On the Accuracy of the Sine Power Lomax Model for Data Fitting

: Every day, new data must be analysed as well as possible in all areas of applied science, which requires the development of attractive statistical models, that is to say adapted to the context, easy to use and efﬁcient. In this article, we innovate in this direction by proposing a new statistical model based on the functionalities of the sinusoidal transformation and power Lomax distribution. We thus introduce a new three-parameter survival distribution called sine power Lomax distribution . In a ﬁrst approach, we present it theoretically and provide some of its signiﬁcant properties. Then the practicality, utility and ﬂexibility of the sine power Lomax model are demonstrated through a comprehensive simulation study, and the analysis of nine real datasets mainly from medicine and engineering. Based on relevant goodness of ﬁt criteria, it is shown that the sine power Lomax model has a better ﬁt to some of the existing Lomax-like distributions.


Introduction
A large part of applied mathematics consists of defining one or more models of a mathematical nature, allowing a sufficiently general consideration of a given phenomenon. In a somewhat schematic way, we can distinguish two kinds of modelling: the deterministic modelling where random variations are not taken into account and stochastic modelling which takes into account these random variations (roughly speaking, 'stochastic' means to be or have a random variable). In the context of stochastic modelling, the random variations are often associated with an underlying probability distribution. Stochastic modelling can be divided into two sub-categories: probabilistic modelling and statistical modelling. The main objective of the probabilistic modelling is to provide a formal framework making it possible to describe the random variations discussed above, and to study the general properties of the phenomena which govern them. More applied, the statistical modelling essentially consists of defining suitable tools aiming to model the observed data taking into account their random nature. This theme is fully developed in [1,2], among others.
Recent developments in stochastic modelling have been driven by the rapid progress and accessibility of computing power. In particular, these have allowed direct applications of existing continuous distributions with some functional complexity for various statistical purposes. Also, these have accelerated the creation of new families of distributions presenting original and practical characteristics. In this regard, we may refer to [3] for a complete overview. Among the latest developments, the families defined by 'trigonometric transformations' of a given distribution have attracted much attention due to their applicability and working capacity in many practical situations. The pioneering works of [4][5][6][7] have focused on the sinusoidal transformation leading to the so-called sine generated (S-G or Sin-G) family. The following equations are the generic definitions of the associated cumulative distribution function (cdf) and probability density function (pdf), respectively: and f S (x; ζ) = π 2 g(x; ζ) cos π 2 G(x; ζ) , x ∈ R. (2) In these equations, G(x; ζ) and g(x; ζ) are the cdf and pdf of a certain continuous distribution with parameter(s) vector denoted by ζ, respectively. They are related to a reference distribution chosen a priori by the practitioner, depending on the context of the study. It is now established that the S-G family (i) offers an attractive alternative to the reference family; one can show that G(x; ζ) ≤ F S (x; ζ) for any x ∈ R, (ii) is of acceptable mathematical complexity without introducing new parameters, and (iii) has the ability to provide flexible statistical models to accommodate data of varying nature. To illustrate these items, in Reference [4], the exponential distribution is used as a reference to define the SE model. It turns out to be well suited to analyse the important bladder cancer patients dataset of [8]. In another study, the inverse Weibull (IW) distribution developed by [9] was considered to be the reference distribution; the sine IW (SIW) model was introduced by [6]. By analyzing the famous guinea pigs dataset by [10], the SIW model is proven to perform better compared to serious and comparable competing models. An open-source R package on the SIW model is developed in [11], facilitating the use of the model beyond these basic purposes. These works inspired the construction of other trigonometric families of distributions, such as the CS-G family by [12], C-G family by [7], TransSC-G family by [13], NS-G family by [14], STL-G family by [15] and SKum-G family by [16].
In this paper, we contribute to the success of the S-G family by applying it to a specific three-parameter survival distribution: the power Lomax (PL) distribution proposed by [17]. We thus introduce the sine PL (SPL) distribution and model. Thus, a retrospective on the PL distribution is necessary to understand the proposed methodology. First, the Lomax distribution was introduced by [18]. It can be presented as a manageable two-parameter heavy-tailed survival distribution with a tuning polynomial decay and also, as a derivation of the Pareto distribution as described in [19] (page 573). It is governed by the cdf and pdf defined by and g L (x; ξ) = αλ(1 + λx) −(α+1) , x > 0, (4) respectively, with G L (x; ξ) = g L (x; ξ) = 0 for x ≤ 0, where ξ = (α, λ), α is a shape parameter and λ is a scale parameter, all the parameters taking strictly positive values. It finds numerous applications in reliability engineering and life testing. The theory, inference and applications of the Lomax distribution have been the subjects of the following inevitable references: [20][21][22][23][24][25][26]. The PL distribution proposed by [17] is obtained by making use of the power transformation to the Lomax distribution, aiming to increase its capabilities on several functional aspects. It corresponds to the distribution of the random variable X = Y 1/β , where Y is a random variable with the Lomax distribution and β > 0. Consequently, based on (3) and (4), the PL distribution is defined by the cdf and pdf defined by and respectively, with G PL (x; ζ) = g PL (x; ζ) = 0 for x ≤ 0, where ζ = (α, β, λ), α is a shape parameter, and β and λ are scale parameters, all the parameters taking strictly positive values. Contrary to the Lomax distribution, it is established in [17] that the PL distribution adapts to both inverted bathtub and decreasing hazard rates. The practical gain is particularly impressive; the PL model is better than ten competing models for analyzing the bladder cancer patients dataset of [8], all based on the Lomax model. For the sake of optimality, some motivated distributions extending or generalizing the PL distribution was introduced, including the type II Topp-Leone PL (TIITLPL) distribution by [27], type I half logistic PL distribution by [28], inverse PL distribution by [29], Marshall-Olkin PL distribution by [30], exponentiated PL distribution by [31] and Kumaraswamy generalized PL distribution (KPL) by [32]. The main strategy of these proposed distributions is to add more parameters to the PL distribution based on exponentiated, transmuted or truncated schemes. Basically, these schemes give better results but add more parameters to the reference distribution; the problem of manipulating all these parameters simultaneously can present a certain difficulty from the modelling point of view. Thus, the immediate motivation of the SPL model is to use the S-G scheme to improve the efficiency of the PL model with the existing parameters. Deeper motivations come after further investigation which is detailed in the next study. To summarize, the functionality and flexibility of the SPL model are particularly attractive for data fitting. Indeed, the corresponding pdf has different kinds of curves such as uni-modal, symmetrical, asymmetrical on right and left, reversed J-shaped curves. Also, the model exhibits decreasing and increasing, inverted bathtub and reversed-J hazard rates. These properties give the SPL model a constant consistency in the precision of the fits unlike many other comparable models. This statement is illustrated in the practical environment by considering nine published datasets mainly from medicine and engineering, and four competing models derived from the Lomax distribution. We organize the rest of the paper as follows. Section 2 is devoted to the definition, characteristics and main properties of the SPL distribution. The parametric estimation related to the SPL model is discussed and illustrated by a comprehensive simulation study in Section 3. Concrete applications to datasets are provided in Section 4. Finally, conclusions are stated in Section 5.

