20	0.458	0.396	0.384	0.374	0.424
ALT	n	1	9	5	13	4	n	9	13	1	5	14
${SB}_1$	10	0.049	0.049	0.048	0.049	0.048	20	0.048	0.048	0.048	0.047	0.047
${SB}_2$	10	0.049	0.049	0.049	0.049	0.049	20	0.051	0.051	0.050	0.049	0.050
${SB}_3$	10	0.051	0.050	0.050	0.049	0.049	20	0.048	0.048	0.048	0.048	0.047
ALT	n	9	1	13	5	14	n	9	1	13	5	18
${EECK}_1$	10	0.065	0.063	0.062	0.060	0.052	20	0.086	0.085	0.082	0.080	0.078
${EECK}_2$	10	0.053	0.052	0.052	0.051	0.048	20	0.057	0.055	0.055	0.053	0.046
${EECK}_3$	10	0.050	0.049	0.050	0.049	0.047	20	0.051	0.050	0.050	0.049	0.044
ALT	n	9	1	13	5	18	n	20	18	9	1	13
${EP}_1$	10	0.083	0.081	0.078	0.077	0.074	20	0.188	0.163	0.130	0.129	0.124
${EP}_2$	10	0.059	0.058	0.057	0.056	0.047	20	0.056	0.061	0.072	0.070	0.069
${EP}_3$	10	0.055	0.054	0.053	0.053	0.045	20	0.045	0.050	0.061	0.060	0.059

,

Table 9. The power of numbered tests (see Table 4) for group E of alternatives (ALTs). The highest values are in bold.

ALT	n	20	18	17	4	19	n	20	18	19	17	4
$P_1$	10	0.775	0.740	0.701	0.684	0.685	20	0.990	0.979	0.970	0.961	0.936
$P_2$	10	0.778	0.741	0.703	0.687	0.685	20	0.991	0.979	0.970	0.960	0.935
$P_3$	10	0.776	0.741	0.704	0.688	0.686	20	0.991	0.980	0.970	0.962	0.937
ALT	n	4	8	18	17	20	n	4	8	18	17	20
${NM}_1$	10	0.137	0.135	0.093	0.092	0.089	20	0.212	0.205	0.166	0.163	0.157
${NM}_2$	10	0.093	0.091	0.064	0.065	0.063	20	0.122	0.117	0.087	0.087	0.086
${NM}_3$	10	0.065	0.064	0.053	0.053	0.051	20	0.071	0.070	0.057	0.056	0.052
ALT	n	4	8	17	18	1	n	4	8	17	18	1
${NDPC}_1$	10	0.693	0.689	0.659	0.645	0.628	20	0.963	0.962	0.974	0.969	0.951
${NDPC}_2$	10	0.107	0.105	0.072	0.073	0.077	20	0.165	0.159	0.127	0.127	0.124
${NDPC}_3$	10	0.081	0.079	0.058	0.056	0.062	20	0.102	0.099	0.075	0.073	0.077
ALT	n	4	8	1	5	18	n	4	8	1	18	5
${PCM}_1$	10	0.153	0.151	0.130	0.125	0.116	20	0.278	0.270	0.244	0.244	0.236
${PCM}_2$	10	0.087	0.086	0.076	0.074	0.066	20	0.135	0.131	0.120	0.109	0.117
${PCM}_3$	10	0.073	0.071	0.056	0.055	0.051	20	0.090	0.087	0.067	0.069	0.065
ALT	n	1	4	8	5	18	n	1	4	8	5	18
${LM}_1$	10	0.864	0.874	0.873	0.859	0.873	20	0.997	0.997	0.997	0.997	0.998
${LM}_2$	10	0.434	0.411	0.409	0.417	0.395	20	0.741	0.732	0.728	0.728	0.739
${LM}_3$	10	0.125	0.129	0.128	0.122	0.109	20	0.211	0.215	0.213	0.206	0.191
ALT	n	20	4	8	18	17	n	20	18	19	17	4
${SB}_1$	10	0.495	0.442	0.437	0.458	0.427	20	0.892	0.844	0.799	0.791	0.746
${SB}_2$	10	0.213	0.232	0.229	0.197	0.186	20	0.502	0.441	0.401	0.394	0.410
${SB}_3$	10	0.048	0.063	0.062	0.048	0.048	20	0.049	0.051	0.042	0.051	0.069
ALT	n	4	8	1	5	12	n	4	19	8	20	18
${SU}_1$	10	0.052	0.052	0.050	0.050	0.049	20	0.052	0.052	0.052	0.051	0.050
${SU}_2$	10	0.051	0.051	0.051	0.051	0.051	20	0.049	0.050	0.049	0.050	0.050
${SU}_3$	10	0.050	0.051	0.052	0.052	0.051	20	0.049	0.049	0.049	0.049	0.048

