Modeling Non-Normal Distributions with Mixed Third-Order Polynomials of Standard Normal and Logistic Variables

Abstract

Continuous data associated with many real-world events often exhibit non-normal characteristics, which contribute to the difficulty of accurately modeling such data with statistical procedures that rely on normality assumptions. Traditional statistical procedures often fail to accurately model non-normal distributions that are often observed in real-world data. This paper introduces a novel modeling approach using mixed third-order polynomials, which significantly enhances accuracy and flexibility in statistical modeling. The main objective of this study is divided into three parts: The first part is to introduce two new non-normal probability distributions by mixing standard normal and logistic variables using a piecewise function of third-order polynomials. The second part is to demonstrate a methodology that can characterize these two distributions through the method of L-moments (MoLMs) and method of moments (MoMs). The third part is to compare the MoLMs- and MoMs-based characterizations of these two distributions in the context of parameter estimation and modeling non-normal real-world data. The simulation results indicate that the MoLMs-based estimates of L-skewness and L-kurtosis are superior to their MoMs-based counterparts of skewness and kurtosis, especially for distributions with large departures from normality. The modeling (or data fitting) results also indicate that the MoLMs-based fits of these distributions to real-world data are superior to their corresponding MoMs-based counterparts.

1. Introduction

In many practical applications of statistics, the accurate modeling of the distribution of data is crucial for effective analysis and reliable decision making. Although several classical statistical procedures (such as the t-test, ANOVA, and linear regression) are based on the normality assumption, many real-world data do not exhibit the true characteristics of a normal distribution [1,2,3,4]. In the field of statistical modeling, the prevalence of non-normal data poses significant challenges for traditional methods in areas such as high-frequency financial trading, economics, public health, psychology, biomedical imaging, and so on. As a result, researchers/practitioners in these fields encounter complex distributional patterns that conventional models struggle to address [5,6].

Since the pioneering work of statisticians like R. A. Fisher, Karl Pearson, and William Gosset (pseudonym “Student”), numerous probability distributions have been introduced to model non-normal data. Some of the well-known non-normal distributions are Student’s t-distribution [7], Burr distribution [8], log-normal distribution [9], Fleishman’s power method distribution [10], Tukey’s g and h distribution [11], generalized lambda distribution [12], skew-normal distribution [13], and so on. Although each of these distributions is unique and is capable of modeling non-normal data with specific degrees of non-normality, one distribution is not “one size fits all” for modeling all types of non-normal data. Therefore, researchers are continuously developing distributions with unique features to model datasets with specific non-normal characteristics (e.g., heavy-tailed data, highly kurtotic data, bimodal data, and so on).

Among other important distributions, Fleishman’s [10] third-order power method distribution, together with its extended fifth-order power method [14] version, has received considerable attention in simulating univariate and multivariate non-normal distributions and in numerous applications (see Section 2). As a more recent modification to Fleishman’s third-order power method distribution, refs. [15,16] introduced two separate families of non-normal distributions via a doubling technique [17] to simulate non-normal data with a wide range of values of method of moments (MoMs)-based skewness and kurtosis and method of $L$-moments (Mo$L$Ms)-based $L$-skewness and $L$-kurtosis.

Historically, approaches to modeling non-normal distributions have evolved significantly from basic transformations to more sophisticated methods like the one presented in this paper. Although traditional approaches to handling non-normal data, such as transformation techniques (e.g., log-transformation and Box–Cox transformation) or the use of alternative distributions (e.g., log-normal, skew-normal, and exponential distributions) laid important groundwork, they have their own limitations that may compromise the reliability and accuracy of statistical inferences [18]. While current modeling techniques struggle with the accurate representation of non-normal data with large values of skewness and kurtosis, our approach utilizing mixed third-order polynomials offers a robust solution, as demonstrated by the simulation results.

A promising approach to addressing the challenge of modeling non-normal distributions involves using a mixture of distributions based on the third-order polynomials of standard normal and logistic variables. These mixture distributions can represent a variety of distributional shapes, accommodating both symmetric and asymmetric features. The use of third-order polynomial terms enables the capture of complex relationships within the data, providing a more nuanced and accurate representation than linear or lower-order polynomial models.

In this paper, we propose two mixtures of polynomial distributions based on piecewise functions of standard normal and logistic variables and explore their applications to approximate and model non-normal distributions. The proposed methodology combines the strengths of both probability distributions: the well-known properties and widespread applicability of normal distribution and the flexibility of the logistic distribution to accommodate skewed and heavy-tailed distributions. By mixing third-order polynomials of normal and logistic variables through piecewise functions, we aim to create a versatile family of distributions that can enhance the precision and adaptability of statistical analysis.

The remainder of this paper is structured as follows. In Section 2, we review the literature on relevant non-normal distributions and the use of power method polynomials in statistical modeling. In Section 3, we detail the theoretical foundation of the proposed third-order mix of polynomials based on piecewise functions, including their mathematical derivations and properties. In Section 4, we present simulation studies and data-fitting examples to demonstrate the application of the proposed distributions when modeling various non-normal data. In Section 5, we discuss the results in the context of potential applications and implications for future research and provide a summary concluding the key findings.

2. Theoretical Framework

The third-order power method of polynomials originally proposed by Fleishman [10] can be defined as follows [4]:

$$p\left(z\right)=c_0+c_{1}z+c_2{z}^2+c_3{z}^3$$

(1)

where $Z~$ $N\left(0, 1\right)$ with $\mathrm{pdf}$ $\varphi \left(z\right)={\left(\sqrt{2\pi }\right)}^{-1}{e}^{-z^2/2}$. The real-valued coefficients $c_{i=0,1,2,3}$ used in Equation (1) can be obtained by solving Equations (2.18)–(2.21) from ([4], p. 15) for the specified values of the method of moments (MoMs)-based parameters of skewness and kurtosis. It is essential to elaborate on these coefficients and their derivation from foundational works in this field, as they play a critical role in the functionality and effectiveness of the proposed polynomial mixtures. This paper builds on these mathematical foundations by proposing an enhanced approach that incorporates logistic variables into the mix, aiming to address and mitigate some of the known limitations of traditional power method polynomials.

The probability density function $\left(\mathrm{pdf}\right)$ and cumulative distribution function $\left(\mathrm{cdf}\right)$ associated with Equation (1) can be expressed in parametric forms as follows:

$$f\left(p\left(z\right)\right)=f\left(p\left(z\right),\varphi \left(z\right)/p^{\prime }\left(z\right)\right)$$

(2)

$$F\left(p\left(z\right)\right)=F\left(p\left(z\right),\Phi \left(z\right)\right)$$

(3)

where $p^{\prime }\left(z\right)$ in Equation (2) is the first derivative of the power method polynomial in Equation (1), which is assumed to be greater than zero for Equation (2) to produce a valid pdf, and $\Phi \left(z\right)$ in Equation (3) is the $\mathrm{cdf}$ of $Z$. Following this framework, it is imperative to critically evaluate the recent applications of similar polynomial-based models for modeling non-normal distributions. While these studies have advanced our understanding, they often fall short of addressing the full spectrum of non-normal characteristics observed in real-world data, such as those involving higher moments of distribution. This paper identifies these gaps, particularly focusing on the limitations of current methods to effectively handle data with extreme values of skewness and kurtosis, and proposes a refined approach that seeks to mitigate these issues.

Power method polynomials have been widely used to simulate univariate and multivariate non-normal distributions with specified values of skewness, kurtosis, and Pearson correlation in a variety of contexts [4]. Some of these contexts include ANOVA [19,20,21], ANCOVA [22,23], regression analysis [24], microarray analysis [25], multivariate analysis [26], item response theory [27], nonparametric statistics [28], and structural equation modeling [29].

Most of the applications of power method polynomials involve the MoMs-based procedure, which has its own limitations. One of the limitations associated with MoMs-based power method polynomials is that distributions with large values of skewness and/or kurtosis can peak at the mode and, thus, may not be representative of real-world data [15,16]. To demonstrate this limitation, Figure 1B shows an extremely peaked pdf of a standard normal-based third-order power method distribution with skewness $\left({\gamma }_3\right)=-3$ and kurtosis $\left({\gamma }_4\right)=39$.

Another limitation associated with the MoMs-based power method distribution in Equation (1) is that it can produce valid $\mathrm{pdf}$s for the combinations of skewness ($\left|{\gamma }_3\right|$) and kurtosis (${\gamma }_4$) (see Figure 2.2 from [4], p. 20), where ${\gamma }_4$ ranges between 0 and 43.2 for standard normal-based power method distributions and between 1.2 and 472.53 for logistic-based PM distributions. Another limitation associated with the MoMs-based power method distribution is that estimates (for example, of skewness, kurtosis, and Pearson correlation) have unfavorable attributes insofar as they can be substantially biased, have high variance, or can be influenced by outliers [15,16,30,31,32,33,34].