Function Anlysis
Here, some mathematics of the SPL distribution are presented. First, by considering (5) and (6) in (1) and (2), we obtain the main distributional functions of the SPL distribution; the corresponding cdf and pdf are given as where ζ = (α, β, λ), and with F SPL (x; ζ) = f SPL (x; ζ) = 0 for x ≤ 0. We recall that ζ = (α, β, λ), α is a shape parameter, and β and λ are scale parameters, all the parameters taking strictly positive values. Considering different values of the parameters, variant forms of the pdf can be obtained. More specifically, by differentiating (8), it can be readily verified that f SPL (x; ζ) is decreasing for β ≤ 1 and unimodal for β > 1. The more representative of them are shown in Figure 1.  From Figure 1, we observe that the pdf of the SPL distribution can be decreasing or unimodal, with a very versatile asymmetry in all the directions. This versatility is an attractive point for the use of the SPL model in data fitting.
We complete this functional study by discussing the hazard rate function (hrf). First, in full generality, the hrf measures the tendency of an item to fail or die depending on the age reached. Therefore, it plays a key role in the classification of survival distributions. Basically, the shapes of hazard rates are either monotonic (increasing or decreasing) or non-monotonic (bathtub or inverted bathtub). The hrf of the SPL distribution is given by and h SPL (x; ζ) = 0 for x ≤ 0. Upon differentiation of (9), it can be seen that h SPL (x; ζ) is increasing for β ≥ 1 and α ≤ 1. It is also conjectured that h SPL (x; ζ) is decreasing for β ≤ 1 and α ≥ 1, and unimodal for β ≥ 1 and α ≥ 1. The graphical study in Figure 2 supports these claims. Figure 2 emphasizes the fact that the proposed SPL distribution possesses increasing and decreasing, and also upside down bathtub hazard rates.
Another important function of the SPL distribution is the quantile function (qf). It is defined as the inverse function of the corresponding cdf. Thus, based on (7), it is specified by As the cdf, the qf determined the SPL distribution. Classically, we can use it for determining the median, as well as the lower and upper quartiles. The qf can also be used to generate values from a random variable with the SPL distribution. Further detail on the quantilebased reliability analysis can be found in [33].