,

Table 10. The power of numbered tests (see Table 4) for group F of alternatives (ALTs). The highest values are in bold.

ALT	n	20	18	9	13	17	n	20	18	19	17	9
$P_1$	10	0.778	0.738	0.735	0.726	0.701	20	0.990	0.980	0.969	0.961	0.953
$P_2$	10	0.777	0.741	0.736	0.727	0.703	20	0.990	0.980	0.970	0.962	0.954
$P_3$	10	0.778	0.739	0.735	0.726	0.701	20	0.990	0.979	0.969	0.961	0.954
ALT	n	9	13	14	10	2	n	9	13	14	10	2
${NM}_1$	10	0.322	0.321	0.312	0.312	0.310	20	0.597	0.595	0.589	0.589	0.577
${NM}_2$	10	0.105	0.101	0.089	0.089	0.089	20	0.152	0.147	0.132	0.132	0.127
${NM}_3$	10	0.056	0.056	0.057	0.057	0.057	20	0.058	0.058	0.060	0.060	0.059
ALT	n	13	9	14	10	2	n	9	14	10	13	2
${NDPC}_1$	10	0.295	0.295	0.289	0.289	0.287	20	0.550	0.549	0.549	0.549	0.536
${NDPC}_2$	10	0.101	0.102	0.099	0.099	0.098	20	0.145	0.144	0.144	0.144	0.137
${NDPC}_3$	10	0.054	0.054	0.053	0.053	0.053	20	0.055	0.056	0.055	0.056	0.054
ALT	n	9	13	14	10	2	n	20	18	9	13	17
${PCM}_1$	10	0.143	0.141	0.130	0.130	0.130	20	0.340	0.318	0.253	0.249	0.269
${PCM}_2$	10	0.067	0.064	0.058	0.058	0.058	20	0.063	0.067	0.084	0.080	0.066
${PCM}_3$	10	0.051	0.050	0.048	0.048	0.048	20	0.041	0.044	0.052	0.051	0.045
ALT	n	9	13	18	17	6	n	18	17	9	13	1
${LM}_1$	10	0.276	0.277	0.299	0.287	0.275	20	0.624	0.589	0.520	0.519	0.529
${LM}_2$	10	0.234	0.236	0.225	0.224	0.237	20	0.475	0.464	0.429	0.430	0.397
${LM}_3$	10	0.131	0.127	0.112	0.115	0.111	20	0.204	0.216	0.226	0.220	0.229
ALT	n	9	13	14	10	2	n	20	9	13	14	10
${SB}_1$	10	0.524	0.518	0.488	0.488	0.485	20	0.916	0.814	0.808	0.792	0.792
${SB}_2$	10	0.129	0.129	0.126	0.126	0.125	20	0.181	0.200	0.200	0.202	0.202
${SB}_3$	10	0.106	0.106	0.106	0.106	0.105	20	0.123	0.149	0.149	0.153	0.153
ALT	n	11	15	14	10	2	n	11	14	10	15	19
${SU}_1$	10	0.051	0.051	0.051	0.051	0.051	20	0.051	0.052	0.051	0.051	0.051
${SU}_2$	10	0.067	0.066	0.065	0.065	0.065	20	0.078	0.077	0.077	0.076	0.078
${SU}_3$	10	0.052	0.052	0.051	0.051	0.051	20	0.054	0.054	0.054	0.054	0.050

,

Table 11. The power of numbered tests (see Table 4) for group G of alternatives (ALTs). The highest values are in bold.