In the context of these limitations, the main objective of this paper is to introduce two families of mixture polynomial (MP) distributions by mixing standard normal- and standard logistic-based third-order polynomials through the method of $L$-moments (MoLMs), specifically, to obviate the problem of (a) the excessive peaking of pdf associated with some power method distributions with substantial departure from normality and (b) bias associated with the MoMs-based estimates of skewness and kurtosis [15,16,35,36]. Another objective of this study was to extend the range of skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ of valid MoMs-based power method distributions that can be used in simulation studies.

3. Methodology

The normal-logistic mixture polynomial (MP) distribution is a piecewise function of standard normal- and logistic-based third-order polynomials, expressed as follows [15]:

$$p\left(v\right)=\left\{\begin{array}{c}z+{\alpha }_{\mathcal{l}} z^3,\\ y+{\alpha }_{\mathcal{r}}{y}^3,\end{array}\right.\begin{array}{c}\mathrm{for} z\le 0\\ \mathrm{for} y\ge 0\end{array}$$

(4)

where ${\alpha }_{\mathcal{l}}\ge 0$, ${\alpha }_{\mathcal{r}}\ge 0$, $Z~N\left(0, 1\right)$, and $Y~Logistic\left(0, \sqrt{\pi /8}\right)$, where the scale parameter,$\sqrt{\pi /8}$, of the logistic distribution adjusts the height of its pdf at $y=0$ to $1/\sqrt{2\pi }$, which is the height of the standard normal pdf at $z=0$.

The pdf of the normal-logistic MP distribution can be defined in parametric form as follows:

$$f\left(p\left(v\right)\right)=\left\{\begin{array}{c}\varphi \left(z\right)/p^{\prime }\left(z\right),\\ \varphi \left(y\right)/p^{\prime }\left(y\right),\end{array}\right.\begin{array}{c}\mathrm{for} z\le 0\\ \mathrm{for} y\ge 0\end{array}$$

(5)

where $\varphi \left(z\right)={\left(\sqrt{2\pi }\right)}^{-1}{e}^{-z^2/2}$ and $\varphi \left(y\right)=\left(\sqrt{8/\pi }{e}^{-\left(\sqrt{8/\pi }y\right)}\right)/{\left(1+e^{-\left(\sqrt{8/\pi }y\right)}\right)}^2$ are the pdfs of standard normal and logistic variables $Z$ and $Y$, respectively, and $p^{\prime }\left(z\right)$ and $p^{\prime }\left(y\right)$ are the first derivatives of $p\left(z\right)=z+{\alpha }_{\mathcal{l}}{z}^3$ and $p\left(y\right)=y+{\alpha }_{\mathcal{r}}{y}^3$, respectively.

The logistic-normal MP distribution is a piecewise function of standard logistic- and normal-based third-order polynomials, and is expressed as follows:

$$p\left(v\right)=\left\{\begin{array}{c}y+{\alpha }_{\mathcal{l}} y^3,\\ z+{\alpha }_{\mathcal{r}} z^3,\end{array}\right.\begin{array}{c}\mathrm{for} y\le 0\\ \mathrm{for} z\ge 0\end{array}$$

(6)

where ${\alpha }_{\mathcal{l}}\ge 0$, ${\alpha }_{\mathcal{r}}\ge 0$, $Y~Logistic\left(0,\sqrt{\pi /8}\right)$ and $Z~N\left(0,1\right)$.

The pdf of the logistic-normal MP distribution can be defined in parametric form as follows:

$$f\left(p\left(v\right)\right)=\left\{\begin{array}{c}\varphi \left(y\right)/p^{\prime }\left(y\right),\\ \varphi \left(z\right)/p^{\prime }\left(z\right),\end{array}\right.\begin{array}{c}\mathrm{for} y\le 0\\ \mathrm{for} z\ge 0\end{array}$$

(7)

where $\varphi \left(y\right)=\left(\sqrt{8/\pi }{e}^{-\left(\sqrt{8/\pi }y\right)}\right)/{\left(1+e^{-\left(\sqrt{8/\pi }y\right)}\right)}^2$ and $\varphi \left(z\right)={\left(\sqrt{2\pi }\right)}^{-1}{e}^{-z^2/2}$ are the pdfs of standard logistic and normal variables $Y$ and $Z$, respectively, and $p^{\prime }\left(y\right)$ and $p^{\prime }\left(z\right)$ are the first derivatives of $p\left(y\right)=y+{\alpha }_{\mathcal{l}}{y}^3$ and $p\left(z\right)=z+{\alpha }_{\mathcal{r}}{z}^3$, respectively.

To demonstrate this methodology, the pdf of a normal-logistic MP distribution based on Equations (4) and (5) is presented in panel A of Figure 1. Specifically, panel A of Figure 1 is the pdf of a normal-logistic MP distribution with mean $\left(\mu \right)=-0.4501$, standard deviation $\left(\sigma \right)=3.1174$, skewness $\left({\gamma }_3\right)=-3$, and kurtosis $\left({\gamma }_4\right)=39$ with corresponding $L$-moment-based parameters of $L$-skewness $\left({\tau }_3\right)=-0.2313$ and $L$-kurtosis $\left({\tau }_4\right)=0.3899$. Also presented in panel B of Figure 1 is the pdf of a standard normal-based third-order power method distribution [4] that has the same values of skewness and kurtosis as that of the distribution in panel A. An inspection of Figure 1A,B indicates that the pdf in panel A is more representative of real-world data, whereas the pdf in panel B shows pointed peak at the mode, even though these two pdfs have the same degree of non-normality (i.e., the same values of skewness = −3 and kurtosis = 39). This example illustrates that Fleishman’s (1978) third-order power method distribution, while widely used, can struggle to accurately fit data with extreme values of skewness and kurtosis. In contrast, the proposed MP distributions produce pdfs that provide a better representation of real-world data with the same values of skewness and kurtosis. This implies that the family of MP distributions offers a compelling alternative for modeling data characterized by excessive skewness and kurtosis.

Table 1. MoMs-based parameters of skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ and Mo$L$Ms-based parameters of $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ along with their corresponding bootstrap estimates, 95% confidence intervals (95% C.I.), and standard errors (SE) for the $\mathrm{pdf}$ in Figure 1A.

$\mathbf{Skewness}:{\mathit{\gamma }}_{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{3}$			$\mathbf{Kurtosis}\mathbf{:}{\mathit{\gamma }}_{\mathbf{4}}\mathbf{=}\mathbf{39}$
${\widehat{\mathit{\gamma }}}_{\mathbf{3}}$	95% C.I.	SE	${\widehat{\mathit{\gamma }}}_{\mathbf{4}}$	95% C.I.	SE
$-1.9686$	$\left(-1.9910,-1.9460\right)$	$0.0113$	$13.73$	$\left(13.59, 13.87\right)$	$0.0721$
$\mathit{L}\mathbf{-}\mathbf{Skewness}\mathbf{:}{\mathit{\tau }}_{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{0.2313}$			$\mathit{L}\mathbf{-}\mathbf{Kurtosis}\mathbf{:}{\mathit{\tau }}_{\mathbf{4}}\mathbf{=}\mathbf{0.3899}$
${\widehat{\mathit{\tau }}}_{\mathbf{3}}$	95% C.I.	SE	${\widehat{\mathit{\tau }}}_{\mathbf{4}}$	95% C.I.	SE
$-0.2235$	$\left(-0.2249,-0.2220\right)$	$0.0008$	$0.3833$	$\left(0.3825, 0.3841\right)$	$0.0004$

indicates that the estimates ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ are much closer to their respective parameters of L-skewness $\left({\tau }_3\right)$ and L-kurtosis $\left({\tau }_4\right)$ than the MoMs-based estimates ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$ of skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$. Specifically, the estimates ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$ are, on average, 65.62% and 35.21% of their respective parameters (${\gamma }_3$ and ${\gamma }_4$). On the other hand, the estimates ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ are, on average, 96.63% and 98.31% of their respective parameters (${\tau }_3$ and ${\tau }_4$). An inspection of Table 1 also indicates that the standard errors (SEs) associated with estimates ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ are much smaller than those associated with estimates ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$. For each bootstrap estimate in Table 1, the 95% bootstrap confidence interval (95% C.I.) and standard error (SE) were based on resampling 25,000 statistics using bootstrap functions of the R [37] package ‘boot’ [38]. Each statistic was based on a sample size of $n=100$.

3.1. L-Moments

For a continuous random variable $V$ from a probability distribution with $\mathrm{pdf}$ $\varphi \left(v\right)$ and $\mathrm{cdf}$ $\Phi \left(v\right)$, the $i$-th probability-weighted moment (PWM), ${\beta }_i$, can be expressed as follows [30]:

$${\beta }_i=\int v{\left\{\Phi \left(v\right)\right\}}^i\varphi \left(v\right)dv$$

(8)

$L$-moments, originally proposed by Hosking [30], are defined as a linear combination of $\mathrm{PWM}$s. Specifically, the first four $L$-moments associated with $V$ can be expressed as follows ([30], p. 107):

$${\lambda }_1={\beta }_0$$

(9)

$${\lambda }_2=2{\beta }_1-{\beta }_0$$

(10)

$${\lambda }_3=6{\beta }_2-6{\beta }_1+{\beta }_0$$

(11)

$${\lambda }_4=20{\beta }_3-30{\beta }_2+12{\beta }_1-{\beta }_0$$

(12)

where ${\beta }_{i=0, 1, 2, 3}$ in Equations (9)–(12) was obtained by evaluating the integral in Equation (8) for $i=0, 1, 2, 3$. The coefficients associated with ${\beta }_i$ in Equations (9)–(12) were obtained from ([31], p. 20).