Moment Analysis
We now conduct a moment analysis. The following result gives a series expansion for the (crude) moments of a random variable with the SPL distribution. Proposition 1. Let r ≥ 1 be an integer and X be a random variable with the SPL distribution. Then, for r < 2αβ, the r-th moment of X exists and can be expanded as where E denotes the mathematical expectation and B(a, b) refers to the standard beta function given since f SPL (x; ζ) = 0 for x ≤ 0. Let us study the mathematical existence of this integral by the Riemann integrability criterion. When x → 0, we have x r f SPL (x; ζ) ∼ (π/2)αβλx r+β−1 , which is integrable over (0, δ) with δ > 0 since β > 0. For the case x → +∞, we have x r f SPL (x; ζ) ∼ (π 2 /4)αβλ −2α x r−2αβ−1 , which is integrable over (δ, +∞) with δ > 0 if and only if r < 2αβ. The desired condition is obtained.
Let us now investigate a linear representation of the cdf expressed in (7), from which we will deduce a series expansion for the pdf as given by (8). By using the Taylor series expansion of the cosine function, for x > 0, we get By applying a first order differentiation with respect to x, the following series expansion of the pdf comes: Please note that we ignored the term in k = 0 since the corresponding term disappears. From (11) and (12), by integrating f SPL (x; ζ) with respect to x, swapping the symbols and ∑ by the dominated convergence theorem, and applying the change of variables y = λx β , we obtain This completes the proof of Proposition 1.
A computational remark is that, for K large enough, a precise approximation of µ r is obtained as Diverse moment measure can be defined from Proposition 1. Here, we restrict our attention on the variance of X basically defined by The first four moments and variance of X for different parameter values are indicated in Table 1 provided αβ > 2.
From Table 1, we see the numerical versatility of the moment measures considered, varying from small to large values; central and dispersion indicators may be negligible or substantial. This confirms the claim about the overall flexibility of the SPL distribution.
Based on similar developments employed in the proof Proposition 1, it is possible to express various series expansions of moment-type functions. Here, we complete our moment analysis by investigating the incomplete moments of the SPL distribution which are involved in the definition of many applied measures and indicators.  Proposition 2. Let r ≥ 1 be an integer, t ≥ 0 and X be a random variable with the SPL distribution. Then, the r-th incomplete moment of X with the truncated value t exists and can be expanded as where I denotes the indicator function and B u (a, b) refers to the truncated beta function given as Proof. First, we have The rest of the development follows the lines of the proof of Proposition 1; From (12) and (13), by integrating f SPL (x; ζ) with respect to x, swapping the symbols and ∑ owing to the dominated convergence theorem, and applying the change of variables y = λx β , we obtain Next, with the change of variables z = y/(1 + y), we get This completes the proof of Proposition 2.
Based on the incomplete moments, we can define the mean residual life function, mean waiting time, mean deviation about the mean, and various inequalities measures (Lorenz curve, Gini index, Bonferroni curve, Atkinson index, Zenga index, Pietra index, etc.). In this regard, we may refer the reader to the book of [34]. However, these measures are beyond the applied line of this paper.