ALT	n	20	18	17	19	4	n	20	18	19	17	4
$P_1$	10	0.801	0.767	0.733	0.727	0.715	20	0.992	0.985	0.978	0.971	0.950
$P_2$	10	0.800	0.765	0.732	0.727	0.716	20	0.992	0.985	0.978	0.971	0.949
$P_3$	10	0.801	0.766	0.733	0.726	0.715	20	0.993	0.984	0.978	0.970	0.950
ALT	n	4	8	18	17	20	n	4	8	20	18	19
${NM}_1$	10	0.095	0.094	0.066	0.065	0.066	20	0.124	0.119	0.094	0.091	0.087
${NM}_2$	10	0.063	0.062	0.052	0.052	0.051	20	0.067	0.066	0.056	0.054	0.056
${NM}_3$	10	0.057	0.057	0.051	0.051	0.050	20	0.061	0.060	0.052	0.051	0.052
ALT	n	4	8	1	5	18	n	4	8	18	20	1
${NDPC}_1$	10	0.174	0.172	0.128	0.127	0.122	20	0.304	0.296	0.238	0.244	0.237
${NDPC}_2$	10	0.095	0.095	0.104	0.101	0.094	20	0.155	0.155	0.168	0.164	0.170
${NDPC}_3$	10	0.095	0.094	0.065	0.065	0.068	20	0.131	0.126	0.094	0.088	0.087
ALT	n	4	8	1	5	18	n	4	8	18	17	1
${PCM}_1$	10	0.273	0.270	0.208	0.207	0.206	20	0.514	0.504	0.407	0.403	0.405
${PCM}_2$	10	0.137	0.137	0.139	0.135	0.130	20	0.241	0.239	0.258	0.250	0.244
${PCM}_3$	10	0.086	0.084	0.061	0.061	0.063	20	0.105	0.101	0.080	0.074	0.072
ALT	n	4	8	17	18	1	n	17	4	8	18	1
${LM}_1$	10	0.573	0.569	0.518	0.512	0.517	20	0.870	0.881	0.877	0.868	0.859
${LM}_2$	10	0.513	0.509	0.456	0.448	0.422	20	0.834	0.835	0.830	0.820	0.778
${LM}_3$	10	0.089	0.090	0.102	0.100	0.117	20	0.172	0.143	0.147	0.165	0.186
ALT	n	4	8	20	18	17	n	4	8	20	18	19
${SB}_1$	10	0.090	0.089	0.064	0.062	0.061	20	0.117	0.113	0.096	0.087	0.087
${SB}_2$	10	0.083	0.081	0.059	0.059	0.058	20	0.103	0.099	0.084	0.077	0.077
${SB}_3$	10	0.068	0.067	0.053	0.053	0.053	20	0.077	0.074	0.061	0.058	0.059
ALT	n	4	8	19	17	5	n	4	8	19	20	18
${SU}_1$	10	0.064	0.063	0.054	0.053	0.053	20	0.071	0.069	0.060	0.060	0.057
${SU}_2$	10	0.059	0.059	0.051	0.051	0.051	20	0.066	0.064	0.054	0.055	0.053
${SU}_3$	10	0.050	0.050	0.050	0.050	0.050	20	0.050	0.050	0.050	0.049	0.049

and

Table 12. The power of numbered tests (see Table 4) for group H of alternatives (ALTs). The highest values are in bold.