The first two $L$-moments, ${\lambda }_1$ and ${\lambda }_2$, in Equations (9) and (10) measure the location and scale of distribution and are the arithmetic mean and one-half of the coefficient of the mean difference (or Gini’s index of spread), respectively. The $L$-moment-based indices of $L$-skewness $({\tau }_3)$ and $L$-kurtosis $({\tau }_4)$ (analogous to skewness and kurtosis) are the ratios defined as ${\tau }_3={\lambda }_3/{\lambda }_2$ and ${\tau }_4={\lambda }_4/{\lambda }_2$, respectively. In general, $L$-moment ratios are bounded in the interval and $-1<{\tau }_{r\ge 3}<1$ as is the index of $L$-skewness (${\tau }_3$), where a symmetric distribution implies that all $L$-moment ratios with odd subscripts are zero [15].

The $L$-moment-based characterizations of distributions have certain advantages over their conventional moment-based counterparts. For example, in terms of parameter estimation, the Mo$L$Ms-based estimates of $L$-skewness, $L$-kurtosis, and $L$-correlation are substantially less biased and more precise than the MoMs-based estimates of skewness, kurtosis, and Pearson correlation. Likewise, in terms of distribution fitting, the Mo$L$Ms-based distributions provide better fits to non-normal data than their MoMs-based counterparts [15,16,32,33,34,35,39,40,41,42,43,44].

According to [30], if the mean (${\lambda }_1$) exists, then all other $L$-moments have finite expectations. To maintain this advantage, it is assumed that the coefficients ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ in Equations (4) and (6) for any distribution are positive (i.e., ${\alpha }_{\mathcal{l}}\ge 0$ and ${\alpha }_{\mathcal{r}}\ge 0$) so that its $i$-th $L$-moment exists and is finite.

3.2. L-Moments for the Normal-Logistic MP Distributions

The derivation of $L$-moments associated with the normal-logistic MP distributions can be obtained by first writing Equation (8) as follows:

$${\beta }_i={{\displaystyle \int }}_{-\infty }^0\left(z+{\alpha }_{\mathcal{l}} z^3\right){\left\{\Phi \left(z\right)\right\}}^i\varphi \left(z\right)dz+{{\displaystyle \int }}_0^{\infty }\left(y+{\alpha }_{\mathcal{r}} y^3\right){\left\{\Phi \left(y\right)\right\}}^i\varphi \left(y\right)dy$$

(13)

where $Z~N\left(0,1\right)$ and $Y~Logistic\left(0,\sqrt{\pi /8}\right)$ have the corresponding pdfs $\varphi \left(z\right)={\left(\sqrt{2\pi }\right)}^{-1}{e}^{-z^2/2}$ and $\varphi \left(y\right)=\left(\sqrt{8/\pi }{e}^{-\left(\sqrt{8/\pi }y\right)}\right)/{\left(1+e^{-\left(\sqrt{8/\pi }y\right)}\right)}^2$ and cdfs $\Phi \left(z\right)=\left(1/2\right)\mathrm{erfc}\left(z/\sqrt{2}\right)$ and $\Phi \left(y\right)={\left(1+e^{-\left(\sqrt{8/\pi }y\right)}\right)}^{-1},$ where $\mathrm{erfc}\left(.\right)$ in the equation for $\Phi \left(z\right)$ is the complementary error function [45] associated with the standard normal distribution.

Evaluating both integrals in Equation (13) for $i=0$ and 1, it is straightforward to derive ${\beta }_0$ and ${\beta }_1$, which can be substituted into Equations (9) and (10) to obtain the first two $L$-moments as follows:

$${\lambda }_1=\left(16\pi \ln \left(2\right)-32-64{\alpha }_{\mathcal{l}}+9{\pi }^2\mathrm{Zeta}\left(3\right){\alpha }_{\mathcal{r}}\right)/\left(32\sqrt{2\pi }\right)$$

(14)

$${\lambda }_2=\left(8\sqrt{2}\left(2+5{\alpha }_{\mathcal{l}}\right)+8\pi +{\alpha }_{\mathcal{r}}{\pi }^4\right)/\left(32\sqrt{2\pi }\right)$$

(15)

For $i=2$ and $3$, the evaluation of the second integral on the right-hand side of Equation (13) is straightforward, but the evaluation of the first integral requires several mathematical manipulations, as shown in [15]. Specifically, the first three pieces on the right-hand side of Equation (16) were derived by substituting Equations (51) and (52) into $I_1+C_{\mathcal{L}}{I}_2$ of Equation (12) from [15]. Note: $C_{\mathcal{L}}$ in Equation (12) in [15] was replaced with ${\alpha }_{\mathcal{l}}$ in this paper. The first piece on the right-hand side of Equation (17) is based on Equation (17) from [15].

$$\begin{array}{cc}\hfill {\beta }_2=& -{\displaystyle \frac{1}{4\sqrt{2\pi }}}+{\displaystyle \frac{{\tan }^{-1}\left(\sqrt{2}\right)}{2{\pi }^{3/2}}}+{\alpha }_{\mathcal{l}}\left({\displaystyle \frac{15 {\tan }^{-1}\left(\sqrt{2}\right)-\sqrt{2}-3\sqrt{2}\pi }{12{\pi }^{3/2}}}\right)\hfill \\ & +{\displaystyle \frac{1}{192}}\sqrt{{\displaystyle \frac{\pi }{2}}}\left(4\left(7+\ln \left(256\right)\right)+3{\alpha }_{\mathcal{r}}\pi \left({\pi }^2+\ln \left(16\right)+6 \mathrm{Zeta}\left(3\right)\right)\right)\hfill \end{array}$$

(16)

$$\begin{gathered} {\beta }_3={\displaystyle \frac{12{\tan }^{-1}\left(\sqrt{2}\right)\left(2+5{\alpha }_{\mathcal{l}}\right)-\left(6+\sqrt{2}\right)\pi -{\alpha }_{\mathcal{l}}\left(\sqrt{2}+\left(15+2\sqrt{2}\right)\pi \right)}{16{\pi }^{3/2}}} \\ +{\displaystyle \frac{1}{768}}\sqrt{{\displaystyle \frac{\pi }{2}}}\left(16\left(7+\ln \left(64\right)\right)+{\alpha }_{\mathcal{r}}\pi \left(6+11{\pi }^2+72 \ln \left(2\right)+54 \mathrm{Zeta}\left(3\right)\right)\right) \end{gathered}$$

(17)

Hence, substituting ${\beta }_0, {\beta }_1, {\beta }_2, {\beta }_3$ into Equations (11) and (12) and simplifying yields the expressions for ${\lambda }_3$ and ${\lambda }_4$, which subsequently yields the expressions for $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ as follows:

$$\begin{gathered} {\tau }_3=\left(4\right(48 {\tan }^{-1}\left(\sqrt{2}\right)+4{\alpha }_{\mathcal{l}}\left(\left(2\sqrt{2}-15\right)\pi +30 {\tan }^{-1}\left(\sqrt{2}\right)-2\sqrt{2}\right) \\ +\pi \left(4\left(\sqrt{2}-6\right)+\sqrt{2}\pi \left(1+{\alpha }_{\mathcal{r}}\pi \ln \left(8\right)\right)\right)\left)\right) \\ /\left(\pi \left(32+80{\alpha }_{\mathcal{l}}+\sqrt{2}\pi \left(8+{\alpha }_{\mathcal{r}}{\pi }^3\right)\right)\right) \end{gathered}$$

(18)

$$\begin{gathered} {\tau }_4=\left(-432\sqrt{2}\left(2+5{\alpha }_{\mathcal{l}}\right)\pi +8{\pi }^2+30{\alpha }_{\mathcal{r}}{\pi }^3+{\alpha }_{\mathcal{r}}{\pi }^5 \\ +480\left({\alpha }_{\mathcal{l}}+6\sqrt{2}{\tan }^{-1}\left(\sqrt{2}\right)+15\sqrt{2}{\alpha }_{\mathcal{l}}{\tan }^{-1}\left(\sqrt{2}\right)\right)\right) \\ /\left(6\pi \left(8\sqrt{2}\left(2+5{\alpha }_{\mathcal{l}}\right)+8\pi +{\alpha }_{\mathcal{r}}{\pi }^4\right)\right) \end{gathered}$$

(19)