Inference of the SPL Model
This section is devoted to the inferential treatment of the SPL distribution for the perspectives of statistical modelling. The maximum likelihood method, as described in full generality in [35], is employed. A mathematical description of this method in the context of the SPL distribution is provided below.
First, let x 1 , x 2 , . . . x n be observations drawn from a random variable X with the SPL distribution. Then the corresponding likelihood function and log-likelihood function are respectively. Then, the maximum likelihood estimates (MLEs) are defined bŷ The components ofζ, sayα,β andλ, form the MLEs of α, β and λ, respectively. The MLEs can be formalized through non-linear equations involving the partial differentiation of the log-likelihood function with respect to the parameters α, β and λ. These partial derivatives are given as Simple analytical expressions forα,β andλ remain impossible, but practice only requires numerical evaluations of them. These numerical values can be easily obtained using specific tools in statistical software as the R software (see [36]). Also, the well-established theory on MLEs ensures that the random version ofζ is asymptotically three-dimensional normal with mean vector ζ and variance-covariance matrix V = {−∇ 2 ζ log L SPL (ζ) | ζ=ζ } −1 , where ∇ ξ denotes the gradient according to ξ.
In particular, the (asymptotic) estimated standard error (SE) ofα is obtained by taking the square-root of the first diagonal component of V, and we can proceed in a similar way to obtain the SEs of the two other parameters. The asymptotic normal distribution is at the basis of diverse statistical tests or confidence intervals. Also, based onξ, f SPL (x;ζ) is the estimated pdf of f SPL (x; ζ). This estimated pdf plays a central role in fitting the normalized histogram of the data, as discussed in the next section on applications.
We now evaluate the accuracy of the MLEs of the SPL model. The data are artificial; they are generated by using the qf as defined by (10) through the inverse transform sampling technique. We conduct 1000 Monte Carlo simulations for each sample size n with n = 50, 100, 200, 300 and 500 to the following different sets of parameters: Set I = (0.5, 1.5, 0.5), Set II = (1.25, 1.25, 0.5), Set III = (1.5, 1.5, 0.5) and Set IV = (1.5, 2.5, 0.5) with reference to the usual order (α, β, λ). In each case, the standard mean MLE (MMLE), bias (Bias) and mean squared error (MSE) are calculated. The results are reported in Table 2.
From Table 2, we see that the maximum likelihood method performs quite well to estimate the parameters for the considered sample sizes. Indeed, as the sample size increases, the biases and the SEs of the MLEs decrease as expected. Also, we observe that when the sample size increases, the MMLEs are closed to the true parameter values.
We now present some useful measures of adequacy by using the notation of the SPL distribution for convenience. Let x 1 , x 2 , . . . , x n be the data and x (1) , x (2) , . . . , x (n) be their ordered values. First, we consider the Cramér-von Mises (W*), Anderson Darling (A*) and Kolmogorov-Smirnov (K-S) statistics (D n ) defined by respectively, where ζ denotes the parameters of the distribution, i.e., ζ = (α, β, λ) for the SPL distribution andζ for its MLE. The p-Value of the K-S test related to D n is also considered. The above definitions can be adapted for any other distribution by changing the definition of the cdf and the notation of the parameters. These adequacy measures are widely used to find out which model is best suited. The model with the minimum value for W* or A*, and maximum value for p-Value, is chosen as the best one that is in adequacy to the data. Also, we consider the Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC) and Hannan-Quinn information criterion (HQIC), defined in the context of the SPL distribution as respectively, where k is the number of parameters so k = 3 for the SPL distribution. As commonly accepted, the model with the minimum value for AIC or CAIC or BIC or HQIC is chosen as the best one that fits the data. Further informations on the use and interpretation of the measures W*, A*, AIC, CAIC, BIC and HQIC can be found in [37].
In this study, we aim to compare the SPL model related to the SPL distribution with the useful and competitive Lomax-type model listed in Table 3.  Table 3. Competitive models of the SPL model.

Models Abbreviations
Cdfs (x > 0) References [18] We can notice that the Lomax model is nested in the TLGL, EL and PL models. The proposed SPL model is completely different in this sense. In addition, conceptually, the TLGL and EL models are closed; they coincide with a reparametrization of the parameters.

Applications of the SPL Model
Based on the above methodology, we apply the SPL model on nine datasets. They differ mainly in size, characteristics or background, but all of them are of modern interest to their respective fields. For each dataset, we proceed as follows: 1.
We briefly present the data, with reference(s).

2.
We provide a table that summarizes the main statistical characteristics of the data.

3.
We assess the quality of the fit measures of the models considered and organize them in a table in order of the model performance.

4.
As complementary work, we indicate the MLES of the model parameters as well as the related SEs.

5.
We end with a visual approach by plotting the histogram of the data and the fitted pdfs, and, in another graph, the probability-probability (PP) plot for the SPL model only.