ALT	n	20	18	9	13	17	n	20	18	19	17	9
$P_1$	10	0.801	0.764	0.757	0.750	0.732	20	0.992	0.984	0.978	0.971	0.963
$P_2$	10	0.801	0.766	0.759	0.752	0.731	20	0.993	0.984	0.979	0.970	0.962
$P_3$	10	0.800	0.767	0.757	0.751	0.735	20	0.993	0.985	0.979	0.971	0.962
ALT	n	14	10	2	13	6	n	14	10	2	13	9
${NM}_1$	10	0.118	0.118	0.117	0.115	0.114	20	0.175	0.175	0.167	0.168	0.168
${NM}_2$	10	0.098	0.098	0.097	0.095	0.095	20	0.131	0.131	0.125	0.123	0.123
${NM}_3$	10	0.058	0.058	0.058	0.058	0.058	20	0.063	0.063	0.061	0.060	0.060
ALT	n	14	10	2	13	15	n	14	10	2	13	15
${NDPC}_1$	10	0.165	0.165	0.164	0.157	0.163	20	0.280	0.280	0.269	0.264	0.269
${NDPC}_2$	10	0.095	0.095	0.094	0.096	0.092	20	0.131	0.131	0.124	0.126	0.122
${NDPC}_3$	10	0.055	0.055	0.055	0.057	0.054	20	0.060	0.060	0.058	0.059	0.057
ALT	n	14	10	2	13	9	n	14	10	2	13	9
${PCM}_1$	10	0.080	0.080	0.079	0.078	0.078	20	0.101	0.101	0.097	0.096	0.095
${PCM}_2$	10	0.063	0.063	0.063	0.063	0.063	20	0.071	0.071	0.069	0.069	0.069
${PCM}_3$	10	0.074	0.074	0.074	0.074	0.074	20	0.092	0.092	0.088	0.088	0.088
ALT	n	18	17	20	13	6	n	18	17	20	9	13
${LM}_1$	10	0.387	0.361	0.373	0.348	0.366	20	0.744	0.693	0.707	0.632	0.638
${LM}_2$	10	0.128	0.132	0.117	0.166	0.159	20	0.239	0.252	0.193	0.278	0.277
${LM}_3$	10	0.133	0.135	0.122	0.092	0.080	20	0.256	0.260	0.206	0.188	0.181
ALT	n	14	10	2	13	9	n	14	10	9	13	2
${SB}_1$	10	0.110	0.110	0.109	0.110	0.110	20	0.165	0.165	0.160	0.160	0.157
${SB}_2$	10	0.079	0.079	0.078	0.077	0.077	20	0.099	0.099	0.094	0.095	0.094
${SB}_3$	10	0.079	0.079	0.078	0.077	0.076	20	0.099	0.099	0.095	0.094	0.094
ALT	n	11	14	10	2	15	n	14	10	11	15	2
${SU}_1$	10	0.051	0.051	0.051	0.051	0.051	20	0.051	0.051	0.051	0.051	0.050
${SU}_2$	10	0.060	0.060	0.060	0.060	0.060	20	0.068	0.067	0.066	0.066	0.066
${SU}_3$	10	0.058	0.058	0.058	0.058	0.058	20	0.062	0.062	0.061	0.061	0.060

. The alternative distributions are indexed such that a higher index value corresponds to a closer resemblance to the normal distribution (e.g., index three in most cases denotes the similarity measure 0.9). The highest values are in bold.

Of course, it is expected that the power of the GoFTs increases as the sample size increases. This basic assumption is not met for the SB (group D) and SU (group E) alternatives. In these cases, the power is close to the significance level. Group D for the SB distribution, due to the choice of parameter values to obtain an appropriate similarity measure, is characterized by excess kurtosis values close to zero and is very similar to group C, for which SB is not defined. Group E for the SU distribution, due to the choice of parameter values to obtain an appropriate similarity measure, is characterized by skewness and excess kurtosis values close to zero and is very similar to group D, for which SU is not defined. It is expected that the power of the GoFTs decreases as the value of the similarity measure (7) increases. This basic assumption is not met for the P (groups A–H), PCM (groups C, D, and H), LM (groups D and H), SB (groups C, D, and H), SU (groups C, D, E, F, and H), and EECK (group C) alternatives.

The average PoT is highest for group B alternatives, followed by groups E and F. This indicates that the GoFTs best detect samples from asymmetric distributions with positive excess kurtosis. The worst thing, as you might expect, is detecting samples from symmetric distributions. The GoFTs perform best on samples from the Pearson distribution and worst on samples from the EECK distribution.