The closed-form formulae for ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$, obtained by solving Equations (18) and (19), can be written in simplified forms as follows:

$$\begin{gathered} {\alpha }_{\mathcal{l}}=\left(\left(\sqrt{2}{\pi }^2\left(2{\widehat{\tau }}_3-1\right)+4\pi \left(2{\widehat{\tau }}_3+6-\sqrt{2}\right)-48d_1\right)\left(6{\pi }^2{\widehat{\tau }}_4-{\pi }^2-30\right)\right) \\ +2\left({\pi }^2{\widehat{\tau }}_3-4d_2\right)\left(\sqrt{2}{\pi }^2\left(6{\widehat{\tau }}_4-1\right)+24\pi \left({\widehat{\tau }}_4+9\right)-720d_1\right) \\ /(4\left(2\sqrt{2}\pi +30d_1-2\sqrt{2}-15\pi -5\pi {\widehat{\tau }}_3\right)\left(6{\pi }^2{\widehat{\tau }}_4-{\pi }^2-30\right) \\ +30\left({\pi }^2{\widehat{\tau }}_3-4d_2\right)\left(\pi \left({\widehat{\tau }}_4+9\right)-30d_1-\sqrt{2}\right)) \end{gathered}$$

(20)

$$\begin{array}{cc}\hfill {\alpha }_{\mathcal{r}}=& \left(8\right(\pi \left(792-60\sqrt{2}+\pi \left(2\pi \left(60+\sqrt{2}\right)+108-17\sqrt{2}\right)\right)-60d_1\left(36+\pi \left(6+7\pi \right)\right)\hfill \\ & +5\pi {\widehat{\tau }}_3\left(24+6\pi \left(\sqrt{2}+30d_1\right)-55{\pi }^2\right)\hfill \\ & +3\pi {\widehat{\tau }}_4\left(16+\pi \left(4+4\sqrt{2}-60d_1+35\pi -4\sqrt{2}\pi \right)\right)\left)\right)\hfill \\ & /\left({\pi }^3\right(60\sqrt{2}\left(1+2d_2\right)+900d_1\left(4d_2-1\right)\hfill \\ & +5\pi {\widehat{\tau }}_3\left(30-6\pi \left(\sqrt{2}+30d_1\right)+55{\pi }^2\right)-2{\pi }^2\left(6{\widehat{\tau }}_4-1\right)\left(\sqrt{2}-15d_1\right)\hfill \\ & +{\pi }^3\left(6{\widehat{\tau }}_4-1\right)\left(2\sqrt{2}-15\right)-30\pi \left(4d_2\left(9+{\widehat{\tau }}_4\right)+2\sqrt{2}-15\right)\left)\right)\hfill \end{array}$$

(21)

where $d_1={\tan }^{-1}\left(\sqrt{2}\right)$ and $d_2=\ln \left(8\right)$, and ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ are the estimates of ${\tau }_3$ and ${\tau }_4$.

3.3. L-Moments for the Logistic-Normal MP Distributions

The derivation of $L$-moments associated with the logistic-normal MP distributions can be obtained by first writing Equation (8) as follows:

$${\beta }_i={{\displaystyle \int }}_{-\infty }^0\left(y+{\alpha }_{\mathcal{l}} y^3\right){\left\{\Phi \left(y\right)\right\}}^i\varphi \left(y\right)dy+{{\displaystyle \int }}_0^{\infty }\left(z+{\alpha }_{\mathcal{r}} z^3\right){\left\{\Phi \left(z\right)\right\}}^i\varphi \left(z\right)dz$$

(22)

where $Y~Logistic\left(0,\sqrt{\pi /8}\right)$ and $Z~N\left(0, 1\right)$ with pdfs and cdfs given in Section 3.2.

Evaluating both integrals in Equation (22) for $i=0$ and 1, it is straightforward to derive ${\beta }_0$ and ${\beta }_1$, which can be substituted into Equations (9) and (10) to obtain the first two $L$-moments as follows:

$${\lambda }_1=\left(32-16\pi \ln \left(2\right)+64{\alpha }_{\mathcal{r}}-9{\alpha }_{\mathcal{l}}{\pi }^2\mathrm{Zeta}\left(3\right)\right)/\left(32\sqrt{2\pi }\right)$$

(23)

$${\lambda }_2=\left(8\sqrt{2}\left(2+5{\alpha }_{\mathcal{r}}\right)+8\pi +{\alpha }_{\mathcal{l}}{\pi }^4\right)/\left(32\sqrt{2\pi }\right)$$

(24)

For $i=2$ and $3$, the evaluation of the first integral on the right-hand side of Equation (22) is straightforward; however, the evaluation of the second integral requires several mathematical manipulations, as shown in [15]. Specifically, the last three pieces on the right-hand side of Equation (25) were derived by substituting Equations (53) and (54) into $I_3+C_{\mathcal{R}}{I}_4$ of Equation (12) from [15]. Note: $C_{\mathcal{R}}$ in Equation (12) in [15] was replaced with ${\alpha }_{\mathcal{r}}$ in this paper. In addition, the second expression on the right-hand side of Equation (26) was based on Equation (18) from [15]. Hence, ${\beta }_2$ and ${\beta }_3$ can be expressed as follows:

$$\begin{array}{cc}\hfill {\beta }_2=& {\displaystyle \frac{\sqrt{\pi }}{192\sqrt{2}}}\left(20-32 \ln \left(2\right)+3{\alpha }_{\mathcal{l}}\pi \left({\pi }^2-4 \ln \left(2\right)-6 \mathrm{Zeta}\left(3\right)\right)\right)\hfill \\ & +{\displaystyle \frac{1}{4\sqrt{2\pi }}}+{\displaystyle \frac{\pi -{\tan }^{-1}\left(\sqrt{2}\right)}{2{\pi }^{3/2}}}+{\alpha }_{\mathcal{r}}\left({\displaystyle \frac{15\left(\pi -{\tan }^{-1}\left(\sqrt{2}\right)\right)+\sqrt{2}\left(1+3\pi \right)}{12{\pi }^{3/2}}}\right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\beta }_3=& {\displaystyle \frac{\sqrt{\pi }}{768\sqrt{2}}}\left(64-96 \ln \left(2\right)+{\alpha }_{\mathcal{l}}\pi \left(6+11{\pi }^2-72 \ln \left(2\right)-54 \mathrm{Zeta}\left(3\right)\right)\right)\hfill \\ & +{\displaystyle \frac{\left(6+\sqrt{2}\right)\pi +{\alpha }_{\mathcal{r}}\left(\left(3\sqrt{2}+\left(15+2\sqrt{2}\right)\pi \right)\right)}{16{\pi }^{3/2}}}\hfill \end{array}$$

(26)

Substituting ${\beta }_0, {\beta }_1, {\beta }_2, {\beta }_3$ into Equations (11) and (12) and simplifying yields the expressions for ${\lambda }_3$ and ${\lambda }_4$, which subsequently yield the expressions for $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ as follows:

$$\begin{array}{cc}\hfill {\tau }_3=& -\left\{4\right.\left(24\sqrt{2}{\tan }^{-1}\left(\sqrt{2}\right)+{\alpha }_{\mathcal{r}}\left(60\sqrt{2}{\tan }^{-1}\left(\sqrt{2}\right)+\left(8-30\sqrt{2}\right)\pi -8\right)\right.\hfill \\ & \left. +\pi \left(4-12\sqrt{2}+\pi +{\alpha }_{\mathcal{l}}{\pi }^2 \ln \left(8\right)\right)\right\}/\left(\pi \left(8\sqrt{2}\left(2+5{\alpha }_{\mathcal{r}}\right)+8\pi +{\alpha }_{\mathcal{l}}{\pi }^4\right)\right)\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\tau }_4=& \left\{-432\sqrt{2}\left(2+5{\alpha }_{\mathcal{r}}\right)\pi \right.+480\left(\left(6+15{\alpha }_{\mathcal{r}}\right)\sqrt{2}{\tan }^{-1}\left(\sqrt{2}\right)+{\alpha }_{\mathcal{r}}\right)\hfill \\ & \left.+8{\pi }^2+30{\alpha }_{\mathcal{l}}{\pi }^3+{\alpha }_{\mathcal{l}}{\pi }^5\right\} /\left(6\pi \left(8\sqrt{2}\left(2+5{\alpha }_{\mathcal{r}}\right)+8\pi +{\alpha }_{\mathcal{l}}{\pi }^4\right)\right)\hfill \end{array}$$

(28)

