Alpha Power Type II-G Family: Adding a Power Parameter of Distributions

Modern classes of distributions stemming from a straightforward, innovative, and well-founded transformation of the baseline distribution are not yet common. Within this manuscript, we put forth a potential candidate in which we present a transformation that is contingent upon the Alpha Power of baseline distribution generated APII-G family, as specified by the subsequent CDF. The general formula for CDF of the new family (APII-G) is defined by

$F(x, \alpha)=\frac{(1+G(x))^\alpha-1}{2^\alpha-1}, \alpha>0$            (2)

where, the function $G(x)$ denotes to the CDF of the baseline distributions, potentially affected by the parameter vector, designated as R . In the field of mathematical functions, it is noted that the function $G(x)$ represents the cumulative distribution function obtained from existing distributions. This specific function is mathematically expressed by Eq. (2), which not only indicates the cumulative distribution function related to APII-G but also provides insight into its properties. In cases where $x_1>x_2$ is true, it can be inferred that $G\left(x_1\right)>$ $G\left(x_2\right)$, so that $F\left(x_1 ; \varphi, \alpha\right)>F\left(x_2 ; \varphi, \alpha\right)$ is a direct result, thus establishing an important relationship within the mathematical framework. Therefore, it can be concluded that the function $F(x ; \varphi, \alpha)$ fulfills the properties of distribution functions of being monotonically increasing. Also, the limit of $F(x: \varphi, \alpha)$ satisfied the following:

$\begin{gathered}\lim _{x \rightarrow-\infty} F(x ; \varphi, \alpha)=\lim _{G(x) \rightarrow 0} F(x ; \varphi, \alpha)=0 \\ \lim _{x \rightarrow \infty} F(x ; \varphi, \alpha)=\lim _{G(x) \rightarrow 1} F(x ; \varphi, \alpha)=1\end{gathered}$

To find the density function, we derive the Eq. (1) and we obtain:

$f(x, \alpha)=\frac{\alpha(1+G(x))^{\alpha-1} g(x)}{2^\alpha-1}, \alpha>0$          (3)

Which fulfills the basic condition for the probability density function as follows:

$\begin{gathered}\int_{-\infty}^{\infty} f(x, \alpha) d x=\int_{-\infty}^{\infty} \frac{\alpha(1+G(x))^{\alpha-1} g(x)}{2^\alpha-1} d x =\frac{1}{2^\alpha-1}\left[(2)^\alpha-1\right]=1\end{gathered}$           (4)

When the well definition of the general form of the transformation has been established, as indicated by Eq. (2), which serves as a representation of the cumulative distribution function, it is important to highlight that through this formulation, the corresponding density function can be derived. The application of this constant transformation is then directed towards two specific distributions, the exponential and the Weibull, by replacing G(x) in the transformation with the cumulative distribution function related to these respective distributions. This substitution is the basis for creating the new distributions for this family because it allows a comprehensive examination of the statistical properties and functions that define the new transformed distributions. When the value of new alpha parameter in the new distributions is equal to 1, it will remain the same as the original distribution that was used.

In this section, we will apply the new APII-G family with the two-parameter Weibull distribution, where we will obtain a new distribution with three parameters which is called APIIW distribution, by replacing G(x) in Eq. (2) with the CDF of the Weibull distribution, as follows:

$F(x ; \lambda, \alpha, \kappa)=\frac{\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^\alpha-1}{2^\alpha-1},(\lambda, \alpha, \kappa, x)>0$        (31)

And the pdf of the APIIW is defined as the following formula:

$f(x ; \lambda, \alpha, \kappa)=\frac{\alpha k\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{x}{\lambda}\right)^k} x^{k-1}}{\lambda^k\left(2^\alpha-1\right)} (x ; \lambda, \alpha, \kappa)>0$        (32)

Some plots of the CDF f(x;λ,α,κ) and PDF f(x;λ,α,κ) of the APIIW model, which is plotted for some different value of the parameters α, κ and λ are sketched in Figure 4 and Figure 5. respectively.

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Figure 4. The CDF of APIIW

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Figure 5. The pdf of APIIW

4.1 The statistical properties of APIIW distribution

We present the functions of reliability, reversed hazard, hazard rate, and the cumulative of the hazard rate of APIIW distribution [17].