Based on the results from Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12, we can conclude that $LF\left(0,1\right)$ and SF tests are the most powerful for the group A of alternatives; $PKS\left(0.9,0.1\right)$ and SF tests are the most powerful for the group B of alternatives; SF test is the most powerful for the group C of alternatives; $PKS\left(0,0\right)$ and $LF\left(0,0\right)$ tests are the most powerful for the group D of alternatives; $LF\left(0,1\right)$ test is the most powerful for the group E of alternatives; $PKS\left(0,0\right)$ and SW tests are the most powerful for the group F of alternatives; $LF\left(0,1\right)$ test is the most powerful for the group G of alternatives; and $PKS\left(0,0\right)$ test is the most powerful for the group H of alternatives.

6. Real Data Examples

In this section, we apply the analyzed GoFTs to empirical datasets to demonstrate their practical utility. Comprehensive details for examples I–XXX are summarized in

Table 13. Real data examples with R sources, sample size, ${\gamma }_1$ and ${\overline{\gamma }}_2$ values.

Ex	Description	R Source	n	${\mathit{\gamma }}_1$	${\overline{\mathit{\gamma }}}_2$
I	Socio-economic data (percentage of draftees receiving the highest mark on the army examination) for 47 French-speaking provinces of Switzerland.	swiss [3]	47	$0.461$	$-0.011$
II	The data give the distances taken to stop.	cars [2]	50	0.782	0.248
III	Socio-economic data (draftees receiving highest mark on army examination) for 47 French-speaking provinces of Switzerland.	Swiss [3]	47	0.461	−0011
IV	Measurements of the height of timber in 31 felled black cherry trees.	trees [2]	31	−0.375	−0.569
V	Displacement of 32 cars (1973–74 models).	mtcars [3]	32	0.400	−1.090
VI	Gross horsepower of 32 cars (1973–74 models).	mtcars [4]	32	0.761	0.052
VII	Rear axle ratio of 32 cars (1973–74 models).	mtcars [5]	32	0.279	−0.565
VIII	The data includes the weight (1000 lbs) of 32 cars (1973–74 models).	mtcars [6]	32	0.444	0.172
IX	Lawyers’ ratings of state judges in the US Superior Court (Preparation for trial).	US Judge Ratings [7]	43	−0.681	0.141
X	Lawyers’ ratings of state judges in the US Superior Court (Judicial integrity).	US Judge Ratings [2]	43	−0.843	0.414
XI	Lawyers’ ratings of state judges in the US Superior Court (Demeanor).	US Judge Ratings [3]	43	−0.948	0432
XII	Daily air quality measurements in New York (wind in mph).	air quality [3]	153	0.344	0.069
XIII	Statistics in arrests per 100,000 residents for the percent urban population in each of the 50 US states.	US Arrests [3]	50	−0.219	−0.784
XIV	A regular time series giving the luteinizing hormone in blood samples at 10 min intervals from a human female, 48 samples.	l h	48	0.284	−0.746
XV	Statistics in arrests per 100,000 residents for assault in each of the 50 US states.	US Arrests [2]	50	0.227	−1.069
XVI	From a survey of the clerical employees of a large financial organization, the data are aggregated from the questionnaires of the approximately 35 employees for each of 30 (randomly selected) departments. The numbers give the percentage proportion of favorable responses to questions in each department (variable: “does not allow special privileges”).	attitude [2]	30	−0.227	−0.514
XVII	As in example XVI (variable “Too critical”).	attitude [5]	30	0.208	−0.431
XVIII	A set of macroeconomic data that provides information on the number of unemployed.	longley [3]	16	0.158	−1.065
XIX	A set of macroeconomic data that provides information on the number of people in the armed forces.	longley [4]	16	−0.404	−0.949
XX	A set of macroeconomic data that provides information on the number of people employed.	longley [7]	16	−0.094	−1.351
XXI	Daily air quality measurements in New York (temperature in degrees F).	air quality [4]	153	−0.374	−0.429
XXII	Measurements on 48 rock samples from a petroleum reservoir (area of pore space, in pixels out of 256 by 256).	rock [1]	48	−0.304	−0.262
XXIII	As in example XVI (variable: “handling of employee complaints”).	attitude [1]	30	−0.377	−0.609
XXIV	A set of macroeconomic data that provides information on the number of unemployed.	longley [3]	16	0.158	−1.065
XXV	The data give the distances taken to stop.	cars [2]	50	0.782	0.248
XXVI	Measurements in centimeters of the sepal length for 50 flowers from each of 3 species of iris. The species are Iris setosa, versicolor, and virginica.	iris [1]	150	0.312	−0.574
XXVII	An experiment to compare yields (as measured by dried weight of plants).	Plant Growth [1]	30	−0.153	−0.659
XXVIII	The data consists of five experiments, each consisting of 20 consecutive ’runs’. The response is the speed of light measurement, suitably coded (km/sec, with 299,000 subtracted).	morley [3]	100	−0.018	0.263
XXIX	The mean annual temperature in degrees Fahrenheit in New Haven, Connecticut.	nhtemp	60	−0.074	0.499
XXX	A classical N, P, K (nitrogen, phosphate, potassium) factorial experiment on the growth of peas in pounds/plot (the plots were (1/70) acre).	npk [5]	24	0.261	−0.290