Data set 1:
We consider a real dataset on the remission times (in months) of a random sample of 128 bladder cancer patients. This dataset is given by Lee  A summary measure of descriptive statistics of dataset 1 is provided in Table 4. We see in Table 4 that the data are right skewed and highly leptokurtic with high variance. With respect to model adequacy, the measures W*, A*, D n , p-Value, AIC, CAIC, BIC and HQIC are reported in Table 5. From Table 5, we observe that the SPL model possesses the lowest values for W*, A*, D n , AIC, CAIC, BIC and HQIC, and the highest value for p-Value compared to the other models. It can be considered the best. The second best model is the PL model.
Please note that for this dataset, the results for the TLGL and EL models are almost identical due to their similar nature, but small numerical variations are observed without rounding.
For additional information, the MLEs of the model parameters as well as their SEs are reported in Table 6. From Table 6, among other, we see that the parameters α, β and λ of the SPL model have been estimated byα = 1.0216200,β = 1.3956063 andλ = 0.0371991, respectively, with quite small SEs. Figure 3 shows two graphics: the histogram of the data fitted by the estimated pdfs, and the PP plot for the SPL model only.
In Figure 3, we observe that the empirical objects are almost perfectly adjusted by the estimated objects. In particular, in the PP plot, the black line is almost confused with the estimated red line related to the SPL model.

Data set 2:
The considered data represent the failure times of the mechanical components of the aircraft windshield. They are taken from [41]. They were recently reviewed by [42].  A summary of descriptive statistics for dataset 2 is provided in Table 7. Based on the information of Table 7, we can say that the data are approximately symmetric and platykurtic, with little dispersion. One more point, we observe that the data have a negative kurtosis value which means that the underlying distributions should have lighter tails.

Fitting of SPL and Lomax distributions
The statistical measures considered for the comparison of the models are given in Table 8. From Table 8, the values of the model adequacy measures and goodness of fit test are clearly in favor of the SPL model. The second best model is the PL model.
The MLEs of the parameters of the SPL model and other models with their SEs are reported in Table 9. In addition, the estimated pdfs over the histogram and PP plot of the SPL model are displayed in Figure 4.  From Figure 4, it is obvious that the light tails of the SPL model are instrumental in having a better fit. In addition, the PP plot underlines this power of adaptation; the black line is almost confused with the estimated red line.

Data set 3:
We now consider a dataset containing 27 observations of time of successive failures of the air conditioning system of jets in a fleet of Boeing 720 as reported in Proschan [43]. Recently, this data was studied by [44] and the data are: 1, 4, 11, 16, 18, 18, 18, 24, 31, 39, 46, 51, 54, 63, 68, 77, 80, 82, 97, 106, 111, 141, 142, 163, 191, 206, 216. Some descriptive measures of dataset 3 are provided in Table 10. From Table 10, we see that the data are right skewed and platykurtic with a high variance. Table 11 indicates the values of the statistical measures considered to compare the models. The analysis of Table 11 ensures that the SPL model is the best with, in particular, p-Value = 0.9399. The second best model is the EL model.
The MLEs of the model parameters as well as their SEs are reported in Table 12. The estimated pdfs over the histogram and the PP plot of the SPL model are shown in Figure 5.

Fitting of SPL and Lomax distributions
Air condition data In Figure 5, the fitted power of the SPL model is flagrant; the corresponding estimated pdf has captured the decreasing roundness shape of the histogram, contrary to the other estimated pdfs. In addition, the red line of the PP plot is generally close to the black line.

Data set 4:
The data represent 69 strength measures for single carbon fibers (and impregnated 1000-carbon fiber tows). They are given by [45] A statistical description of dataset 4 is given in Table 13.  Table 13 shows that the data are almost symmetric and leptokurtic, with a low variance. The fitting performance of the considered models are investigated numerically in Table 14. From Table 14, we see that the SPL model is more relevant for the fit of the dataset than the other models. Indeed, it has the lowest value for all the statistical measures considered, except for the p-Value where it has the highest value. The second best model is the PL model. Table 15 contains the MLEs of the considered models along with their SEs. The fitted histogram of the data is shown in Figure 6, along with the PP plot of the SPL model. From Figure 6, the curve of the estimated pdf of the SPL model is close to the shape of the histogram and has captured the 'elbow phenomena' in the right. The corresponding PP plot is also convincing.