The survival function of random variable $X \sim \operatorname{APIIW}(\alpha, \lambda, k)$ is defined as follows:

$\bar{F}(x ; \alpha, \lambda, k)=1-F(x ; \alpha, \lambda, k)=\frac{2^\alpha-\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^\alpha}{2^\alpha-1}$       (33)

The function of reverse hazard $r(x: \alpha, \lambda)$ to the APIIW distribution is defined by the following:

$r(x: \alpha, \lambda, k)=\frac{f(x: \alpha, \lambda, k)}{F(x: \alpha, \lambda, k)}=\frac{\alpha k\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{x}{\lambda}\right)^k} x^{k-1}}{\lambda^k\left(\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^\alpha-1\right)}$          (34)

The function of hazard $h(x: \alpha, \lambda, k)$ of the APIIW distribution is defined by the following:

$h(x: \alpha, \lambda, k)=\frac{f(x: \alpha, \lambda, k)}{\bar{F}(x: \alpha, \lambda, k)}=\frac{\alpha k\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{x}{\lambda}\right)^k} x^{k-1}}{\lambda^k\left(2^\alpha-\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^\alpha\right)}$         (35)

And the function of cumulative hazard $H(x: \alpha, \lambda, k)$ of the APIIW distribution is defined by the following:

$\begin{aligned} & H(x: \alpha, \lambda, k)=-\operatorname{Ln}[1-F(x: \alpha, \lambda, k)] =-\operatorname{Ln}\left[\frac{2^\alpha-\left(2-e^{\left.-\left(\frac{x}{\lambda}\right)^k\right)^\alpha}\right.}{2^\alpha-1}\right]\end{aligned}$       (36)

4.2 Moments of APIIW distribution

We will define the $r^{t h}$ moment (at the point of origin) for APIIW distribution by the following theorem.

Theorem:

If X is a variable to the APIIW distribution, then moment of the order $r$ (about the point of origin) is defined by the following.

$\grave{\mu}_r=\frac{\alpha \lambda^r}{\left(2^\alpha-1\right)} \sum_{n=0}^{\infty} \frac{(-1)^n(\alpha-1)!2^{\alpha-(n+1)}}{(\alpha-1-\mathrm{n})!n!} \frac{\Gamma\left(\frac{r+k}{k}\right)}{(n+1)^{\frac{r+k}{k}}}$      (37)

Proof:

By definition the moment of the order $r$ (about the point of origin) $\dot{\mu}_r$ to the variable $X$ of APIIW distribution.

$\begin{gathered}\grave{\mu}_r=E\left(x^r\right)=\int_0^{\infty} x^r f(x: \alpha, \lambda, k) d x \\ =\int_0^{\infty} \frac{\alpha k\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{x}{\lambda}\right)^k} x^{r+k-1}}{\lambda^k\left(2^\alpha-1\right)} d x \\ =\frac{\alpha k}{\lambda^k\left(2^\alpha-1\right)} \int_0^{\infty}\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{x}{\lambda}\right)^k} x^{r+k-1} d x \\ x, \alpha, \lambda, k>0, r=1,2,3, \ldots\end{gathered}$          (38)

Let $e^{-\left(\frac{x}{\lambda}\right)^k}=y$, if $x \rightarrow o$ then $y \rightarrow 1$, and if $x \rightarrow \infty$ then $y \rightarrow 0, d x=\frac{-\lambda}{k y}(-\ln y)^{\frac{1}{k}-1} d y$. So

$\grave{\mu}_r=\frac{\alpha \lambda^r}{\left(2^\alpha-1\right)} \int_0^1(-\ln y)^{\frac{r+k}{k}-1}(2-y)^{\alpha-1} d y$          (39)

Now, Taking the Maclaurin series of formula $(2-y)^{\alpha-1}$, we obtain the following series:

$(2-y)^{\alpha-1}=\sum_{n=0}^{\infty} \frac{(-1)^n(\alpha-1)!2^{\alpha-(n+1)}}{(\alpha-1-\mathrm{n})!n!} y^n$        (40)

We replace $(2-y)^{\alpha-1}$ in Eq. (39) with Eq. (40)

$\begin{gathered}\grave{\mu}_r=\frac{\alpha \lambda^r}{\left(2^\alpha-1\right)} \sum_{n=0}^{\infty} \frac{(-1)^n(\alpha-1)!2^{\alpha-(n+1)}}{(\alpha-1-\mathrm{n})!n!} \\ \int_0^1(-\ln y)^{\frac{r+k}{k}-1} y^n d y\end{gathered}$          (41)

Using the infinitive form in reference [11] and comparing it with Eq. (40), the integration result becomes as follows:

$\grave{\mu}_r=\frac{\alpha \lambda^r(\alpha-1)!}{\left(2^\alpha-1\right)} \sum_{n=0}^{\infty} \frac{(-1)^n 2^{\alpha-(n+1)}}{(\alpha-1-n)!n!} \frac{\Gamma\left(\frac{r+k}{k}\right)}{(n+1)^{\frac{r+k}{k}}}$           (42)

Now, by using theorem (4.2.1) we calculate the mean and variance as follows.