.

“When fitting the normal distribution to the data, we compute p-value for the analyzed GoFTs using ${10}^5$ simulated statistic values (see

Table 14. The p-values for the GoFTs related to examples I–X.

GoFT	I	II	III	IV	V	VI	VII	VIII	IX	X
$LF\left(0,0\right)$	0.29	0.039	0.298	0.216	0.002	0.020	0.024	0.212	0.217	0.036
$LF\left(1,0\right)$	0.507	0.102	0.517	0.172	0.016	0.084	0.115	0.336	0.148	0.019
$LF\left(1,1\right)$	0.29	0.049	0.294	0.364	0.005	0.030	0.055	0.091	0.254	0.032
$LF\left(0,1\right)$	0.194	0.025	0.193	0.574	0.002	0.013	0.020	0.082	0.444	0.089
$LF\left(0.1,0.1\right)$	0.287	0.039	0.296	0.226	0.002	0.020	0.026	0.194	0.219	0.035
$LF\left(0.9,0.1\right)$	0.457	0.084	0.459	0.182	0.011	0.064	0.089	0.278	0.157	0.020
$LF\left(0.9,0.9\right)$	0.286	0.047	0.292	0.344	0.004	0.028	0.050	0.098	0.248	0.032
$LF\left(0.1,0.9\right)$	0.204	0.027	0.205	0.495	0.002	0.014	0.021	0.087	0.387	0.072
$PKS\left(0,0\right)$	0.533	0.095	0.531	0.132	0.012	0.073	0.084	0.472	0.139	0.020
$PKS\left(1,0\right)$	0.515	0.269	0.520	0.150	0.084	0.299	0.337	0.378	0.126	0.017
$PKS\left(1,1\right)$	0.493	0.116	0.496	0.238	0.022	0.102	0.165	0.248	0.166	0.018
$PKS\left(0,1\right)$	0.278	0.042	0.290	0.274	0.003	0.023	0.035	0.137	0.230	0.033
$PKS\left(0.1,0.1\right)$	0.527	0.096	0.526	0.139	0.012	0.075	0.089	0.440	0.140	0.020
$PKS\left(0.9,0.1\right)$	0.527	0.223	0.528	0.152	0.061	0.235	0.306	0.383	0.128	0.017
$PKS\left(0.9,0.9\right)$	0.503	0.113	0.512	0.223	0.021	0.098	0.152	0.263	0.161	0.018
$PKS\left(0.1,0.9\right)$	0.306	0.049	0.323	0.240	0.004	0.029	0.044	0.162	0.205	0.028
$CM$	0.37	0.049	0.354	0.438	0.023	0.054	0.050	0.166	0.274	0.072
$AD$	0.379	0.051	0.364	0.439	0.022	0.059	0.054	0.106	0.233	0.048
$SF$	0.291	0.044	0.284	0.520	0.052	0.057	0.124	0.106	0.157	0.029
$SW$	0.265	0.039	0.257	0.405	0.021	0.050	0.109	0.093	0.171	0.022

,

Table 15. The p-values for the GoFTs related to examples XI–XX.