The closed-form formulae for ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$, obtained by solving Equations (27) and (28), can be written in simplified forms as follows:

$$\begin{array}{cc}\hfill {\alpha }_{\mathcal{l}}=& -\left(\right(960\left(2\pi {\widehat{\tau }}_3\left(2\sqrt{2}+\pi \right)+\pi \left(\pi +4-12\sqrt{2}\right)+24\sqrt{2}{d}_1\right)(\sqrt{2}\pi \left({\widehat{\tau }}_4+9\right)\hfill \\ & -30\sqrt{2}{d}_1-2)\hfill \\ & -64\left(5\sqrt{2}\pi {\widehat{\tau }}_3+30\sqrt{2}{d}_1+4\pi -15\sqrt{2}\pi -4\right)(12\sqrt{2}\pi \left({\widehat{\tau }}_4+9\right)\hfill \\ & +{\pi }^2\left(6{\widehat{\tau }}_4-1\right)-360\sqrt{2}{d}_1\left)\right)\hfill \\ & /(-8{\pi }^3\left(5\sqrt{2}\pi {\widehat{\tau }}_3+30\sqrt{2}{d}_1+4\pi -15\sqrt{2}\pi -4\right)(6{\pi }^2{\widehat{\tau }}_4-{\pi }^2\hfill \\ & -30)+240\left(4d_2{\pi }^3+{\pi }^5{\widehat{\tau }}_3\right)\left(\sqrt{2}\pi \left({\widehat{\tau }}_4+9\right)-30\sqrt{2}{d}_1-2\right)))\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\alpha }_{\mathcal{r}}=& (8d_2\pi \left(\pi -108\sqrt{2}\right)+\pi \left({\pi }^2+30\right)\left(12\sqrt{2}-\pi -4\right)\hfill \\ & -24\sqrt{2}{d}_1\left({\pi }^2+30-120d_2\right)\hfill \\ & -20\pi {\widehat{\tau }}_3\left(6\sqrt{2}+\pi \left(11\sqrt{2}\pi -36\sqrt{2}{d}_1+3\right)\right)\hfill \\ & +6\pi {\widehat{\tau }}_4\left(\pi \left(24\sqrt{2}{d}_1+\pi \left(4-12\sqrt{2}+\pi \right)\right)\right))\hfill \\ & /(1800\sqrt{2}{d}_1\left(1-4d_2\right)-240\left(1+2d_2\right)\hfill \\ & +60\pi \left(4\sqrt{2}{d}_2\left({\widehat{\tau }}_4+9\right)+5\sqrt{2}{\widehat{\tau }}_3-15\sqrt{2}+4\right)\hfill \\ & +2{\pi }^3\left(275\sqrt{2}{\widehat{\tau }}_3+\left(15\sqrt{2}-4\right)\left(6{\widehat{\tau }}_4-1\right)\right)\hfill \\ & -4{\pi }^2\left(2+30{\widehat{\tau }}_3-12{\widehat{\tau }}_4+15\sqrt{2}{d}_1\left(30{\widehat{\tau }}_3+6{\widehat{\tau }}_4-1\right)\right))\hfill \end{array}$$

(30)

where $d_1={\tan }^{-1}\left(\sqrt{2}\right)$ and $d_2=\ln \left(8\right)$, and ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ are the estimates of ${\tau }_3$ and ${\tau }_4$.

Hence, for the specified values of $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ associated with the normal-logistic and logistic-normal MP distributions, the systems of Equations (18), (19), (27) and (28) can be simultaneously solved for the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$. These solved values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ can be substituted into Equations (14), (15), (23) and (24), respectively, to obtain the corresponding values of $L$-mean $\left({\lambda }_1\right)$ and $L$-scale $\left({\lambda }_2\right)$.

Figure 2A,B show Mo$L$Ms-based boundary graphs of $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ for the two MP distributions to help practitioners choose a specific combination of ${\tau }_3$ and ${\tau }_4$ for simulating data. Specifically, Figure 2A presents the boundary graph for possible combinations of $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ in Equations (18) and (19), which are associated with normal-logistic MP distributions. The graph in Figure 2A was drawn by setting ${\alpha }_{\mathcal{r}}=0$ with ${\alpha }_{\mathcal{l}}\in \left[0,\infty \right)$ for the part on the left side of the vertical axis and by setting ${\alpha }_{\mathcal{l}}=0$ with ${\alpha }_{\mathcal{r}}\in \left[0,\infty \right)$ for the part on the right side. The minimum value of ${\tau }_4$ in Figure 2A is shown as $min\left({\tau }_4\right)\approx 0.1458$, where ${\alpha }_{\mathcal{l}}={\alpha }_{\mathcal{r}}=0$ and ${\tau }_3\approx 0.0412$. The maximum value of ${\tau }_4$ is shown as $max\left({\tau }_4\right)\approx 0.5728$ on the left side of the vertical axis and $max\left({\tau }_4\right)\approx 0.6733$ on the right side, which are associated with the pdfs of symmetric distributions of the forms $\left(2/5\right)z^3$ [46] and $\left(2/5\right)y^3$, where $y~logistic\left(0,\sqrt{\pi /8}\right)$. The value of ${\tau }_3$ ranges from ${\tau }_3\approx -0.7899$ on the left side of the vertical axis to ${\tau }_3\approx 0.8428$ on the right side. The graph in Figure 2B, a mirror image of the graph in Figure 2A, can be used for possible combinations of ${\tau }_3$ and ${\tau }_4$ in Equations (27) and (28) associated with logistic-normal MP distributions.

Similarly, for the specified values of skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ associated with the normal-logistic and logistic-normal MP distributions, the systems of Equations (A4), (A5), (A9) and (A10) from Appendix A and Appendix B, respectively, can be simultaneously solved for the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$. These solved values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ can be substituted into Equations (A2), (A3), (A7) and (A8), respectively, to obtain the corresponding values of the mean $\left(\mu \right)$ and variance $\left({\sigma }^2\right)$.

Presented in Figure A1A of Appendix C is the boundary graph of possible combinations of skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ in Equations (A4) and (A5) associated with the normal-logistic MP distributions. The lower boundary point for the graph in Figure A1A is $\left({\gamma }_3\approx 0.3361,{\gamma }_4\approx 0.7685\right)$, which is associated with ${\alpha }_{\mathcal{l}}={\alpha }_{\mathcal{r}}=0$. The maximum value of ${\gamma }_4$ is shown as $max\left({\gamma }_4\right)\approx 97.54$ on the left side of the vertical axis and $max\left({\gamma }_4\right)\approx 1014.96$ on the right side. The graph in Figure A1B, a mirror image of the graph in Figure A1A, can be used for possible combinations of ${\gamma }_3$ and ${\gamma }_4$ in Equations (A9) and (A10) associated with the logistic-normal MP distributions.

To demonstrate this methodology, the $\mathrm{pdfs}$ and $\mathrm{cdfs}$ of two normal-logistic (Distributions 1 and 2) and one logistic-normal (Distribution 3) MP distributions are displayed in Figure 3.

4. Results

4.1. Monte Carlo Simulation Results: Estimation of Parameters

To demonstrate the advantages of the Mo$L$Ms-based procedure over the MoMs-based procedure, the results of the Monte Carlo simulation in the context of parameter estimation for the three distributions of Figure 3 are shown in

Table 2. Simulation results for the Mo$L$Ms-based $L$-skewness (${\tau }_3$) and $L$-kurtosis (${\tau }_4$).

Dist.	Parameter	Estimate	95% C.I.	SE	RB%
$n=25$ (ET = 30.7 min *)
1	${\tau }_3=-0.0967$	${\widehat{\tau }}_3=-0.0889$	$\left(-0.0917,-0.0861\right)$	0.0014	$-8.07$
	τ4 = 0.3716	τ4 = 0.3491	(0.3475, 0.3506)	0.0008	$-6.05$
2	τ3 = −0.2973	τ3 = −0.2567	(−0.2595, −0.2538)	0.0014	$-13.66$
	τ4 = 0.4224	τ4 = 0.3968	(0.3952, 0.3985)	0.0008	$-6.06$
3	τ3 = 0.3986	τ3 = 0.3535	(0.3502, 0.3567)	0.0017	$-11.31$
	τ4 = 0.5180	τ4 = 0.4913	(0.4896, 0.4931)	0.0009	$-5.15$
$n=200$ (ET = 33.3 min *)
1	${\tau }_3=-0.0967$	${\widehat{\tau }}_3=-0.0954$	$\left(-0.0966,-0.0943\right)$	0.0006	$-1.34$
	${\tau }_4=0.3716$	${\widehat{\tau }}_4=0.3685$	$\left(0.3680, 0.3691\right)$	0.0003	$-0.83$
2	${\tau }_3=-0.2973$	${\widehat{\tau }}_3=-0.2910$	$\left(-0.2921,-0.2899\right)$	$0.0006$	$-2.12$
	${\tau }_4=0.4224$	${\widehat{\tau }}_4=0.4188$	$\left(0.4182, 0.4195\right)$	0.0003	$-0.85$
3	${\tau }_3=0.3986$	${\widehat{\tau }}_3=0.3932$	$\left(0.3919, 0.3945\right)$	$0.0007$	$-1.35$
	${\tau }_4=0.5180$	${\widehat{\tau }}_4=0.5145$	$\left(0.5139, 0.5152\right)$	0.0003	$-0.68$
$n=500$ (ET = 53.9 min *)
1	${\tau }_3=-0.0967$	${\widehat{\tau }}_3=-0.0963$	$\left(-0.0970,-0.0955\right)$	0.0004	$-0.41$
	${\tau }_4=0.3716$	${\widehat{\tau }}_4=0.3702$	$\left(0.3699, 0.3706\right)$	0.0002	$-0.38$
2	${\tau }_3=-0.2973$	${\widehat{\tau }}_3=-0.2948$	$\left(-0.2955,-0.2941\right)$	$0.0004$	$-0.84$
	${\tau }_4=0.4224$	${\widehat{\tau }}_4=0.4209$	$\left(0.4205, 0.4212\right)$	0.0002	$-0.36$
3	${\tau }_3=0.3986$	${\widehat{\tau }}_3=0.3964$	$\left(0.3956, 0.3973\right)$	$0.0004$	$-0.55$
	${\tau }_4=0.5180$	${\widehat{\tau }}_4=0.5165$	$\left(0.5161, 0.5169\right)$	0.0002	$-0.29$
$n=1000$ (ET = 83.6 min *)
1	${\tau }_3=-0.0967$	${\widehat{\tau }}_3=-0.0963$	$\left(-0.0969,-0.0958\right)$	0.0003	$-0.41$
	${\tau }_4=0.3716$	${\widehat{\tau }}_4=0.3710$	$\left(0.3708, 0.3713\right)$	0.0001	$-0.16$
2	${\tau }_3=-0.2973$	${\widehat{\tau }}_3=-0.2959$	$\left(-0.2964,-0.2954\right)$	$0.0003$	$-0.47$
	${\tau }_4=0.4224$	${\widehat{\tau }}_4=0.4217$	$\left(0.4214, 0.4220\right)$	0.0001	$-0.17$
3	${\tau }_3=0.3986$	${\widehat{\tau }}_3=0.3976$	$\left(0.3970, 0.3982\right)$	$0.0003$	$-0.25$
	${\tau }_4=0.5180$	${\widehat{\tau }}_4=0.5173$	$\left(0.5171, 0.5176\right)$	0.0001	$-0.14$