Data set 5:
We now consider a dataset containing 100 observations on breaking stress of carbon fibers (in Gba). It was studied by [46]  A summary of descriptive statistics for these data is presented in Table 16. From Table 16, we see that the data are approximately symmetric and platykurtic with a low variability.
The statistical measures considered for the comparison of the models are given in Table 17. In our framework, Table 17 attests to the superior adequacy of the SPL model. The MLEs of the model parameters and their SEs are reported in Table 18. A visual work is performed in Figure 7, showing the histogram and PP plot of the SPL model. In Figure 7, the flexible skewness of the SPL model is clearly the key, allowing the symmetrical nature of the data to be fully captured. The observation of the PP plot confirm the high quality of the fit of the SPL model.
The goodness of fit measures of the considered models are calculated and collected in Table 20. From Table 20, the SPL model shows the best results, far superior to those of the competition. The second best model is the EL model.
The MLEs of the model parameters along with their SEs are reported in Table 21.  Figure 8 illustrates the nice fit of the SPL model by two different graphical approaches.

Fitting of SPL and Lomax distributions
Diaster data From Figure 8, we observe that the adjustment of the SPL model proposes a slope more adapted to the form of the histogram of the data, compared to those of the other models. A nice result in the PP plot is also observed.

Data set 7:
The data are measures of life of Kevlar 373/epoxy fatigue fractures that are subjected to constant pressure (at the 90% stress level) until all has failed. These data was recently studied by [13] Table 22 presents a brief summary of descriptive statistics for these data. From Table 22, it can be deduced that the data are right skewed and leptokurtic, with a low variability.
According to Table 23, for the purpose of optimal data fit, the SPL model is more pertinent than the other models. The second best model is the PL model.  The histogram and PP plot of the data with the model fits are shown in Figure 9. From Figure 9, in the fitting exercise, we see that the SPL model is slightly better than the competing models. A favorable PP plot is also observed.

Fitting of SPL and Lomax distributions
Data set 8: Data on service times for a particular model windshield are now considered. They are given from [41]. The unit for measurement is 1000 h and the data are: 0.046,  We see in Table 25 that the data are right skewed and platykurtic, with a moderate variability. Table 26 indicates that the SPL model is the most appropriate fitted model. The second best model is the PL model. Some additional elements are now given. The MLEs of the models along with their SEs are shown in Table 27. We visually see the adjustability of the SPL model in Figure 10.

Fitting of SPL and Lomax distributions
Service times data From Figure 10, it is evident that the histogram of the data is better fitted by the estimated pdf of the SPL model. The red line of the PP plot is relatively close to the black line, confirming the SPL model fitting power.
Data set 9: Data relating to the strengths of 1.5 cm glass fibres which was obtained by workers at the UK National Physical Laboratory are now used. They were previously analysed by [48]. A first statistical approach of these data is proposed in Table 28. From Table 28, we observe that the data are left skewed and platykurtic, with almost negligible dispersion.
The goodness of fit measures of the considered models are calculated and collected in Table 29. According to Table 29, we assert that the SPL model has a better goodness of fit than the other models. The second best model is the PL model.
The MLEs of the model parameters and their SEs are shown in Table 30. Estimated pdfs over the histogram of the data and PP plot of the SPL model are shown in Figure 11. From Figure 11, unsurprisingly in view of Table 30, the SPL model shows the best fit curve of the histogram. A nice fit of the SPL model is also validated by the PP plot.

Conclusions
The main contribution of the article is to propose a new efficient statistical modelling strategy through a flexible trigonometric extension of the famous power Lomax model. In this regard, we use the functionalities of the sine generalized (S-G) family of distributions and introduce the sine power Lomax (SPL) distribution. We exhibited some of its interesting characteristics, with an emphasis on the modelling ability of the corresponding probability density and hazard rate functions, and discussed the moments and incomplete moments. Simulations and applications illustrate the usefulness of the considered SPL model. In particular, we carried out nine practical datasets for the evaluation of the SPL model with the main existing models derived from the Lomax model. Whenever the data is symmetric or skewed, the SPL model performs better than the competing models considered. Thus, the results obtained are quite satisfactory, showing that the SPL model can be used fairly to efficiently analyse a large panel of datasets.
Author Contributions: V.B.V.N., R.V.V. and C.C. have contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.