GoFT	XI	XII	XIII	XIV	XV	XVI	XVII	XVIII	XIX	XX
$LF\left(0,0\right)$	0.009	0.016	0.442	0.220	0.028	0.484	0.449	0.525	0.128	0.411
$LF\left(1,0\right)$	0.005	0.022	0.411	0.397	0.089	0.398	0.728	0.819	0.050	0.338
$LF\left(1,1\right)$	0.009	0.011	0.694	0.205	0.050	0.691	0.363	0.888	0.077	0.723
$LF\left(0,1\right)$	0.030	0.009	0.804	0.135	0.022	0.784	0.256	0.471	0.159	0.439
$LF\left(0.1,0.1\right)$	0.009	0.015	0.463	0.217	0.029	0.499	0.435	0.553	0.119	0.431
$LF\left(0.9,0.1\right)$	0.005	0.020	0.431	0.347	0.073	0.420	0.648	0.836	0.053	0.355
$LF\left(0.9,0.9\right)$	0.009	0.012	0.665	0.204	0.047	0.666	0.367	0.847	0.079	0.679
$LF\left(0.1,0.9\right)$	0.023	0.010	0.751	0.144	0.023	0.805	0.271	0.492	0.168	0.459
$PKS\left(0,0\right)$	0.005	0.027	0.314	0.416	0.071	0.326	0.702	0.793	0.069	0.249
$PKS\left(1,0\right)$	0.004	0.042	0.351	0.632	0.089	0.352	0.781	0.794	0.048	0.314
$PKS\left(1,1\right)$	0.005	0.019	0.541	0.395	0.118	0.515	0.710	0.780	0.042	0.531
$PKS\left(0,1\right)$	0.009	0.013	0.557	0.208	0.037	0.573	0.393	0.690	0.094	0.536
$PKS\left(0.1,0.1\right)$	0.005	0.026	0.331	0.411	0.074	0.338	0.726	0.791	0.065	0.262
$PKS\left(0.9,0.1\right)$	0.004	0.037	0.358	0.640	0.091	0.356	0.786	0.795	0.048	0.315
$PKS\left(0.9,0.9\right)$	0.005	0.020	0.512	0.394	0.111	0.487	0.709	0.823	0.043	0.478
$PKS\left(0.1,0.9\right)$	0.007	0.015	0.518	0.234	0.043	0.523	0.444	0.767	0.075	0.470
$CM$	0.022	0.052	0.590	0.316	0.064	0.588	0.645	0.712	0.136	0.485
$AD$	0.015	0.054	0.544	0.351	0.053	0.569	0.738	0.665	0.107	0.398
$SF$	0.011	0.111	0.595	0.447	0.102	0.589	0.848	0.677	0.175	0.452
$SW$	0.006	0.117	0.439	0.271	0.040	0.554	0.897	0.481	0.112	0.260

and

Table 16. The p-values for the GoFTs related to examples XXI–XXX.