Note. Dist. = distribution. * ET = execution time for the algorithm.

and

Table 3. Simulation results for the MoMs-based skewness (${\gamma }_3$) and kurtosis (${\gamma }_4$).

Dist.	Parameter	Estimate	95% C.I.	SE	RB%
$n=25$ (ET = 40.6 min *)
1	${\gamma }_3=0$	${\widehat{\gamma }}_3=-0.4799$	$\left(-0.5000,-0.4602\right)$	0.0101	-----
	${\gamma }_4=50$	${\widehat{\gamma }}_4=3.9515$	$\left(3.903, 3.999\right)$	0.0245	$-92.10$
2	${\gamma }_3=-4$	${\widehat{\gamma }}_3=-1.4309$	$\left(-1.450,-1.412\right)$	0.0097	$-64.23$
	${\gamma }_4=48$	${\widehat{\gamma }}_4=5.2448$	$\left(5.19, 5.30\right)$	0.0279	$-89.07$
3	${\gamma }_3=5$	${\widehat{\gamma }}_3=1.8093$	$\left(1.788, 1.830\right)$	0.0106	$-63.81$
	${\gamma }_4=70$	${\widehat{\gamma }}_4=6.8858$	$\left(6.826, 6.946\right)$	0.0306	$-90.16$
$n=200$ (ET = 53.7 min *)
1	${\gamma }_3=0$	${\widehat{\gamma }}_3=-0.6283$	$\left(-0.6553,-0.6013\right)$	0.0138	-----
	${\gamma }_4=50$	${\widehat{\gamma }}_4=15.774$	$\left(15.58, 15.96\right)$	0.0970	$-68.45$
2	${\gamma }_3=-4$	${\widehat{\gamma }}_3=-2.9557$	$\left(-2.979,-2.932\right)$	0.0120	$-26.11$
	${\gamma }_4=48$	${\widehat{\gamma }}_4=21.586$	$\left(21.37, 21.80\right)$	0.1094	$-55.03$
3	${\gamma }_3=5$	${\widehat{\gamma }}_3=3.6669$	$\left(3.640, 3.694\right)$	0.0138	$-26.66$
	${\gamma }_4=70$	${\widehat{\gamma }}_4=28.616$	$\left(28.36, 28.87\right)$	0.1317	$-59.12$
$n=500$ (ET = 85.4 min *)
1	${\gamma }_3=0$	${\widehat{\gamma }}_3=-0.5462$	$\left(-0.5737,-0.5190\right)$	0.0140	-----
	${\gamma }_4=50$	${\widehat{\gamma }}_4=22.300$	$\left(22.00, 22.61\right)$	0.1566	$-55.40$
2	${\gamma }_3=-4$	${\widehat{\gamma }}_3=-3.4251$	$\left(-3.448,-3.402\right)$	0.0117	$-14.37$
	${\gamma }_4=48$	${\widehat{\gamma }}_4=30.131$	$\left(29.83, 30.44\right)$	0.1562	$-37.23$
3	${\gamma }_3=5$	${\widehat{\gamma }}_3=4.2306$	$\left(4.203, 4.258\right)$	0.0140	$-15.39$
	${\gamma }_4=70$	${\widehat{\gamma }}_4=40.569$	$\left(40.19, 40.96\right)$	0.1964	$-42.04$
$n=1000$ (ET = 290.4 min*)
1	${\gamma }_3=0$	${\widehat{\gamma }}_3=-0.4191$	$\left(-0.4463,-0.3916\right)$	0.0139	-----
	${\gamma }_4=50$	${\widehat{\gamma }}_4=27.755$	$\left(27.32, 28.20\right)$	0.2225	$-44.49$
2	${\gamma }_3=-4$	${\widehat{\gamma }}_3=-3.6514$	$\left(-3.673,-3.630\right)$	0.0110	$-8.71$
	${\gamma }_4=48$	${\widehat{\gamma }}_4=35.904$	$\left(35.55, 36.28\right)$	0.1863	$-25.20$
3	${\gamma }_3=5$	${\widehat{\gamma }}_3=4.5427$	$\left(4.516, 4.569\right)$	0.0135	$-9.15$
	${\gamma }_4=70$	${\widehat{\gamma }}_4=49.293$	$\left(48.82, 49.78\right)$	0.2438	$-29.58$

Note. Dist. = distribution. * ET = execution time for the algorithm.

. Figure 3 shows the $\mathrm{pdfs}$ and $\mathrm{cdfs}$ of two normal-logistic (Distributions 1 and 2) and one logistic-normal (Distribution 3) MP distributions. Also displayed below each distribution in Figure 3 are the Mo$L$Ms- and MoMs-based parameters of $L$-skewness $\left({\tau }_3\right)$, $L$-kurtosis $\left({\tau }_4\right)$, skewness $\left({\gamma }_3\right)$, and kurtosis $\left({\gamma }_4\right)$, along with the solved values of the shape parameters (${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$). The pdfs in Figure 3 were plotted by first substituting the solved values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ into Equations (4) and (6), respectively, to simulate the normal-logistic and logistic-normal MP distribution and then substituting these values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ into Equations (5) and (7) to plot the parametric forms of the pdfs associated with these three distributions.

Presented in Table 2 and Table 3, respectively, are the parameter values of the Mo$L$Ms-based $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$, the MoMs-based skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$, their corresponding bootstrap estimates, 95% confidence intervals (95% C.I.), and standard errors (SEs). To obtain these results, an R [37] algorithm consisting of bootstrap functions from the ‘boot’ [38] package was written to simulate 25,000 independent samples of sizes $n$= 25, 200, 500, and 1000 to compute the Mo$L$Ms-based estimates (${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$) of $L$-skewness and $L$-kurtosis (${\tau }_3$ and ${\tau }_4$) and the MoMs-based estimates (${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$) of skewness and kurtosis $\left({\gamma }_3\mathrm{and}{\gamma }_4\right)$ for each of the $\left(3\times 25,000\right)$ samples based on the values of shape parameters $({\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}})$ given for each distribution in Figure 3. The estimates (${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$) of ${\tau }_3\mathrm{and}{\tau }_4$ were computed by substituting the sample estimates of probability-weighted moments (PWMs) from ([30], pp. 113–115) into Equations (9)–(12) to obtain the sample estimates of $L$-moments, and these estimates of $L$-moments were subsequently substituted into the formulae for the estimates ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ of ${\tau }_3\mathrm{and}{\tau }_4$ from Section 3.1. The estimates (${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$) of ${\gamma }_3$ and ${\gamma }_4$ were computed based on Fisher’s $k$-statistics formulae ([47], pp. 47–48; [48], p. 2211; [49], p. 7). From each independent sample of 25,000 statistics, bootstrapped average estimates (Estimate), associated 95% confidence intervals (95% C.I.), and standard errors (SEs) were computed for each type of estimate using 25,000 resamples via the bootstrap functions of the ‘boot’ [38] package. If a parameter was outside its associated 95% C.I., then the percentage of relative bias (RB%) was computed for the estimate as follows:

RB% = 100 × (Estimate − Parameter)/Parameter

(31)

Also displayed in Table 2 and Table 3 (after the sample size, n) are the execution times of the algorithms for the Mo$L$Ms- and MoMs-based Monte Carlo simulation results. A comparison of these execution times clearly indicates that the Mo$L$Ms-based parameter estimation is computationally less expensive than the MoMs-based estimation, notably for large sample sizes ($n$ = 500 and 1000). For example, for $n$ = 1000, the execution time of 83.6 min for the Mo$L$Ms-based algorithm was substantially lower than the execution time of 290.4 min for the MoMs-based algorithm.