GoFT	XI	XII	XIII	XIV	XV	XVI	XVII	XVIII	XIX	XX
$LF\left(0,0\right)$	0.017	0.484	0.116	0.522	0.038	0.005	0.738	0.100	0.398	0.824
$LF\left(1,0\right)$	0.011	0.340	0.056	0.818	0.102	0.011	0.535	0.145	0.332	0.694
$LF\left(1,1\right)$	0.015	0.481	0.085	0.888	0.048	0.008	0.727	0.074	0.244	0.868
$LF\left(0,1\right)$	0.026	0.734	0.265	0.473	0.026	0.004	0.964	0.058	0.215	0.769
$LF\left(0.1,0.1\right)$	0.017	0.481	0.111	0.551	0.038	0.005	0.732	0.096	0.378	0.843
$LF\left(0.9,0.1\right)$	0.012	0.358	0.060	0.835	0.084	0.010	0.559	0.129	0.350	0.718
$LF\left(0.9,0.9\right)$	0.015	0.478	0.087	0.848	0.047	0.008	0.723	0.076	0.256	0.864
$LF\left(0.1,0.9\right)$	0.023	0.674	0.213	0.492	0.027	0.005	0.927	0.062	0.228	0.790
$PKS\left(0,0\right)$	0.012	0.349	0.068	0.791	0.095	0.009	0.562	0.167	0.399	0.742
$PKS\left(1,0\right)$	0.009	0.292	0.050	0.793	0.267	0.022	0.478	0.265	0.281	0.644
$PKS\left(1,1\right)$	0.010	0.347	0.050	0.776	0.115	0.014	0.552	0.128	0.284	0.714
$PKS\left(0,1\right)$	0.016	0.474	0.096	0.688	0.042	0.006	0.717	0.085	0.310	0.866
$PKS\left(0.1,0.1\right)$	0.012	0.346	0.064	0.789	0.096	0.009	0.554	0.163	0.384	0.728
$PKS\left(0.9,0.1\right)$	0.009	0.297	0.051	0.794	0.223	0.020	0.483	0.235	0.288	0.648
$PKS\left(0.9,0.9\right)$	0.010	0.345	0.051	0.819	0.112	0.014	0.544	0.131	0.293	0.703
$PKS\left(0.1,0.9\right)$	0.015	0.437	0.082	0.764	0.048	0.007	0.668	0.094	0.340	0.823
$CM$	0.022	0.698	0.227	0.710	0.046	0.049	0.959	0.223	0.217	0.871
$AD$	0.015	0.713	0.250	0.663	0.048	0.022	0.971	0.256	0.274	0.892
$SF$	0.026	0.659	0.353	0.674	0.043	0.028	0.971	0.306	0.347	0.816
$SW$	0.010	0.557	0.257	0.480	0.038	0.010	0.885	0.513	0.598	0.860

).” The lowest p-value for the analyzed tests is in bold. The non-normality is the most pronounced by parameterized GoFTs, namely $LF\left(0,0\right)$ (example V), $LF\left(0,1\right)$ (examples I–III, V–II, XII, XIV, XV, XVII, XVIII, XXIV–XXVI, XXVIII and XXIX), $LF\left(0.1,0.1\right)$ (example V), $LF\left(0.1,0.9\right)$ (example V), $PKS\left(0,0\right)$ (examples IV, XIII, XVI and XX), $PKS\left(1,0\right)$ (examples IX, X, XI, XXII, XXIII, XXVII and XXX), $PKS\left(1,1\right)$ (examples XIX and XXIII) and $PKS\left(0.9,0.1\right)$ (examples X, XI and XXI). The obtained results indicate that the new proposal is very universal and can handle various data regarding different groups of skewness and kurtosis signs and sample sizes from 16 to 153.

7. Conclusions

The analyzed GoFTs detect samples from asymmetric distributions with positive excess kurtosis most effectively, and those from symmetric distributions with positive excess kurtosis least effectively. The mentioned tests detect samples from a Pearson distribution most effectively and those from an EP distribution least effectively.

The parameterized GoFT, called $LF\left(0,1\right)$, stands out in the alternative groups A, E and G. The SW test is the best test for the analyzed groups when the alternative distribution is the P distribution (except for group C). The SF test performs excellently for the alternative distributions in group C. The AD test has the highest power only very rarely, for example, when the alternative NDPC distribution is for groups B and C, and the similarity measures are 0.5 and 0.9, respectively.

The new parameterized KS GoFT distribution with parameters $\left(\alpha =0.9,\beta =0.1\right)$ stands out among the alternative distributions from group B, characterized by negative skewness and positive excess kurtosis. The $PKS\left(\alpha =0,\beta =0\right)$ stands out for symmetric alternative distributions with negative excess kurtosis (group D), left-skewed alternative distributions with negative excess kurtosis (group F), and left-skewed alternative distributions with zero excess kurtosis (group H). Among the mentioned distributions, as can be seen in the Appendix, there are alternative distributions: asymmetric short-tailed and long-tailed, symmetric short-tailed and long-tailed, and symmetric distributions close to the Gaussian distribution, i.e., they meet the division proposed in [22], Krauczi [23], and Torabi et al. [21]

The good performance of the parameterized GoFTs, including the new proposal, was demonstrated by analyzing thirty real datasets.

In our opinion, future research should focus on alternative distributions which, thanks to appropriately selected parameter values, are close to the normal distribution, i.e., their similarity measure to the Gaussian distribution is at least 0.9.