An inspection of Table 2 and Table 3 illustrates that each of the Mo$L$Ms-based estimates is superior to its corresponding MoMs-based counterpart in terms of both smaller SE and smaller RB%. These characteristics are mostly pronounced in the context of smaller sample sizes and higher-order moments. For example, for Distribution 3, for $n=25$, the simulated Mo$L$Ms-based estimates (${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$) were, on average, 88.69% and 94.85% of their respective parameters (${\tau }_3$ and ${\tau }_4$). On the other hand, the simulated MoMs-based estimates (${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$) were, on average, 36.19% and 9.84% of their respective parameters (${\gamma }_3$ and ${\gamma }_4$). Thus, the relative biases (RB%) of the Mo$L$Ms-based estimates are substantially smaller than those associated with MoMs-based estimates.

Additionally, it can be verified from an inspection of Table 2 and Table 3 that the standard errors (SEs) associated with the Mo$L$Ms-based estimates (${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$) are much smaller than the SEs associated with the MoMs-based estimates (${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$). When comparing SEs, we used the relative standard error (RSE) for each type of estimate, where RSE = 100 × SE/Estimate. For example, for Distribution 3, with $n=500$, the RSE values associated with Mo$L$Ms-based estimates ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ were 0.10% and 0.04%, respectively. On the other hand, the RSE values associated with MoMs-based estimates ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$ were 0.33% and 0.48%, respectively.

4.2. Data-Fitting Results

Presented in Figure 4, Figure 5, Figure 6 and Figure 7 are examples of data fitting with normal-logistic MP distributions. Specifically, Figure 4 and Figure 5 display pdfs of (panel A) Mo$L$Ms- and (panel B) MoMs-based normal-logistic MP distributions superimposed over histograms of random samples of size $n=500$ drawn, respectively, from Student’s $t$-distribution with three degrees of freedom ($t_{df=3}$) (Figure 4) and Type 1 stable distribution with parameters of stability of $\alpha =1.7$, skewness $\beta =-0.4$, location $\mu =0$, and scale $\sigma =1$ $\left(S_{Type=1}\left(\alpha =1.7,\beta =-0.4,\mu =0,\sigma =1\right)\right)$ (Figure 5) using Mathematica Version 13.2 [45]. The random seed for drawing random samples from each distribution was set to 8924 (as ‘SeedRandom[8924]’) for reproducibility purposes.

Presented in Figure 6 and Figure 7 are the pdfs of (panel A) Mo$L$Ms- and (panel B) MoMs-based normal-logistic MP distributions superimposed over the histograms of the daily return rate (percent change) of Apple (AAPL) (Figure 6) and Amazon (AMZN) (Figure 7) stocks for the last 10 years between 18 August 2014 and 16 August 2024. These data were downloaded on 18 August 2024 from the websites https://www.nasdaq.com/market-activity/stocks/aapl/historical and https://www.nasdaq.com/market-activity/stocks/amzn/historical, respectively.

To superimpose the pdfs of Mo$L$Ms-based normal-logistic MP distributions in panel A of Figure 4, Figure 5, Figure 6 and Figure 7, the sample estimates ${\widehat{\lambda }}_1$, ${\widehat{\lambda }}_2$, ${\widehat{\tau }}_3,$ and ${\widehat{\tau }}_4$ of $L$-mean $\left({\lambda }_1\right)$, $L$-scale $\left({\lambda }_2\right)$, $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ were computed from each sample dataset using Equation (3.1) from [30]. The values of ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ were substituted into the right-hand sides of Equations (18) and (19), respectively, which were subsequently solved for the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ using the “FindRoot[]” function of Mathematica Version 13.2 [45]. These solved values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ were substituted into Equation (4) to obtain the quantile function, $p\left(v\right)$, of each normal-logistic MP distribution. This quantile function, $p\left(v\right)$, was transformed as $x={\widehat{\lambda }}_1+{\widehat{\lambda }}_2\left(p\left(v\right)-{\lambda }_1\right)/{\lambda }_2$, where (${\widehat{\lambda }}_1, {\widehat{\lambda }}_2$) and (${\lambda }_1, {\lambda }_2$) are the values of the $L$-mean and $L$-scale computed from the actual data and the normal-logistic MP distribution. This transformed quantile $x$ was used in Equation (5) to obtain the pdf of each Mo$L$Ms-based normal-logistic MP distribution.

Alternatively, the closed-form formulae in Equations (20) and (21) can also be used to obtain the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ associated with an Mo$L$Ms-based pdf of normal-logistic MP distribution. However, the values of ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$, which need to be substituted into the right-hand sides of Equations (20) and (21), must be from the boundary region of Figure 2A.

To superimpose the pdfs of MoMs-based normal-logistic MP distributions in panel (B) of Figure 4, Figure 5, Figure 6 and Figure 7, the sample estimates $\widehat{\mu }$, $\widehat{\sigma }$, ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$ of the mean $\left(\mu \right)$, standard deviation $\left(\sigma \right)$, skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ were obtained from each sample dataset using Fisher’s $k$-statistics ([47], pp. 47–48; [48], p. 2211; [49], p. 7). The values of ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$ were substituted into the right-hand sides of Equations (A4) and (A5) from Appendix A, respectively, which were subsequently solved for the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ using the “FindRoot[]” function of Mathematica Version 13.2 [45]. These solved values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ were substituted into Equation (4) to obtain the quantile function, $p\left(v\right)$, of each normal-logistic MP distribution. This quantile function, $p\left(v\right)$, was transformed as $x=\widehat{\mu }+\widehat{\sigma }\left(p\left(v\right)-\mu \right)/\sigma$, where ($\widehat{\mu }, \widehat{\sigma }$) and ($\mu , \sigma$) are the values of the mean and standard deviation computed from the actual data and the normal-logistic MP distribution. This transformed quantile $x$ was used in Equation (5) to obtain the pdf of each MoMs-based normal-logistic MP distribution.

To superimpose the pdf of a Mo$L$Ms-based logistic-normal MP distribution, Equations (27) and (28) can be simultaneously solved for the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ after substituting the sample estimates of ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ of $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ on the right-hand sides of these equations. The solved values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ can be substituted into Equations (6) and (7) to obtain the pdf of Mo$L$Ms-based logistic-normal MP distribution. Alternatively, the closed-form formulae in Equations (29) and (30) can be used to obtain the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ associated with an Mo$L$Ms-based pdf of logistic-normal MP distribution. However, the values of ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$, to be substituted into the right-hand sides of Equations (29) and (30), must be taken from the boundary region of Figure 2B.

Likewise, to superimpose the pdf of an MoMs-based logistic-normal MP distribution, Equations (A9) and (A10) from Appendix B can be simultaneously solved for the values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ after substituting the sample estimates of ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$ of skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ on the right-hand sides of these equations. The solved values of ${\alpha }_{\mathcal{l}}$ and ${\alpha }_{\mathcal{r}}$ can be substituted into Equations (6) and (7) to obtain the pdf of the MoMs-based logistic-normal MP distribution.

Table 4. Chi-square goodness-of-fit statistics for the MoMs- and MoMs-based approximations of normal-logistic MP distributions over the histograms of the daily return rate (percent change) of Apple stock (AAPL) for the 10 years (n = 2517) shown in Figure 6A,B.

Percent	EF	OF (MoLMs)	OF (MoMs)	Class Interval (MoLMs)	Class Interval (MoMs)
10	251.7	261	257	(≤−1.5213)	(≤−1.5514)
20	251.7	237	208	(−1.5213, −0.8818)	(−1.5514, −0.9304)
30	251.7	248	255	(−0.8818, −0.4885)	(−0.9304, −0.5317)
40	251.7	250	251	(−0.4885, −0.1799)	(−0.5317, −0.2110)
50	251.7	273	278	(−0.1799, 0.0975)	(−0.2110, 0.0808)
60	251.7	258	277	(0.0975, 0.3743)	(0.0808, 0.3724)
70	251.7	222	233	(0.3743, 0.6783)	(0.3724, 0.6917)
80	251.7	258	262	(0.6783, 1.0562)	(0.6917, 1.0861)
90	251.7	260	257	(1.0562, 1.6493)	(1.0861, 1.6961)
100	251.7	250	239	(>1.6493)	(>1.6961)
				χ2 = 7.1756 p = 0.2079	χ2 = 15.5983 p = 0.0081

Note: EF = expected frequency; OF = observed frequency.

and

Table 5. Chi-square goodness-of-fit statistics for the Mo$L$Ms- and MoMs-based approximations of normal-logistic MP distributions over the histograms of the daily return rate (percent change) of Amazon stock (AMZN) for the 10 years (n = 2517) shown in Figure 7A,B.

Percent	EF	OF (MoLMs)	OF (MoMs)	$\mathbf{Class}\mathbf{Interval}(\mathbf{Mo}\mathit{L}$Ms)	Class Interval (MoMs)
10	251.7	262	252	(≤−1.8470)	(≤−1.8809)
20	251.7	229	223	(−1.8470, −1.1266)	(−1.8809, −1.1619)
30	251.7	262	263	(−1.1266, −0.6746)	(−1.1619, −0.6999)
40	251.7	252	257	(−0.6746, −0.3157)	(−0.6999, −0.3281)
50	251.7	260	271	(−0.3157, 0.0087)	(−0.3281, 0.0104)
60	251.7	260	272	(0.0087, 0.3327)	(0.0104, 0.3484)
70	251.7	232	239	(0.3327, 0.6882)	(0.3484, 0.7178)
80	251.7	261	257	(0.6882, 1.1291)	(0.7178, 1.1711)
90	251.7	244	242	(1.1291, 1.8178)	(1.1711, 1.8626)
100	251.7	255	241	(>1.8178)	(>1.8626)
				${\chi }^2=5.6023$ $p=0.3469$	${\chi }^2=8.5900$ $p=0.1266$

Note: EF = expected frequency; OF = observed frequency.

show chi-square goodness-of-fit statistics for comparing the Mo$L$Ms-based fits (panel A) of normal-logistic MP distributions superimposed over the histograms in Figure 6 and Figure 7, respectively, with MoMs-based fits (panel B).

An inspection of Figure 4, Figure 5, Figure 6 and Figure 7 and Table 4 and Table 5 illustrates the superiority of Mo$L$Ms-based fits of normal-logistic MP distributions over the histograms of real-world datasets. Specifically, the chi-square statistics, together with their corresponding $p$-values, indicate that Mo$L$Ms-based fits of normal-logistic MP distributions are substantially better than MoMs-based fits. The chi-square goodness of fit statistics, along with their $p$-values in Table 4 and Table 5, were computed using five degrees of freedom ($\mathrm{df}$), where $\mathrm{df}$ = 10 (number of class intervals)—4 (number of estimates)—1 (sample size).

5. Discussion and Conclusions

This paper introduced two families of mixture polynomial (MP) distributions, namely, the normal-logistic and logistic-normal MP distributions, via the method of $L$-moments (Mo$L$Ms) and the method of moments (MoMs). The systems of equations for each method (Mo$L$Ms and MoMs) were derived, and corresponding boundary graphs were plotted (Figure 2 and Figure A1 of Appendix C). Based on Figure 2A, the lower boundary point for the Mo$L$Ms-based normal-logistic MP distributions is $\left({\tau }_3\approx 0.0412,{\tau }_4\approx 0.1458\right)$, which is associated with ${\alpha }_{\mathcal{l}}={\alpha }_{\mathcal{r}}=0$, whereas the upper boundary points corresponding to the negative and positive axes of ${\tau }_3$ are $\left({\tau }_3\approx -0.7899,{\tau }_4\approx 0.5728\right)$ and $\left({\tau }_3\approx 0.8428,{\tau }_4\approx 0.6733\right)$, respectively. Based on Figure 2B, the lower boundary point for the Mo$L$Ms-based logistic-normal MP distributions is $\left({\tau }_3\approx -0.0412,{\tau }_4\approx 0.1458\right)$, whereas the upper boundary points corresponding to the negative and positive axes of ${\tau }_3$ are $\left({\tau }_3\approx -0.8428,{\tau }_4\approx 0.6733\right)$ and $\left({\tau }_3\approx 0.7899,{\tau }_4\approx 0.5728\right)$, respectively. Furthermore, based on Figure A1A of Appendix C, the lower boundary point for MoMs-based normal-logistic MP distributions is $\left({\gamma }_3\approx 0.3361,{\gamma }_4\approx 0.7685\right)$, whereas the upper boundary points corresponding to the negative and positive axes of ${\gamma }_3$ are $\left({\gamma }_3\approx -7.5780,{\gamma }_4\approx 97.5394\right)$ and $\left({\gamma }_3\approx 19.7792,{\gamma }_4\approx 1014.96\right)$, respectively. Based on Figure A1B of Appendix C, the lower boundary point for the MoMs-based logistic-normal MP distributions is $\left({\gamma }_3\approx -0.3361,{\gamma }_4\approx 0.7685\right)$, whereas the upper boundary points corresponding to the negative and positive axes of ${\gamma }_3$ are $\left({\gamma }_3\approx -19.7792,{\gamma }_4\approx 1014.96\right)$ and $\left({\gamma }_3\approx 7.5780,{\gamma }_4\approx 97.5394\right)$, respectively.

The advantage of the Mo$L$Ms-based procedure over the MoMs-based procedure can be expressed in the context of parameter estimation and data fitting. The Mo$L$Ms-based estimates of $L$-skewness and $L$-kurtosis are far less biased than the MoMs-based estimates of skewness and kurtosis when samples are drawn from distributions with more severe departures from normality [15,16,30,31,32,35]. The simulation results in Table 2 and Table 3 clearly indicate the superiority of the Mo$L$Ms-based estimates (${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$) of $L$-skewness $\left({\tau }_3\right),$ and $L$-kurtosis $\left({\tau }_4\right)$ over corresponding MoMs-based estimates (${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$) of skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ in terms of much smaller relative biases (RB%) and smaller standard errors (SEs) in the context of normal-logistic and logistic-normal MP distributions in Figure 3. For example, for a sample of size $n=25$, the estimates ${\widehat{\tau }}_3$ and ${\widehat{\tau }}_4$ for Distribution 3 were, on average, 88.69% and 94.85% of their respective parameters, whereas the estimates ${\widehat{\gamma }}_3$ and ${\widehat{\gamma }}_4$ were, on average, 36.19% and 9.84% of their respective parameters.

Another advantage of Mo$L$Ms-based estimates over their MoMs-based counterparts can be expressed by comparing their relative standard errors (RSEs), where $\mathrm{RSE}=100\times$ (St. Error/Estimate). From Table 2 and Table 3, it is evident that the estimates of ${\tau }_3$ and ${\tau }_4$ are more efficient, as their RSEs are considerably smaller than the RSEs associated with the MoMs-based estimates of ${\gamma }_3$ and ${\gamma }_4$. For example, in terms of Distribution 2 in Figure 3, an inspection of Table 2 and Table 3 (for $n=1000$) indicates that the RSE measures of $\mathrm{RSE} \left({\widehat{\tau }}_3\right)=0.10\%$ and $\mathrm{RSE} \left({\widehat{\tau }}_4\right)=0.02\%$ are considerably smaller than the RSE measures of $\mathrm{RSE} \left({\widehat{\gamma }}_3\right)=0.30\%$ and $\mathrm{RSE} \left({\widehat{\gamma }}_4\right)=0.52\%$. This comparison of RSEs demonstrates that the Mo$L$Ms-based estimates of $L$-skewness and $L$-kurtosis have higher precision than the MoMs-based estimates of skewness and kurtosis.

Another advantage of this study is that the proposed new family of normal-logistic and logistic-normal MP distributions provides researchers with a much wider selection of non-normal distributions that can be used in simulation studies. For example, Figure 2A,B show a much wider range of Mo$L$Ms-based $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ for the proposed normal-logistic and logistic-normal MP distributions than that for traditional power method distributions [15,35]. Likewise, Figure A1A,B of Appendix C indicate a much wider range of MoMs-based skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ for the normal-logistic and logistic-normal MP distributions than that for traditional power method distributions [4,10,15,16].

In addition, the proposed MP distributions provide much wider selections of $L$-skewness $\left({\tau }_3\right)$ and $L$-kurtosis $\left({\tau }_4\right)$ and skewness $\left({\gamma }_3\right)$ and kurtosis $\left({\gamma }_4\right)$ compared to the standard normal- and logistic- based double power method distributions [15] and uniform and triangular- based double power method distributions [16].

Furthermore, the Mo$L$Ms-based MP distributions are also superior to MoMs-based distributions in terms of their lower computational cost during parameter estimation. A simple inspection of Table 2 and Table 3 indicates that the algorithm for Mo$L$Ms-based Monte Carlo simulation results for each sample size under the same conditions (e.g., using the loop, number of replications, computation of bootstrap estimates with relevant 95% confidence intervals, and standard errors), takes a relatively shorter execution time than the corresponding MoMs-based algorithm. For example, for $n$ = 1000, the execution time of 83.6 min for the Mo$L$Ms-based algorithm was substantially lower than the execution time of 290.4 min for the MoMs-based algorithm.

One of the limitations of this study was that we did not consider multivariate aspects of normal-logistic and logistic-normal MP distributions via the multivariate measures of Mo$L$Ms-based $L$-skewness, $L$-kurtosis, and MoMs-based skewness and kurtosis [50,51,52,53]. In this context, we suggest that the development and evaluation of methodologies for modeling multivariate data through multivariate measures of Mo$L$Ms-based $L$-skewness, $L$-kurtosis, and MoMs-based skewness and kurtosis can be the subjects of future research projects.

In conclusion, the Mo$L$Ms-based families of normal-logistic and logistic-normal MP distributions are more attractive alternatives than MoMs-based families because of their capability of producing more precise estimates of the parameters and providing better approximations to the empirical distributions of real-world data. Finally, Mathematica [45] and R [37] algorithms are available from the first author to implement the Mo$L$Ms- and MoMs-based procedures.