$E(x)=\frac{\alpha \lambda}{\left(2^\alpha-1\right)} \sum_{n=0}^{\infty} \frac{\Gamma\left(\frac{1+k}{k}\right)(-1)^n(\alpha-1)!2^{\alpha-(n+1)}}{(\alpha-1-\mathrm{n})!n!(n+1)^{\frac{1+k}{k}}}$         (43)

$\begin{gathered}\operatorname{Var}(x)= \\ \frac{\alpha \lambda^2}{\left(2^\alpha-1\right)} \sum_{n=0}^{\infty} \frac{(-1)^n(\alpha-1)!2^{\alpha-(n+1)}}{(\alpha-1-\mathrm{n})!n!} \frac{\Gamma\left(\frac{2+k}{k}\right)}{(n+1)^{\frac{2+k}{k}}} \\ -\frac{\alpha^2 \lambda^2}{\left(2^\alpha-1\right)^2}\left(\sum_{n=0}^{\infty} \frac{(-1)^n(\alpha-1)!2^{\alpha-(n+1)}}{(\alpha-1-\mathrm{n})!n!} \frac{\Gamma\left(\frac{1+k}{k}\right)}{(n+1)^{\frac{1+k}{k}}}\right)^2\end{gathered}$        (44)

The moment generating function mgf, denoted by the symbol $M_x(t)$, of APIIW distribution can be find by the following:

$M_x(t)=E\left(e^{t x}\right)=\int_0^{\infty} e^{t x} f(x: \alpha, \lambda, k)$        (45)

where, $f(x: \alpha, \lambda, k) d x$ is pdf of APIIW distribution. By Taylor series for $e^{t x}$ yields, which is $e^{t x}=\sum_{s=0}^{\infty} \frac{t^s x^s}{s!}$.

$M_x(t)=\sum_{s=0}^{\infty} \frac{t^s}{s!} \int_0^{\infty} x^s f(x: \alpha, \lambda, k) d x=\sum_{s=0}^{\infty} \frac{t^s}{s!} \dot{\mu}_s$         (46)

And by substituting Eq. (42) into Eq. (46) then

$\begin{gathered}M_x(t)=\frac{\alpha}{\left(2^\alpha-1\right)} \\ \sum_{s=0}^{\infty} \sum_{n=0}^{\infty} \frac{\lambda^s t^s(-1)^n(\alpha-1)!2^{\alpha-(n+1)} \Gamma\left(\frac{s+k}{k}\right)}{s!(\alpha-1-\mathrm{n})!n!(n+1)^{\frac{s+k}{k}}}\end{gathered}$        (47)

4.3 Maximum Likelihood Estimation of APIIW

In this subsection, we find MLE of the two parameters $\alpha, \lambda$ and $k$. If $x_1, x_2, x_3 \ldots, x_n$ denote a sample of APIIW, the function of likelihood is defined by references [18, 19].

$\begin{gathered}L(\alpha, \lambda, k ; x)=\prod_{i=1}^n f\left(x_i: \alpha, \lambda, k\right) \\ =\prod_{i=1}^n \frac{\alpha k\left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{x_i}{\lambda}\right)^k} x^{k-1}}{\lambda^k\left(2^\alpha-1\right)} \\ =\frac{\alpha^n k^n \prod_{=1}^n\left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)^{\alpha-1} \prod_{i=1}^n x_i^{k-1} e^{-\sum_{i=1}^n\left(\frac{x_i}{\lambda}\right)^k}}{\lambda^{n k}\left(2^\alpha-1\right)^n}\end{gathered}$        (48)

Then the logarithm of the likelihood in Eq. (48) will have the following:

$\begin{gathered}\ln L(\alpha, \lambda, k ; x)=\mathrm{n} \ln \alpha+\mathrm{n} \ln k-n \ln \left(2^\alpha-1\right) \\ -\mathrm{n} \operatorname{k} \ln \lambda-\sum_{i=1}^n\left(\frac{x_i}{\lambda}\right)^k+(\mathrm{k}-1) \sum_{i=1}^n \ln \left(x_i\right) \\ +(\alpha-1) \sum_{i=1}^n \ln \left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)\end{gathered}$        (49)

by taking the partial derivatives of $\ln L(\alpha, \lambda, k ; x))$ with respect to the parameter $\alpha, \lambda$ and $k$ respectful, as follows:

$\begin{gathered}\frac{\partial \ln L(\alpha, \lambda, k ; x)}{\partial \lambda}=\frac{-k n}{\lambda}+\frac{k}{\lambda^{k+2}} \sum_{i=1}^n\left(x_i\right)^k \\ -\frac{(\alpha-1) \mathrm{k}}{\lambda^{k+2}} \sum_{i=1}^n \frac{\left(x_i\right)^k e^{-\left(\frac{x_i}{\lambda}\right)^k}}{\left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)}\end{gathered}$       (50)

By setting Eq. (50) equal to zero, we get

$\frac{-k n}{\lambda}+\frac{k}{\lambda^{k+2}} \sum_{i=1}^n\left(x_i\right)^k-\frac{(\alpha-1) \mathrm{k}}{\lambda^{k+2}} \sum_{i=1}^n \frac{\left(x_i\right)^k e^{-\left(\frac{x_i}{\lambda}\right)^k}}{\left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)}=0$       (51)

$\frac{\partial \ln (L(\alpha, \lambda, k ; x))}{\partial \alpha}=\frac{n}{\alpha}-\frac{n 2^\alpha \ln 2}{2^\alpha-1}+\sum_{i=1}^n \ln \left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)$          (52)

Also, by setting Eq. (52) equal to zero, we get

$\frac{n}{\alpha}-\frac{n 2^\alpha \ln 2}{2^\alpha-1}+\sum_{i=1}^n \ln \left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)=0$      (53)

$\begin{gathered}\frac{\partial \ln (L(\alpha, \lambda, k ; x))}{\partial k}=\frac{n}{k} \\ +(\alpha-1) \sum_{i=1}^n \frac{e^{-\left(\frac{x_i}{\lambda}\right)^k}\left(\frac{x_i}{\lambda}\right)^k \ln \left(\frac{x_i}{\lambda}\right)}{2-e^{-\left(\frac{x_i}{\lambda}\right)^k}} \\ +\sum_{i=1}^n \ln \left(x_i\right)-\sum_{i=1}^n\left(\frac{x_i}{\lambda}\right)^k \ln \left(\frac{x_i}{\lambda}\right)-n \ln (\lambda)\end{gathered}$        (54)

Also, by setting Eq. (54) equal to zero, we get

$\begin{gathered}\frac{n}{k}+(\alpha-1) \sum_{i=1}^n \frac{e^{-\left(\frac{x_i}{\lambda}\right)^k}\left(\frac{x_i}{\lambda}\right)^k \ln \left(\frac{x_i}{\lambda}\right)}{2-e^{-\left(\frac{x_i}{\lambda}\right)^k}} \\ +\sum_{i=1}^n \ln \left(x_i\right)-\sum_{i=1}^n\left(\frac{x_i}{\lambda}\right)^k \ln \left(\frac{x_i}{\lambda}\right)-n \ln (\lambda)=0\end{gathered}$          (55)

By solving Eq. (51), Eq. (53) and Eq. (55), by using numerical methods, then obtain the MLE for all the parameters $\alpha, \lambda$ and $k$.

4.4 Order statistics of APIIW distribution

If $Y_1, Y_2, \ldots, Y_n$ denotes the order statistic to the random sample $X_1, X_2, \ldots, X_n$ taken from the population distributing by the APIIW distribution with the $\operatorname{CDF} F_X(x)$ and pdf $f_X(x)$, then the pdf of random variable $Y_j$ of the order $j$ is defined by following:

$\begin{aligned} & f_{Y_j}(x)=\frac{n!\alpha k\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{x}{\lambda}\right)^k} x^{k-1}}{(j-1)!(n-j)!\lambda^k\left(2^\alpha-1\right)} \\ & \left(\frac{\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^\alpha-1}{2^\alpha-1}\right)^{j-1}\left(\frac{2^\alpha-\left(2-e^{-\left(\frac{x}{\lambda}\right)^k}\right)^\alpha}{2^\alpha-1}\right)^{n-j}\end{aligned}$      (56)

And, if $1 \leq i \leq j \leq n$ then the joint pdf of $Y_i$ and $Y_j$ with the order random variable $u$ and $v$, is

$\begin{aligned}& f_{Y_i, Y_j}(u, v) f_{X_{(i)}, X_{(j)}}(u, v)=\frac{n!}{(i-1)!(j-i-1)!(n-j)!} \left(\frac{\left(2-e^{-\left(\frac{u}{\lambda}\right)^k}\right)^\alpha-1}{2^\alpha-1}\right) \left(\frac{\left(2-e^{-\left(\frac{v}{\lambda}\right)^k}\right)^\alpha-1}{2^\alpha-1}-\frac{\left(2-e^{-\left(\frac{u}{\lambda}\right)^k}\right)^\alpha-1}{2^\alpha-1}\right)^{j-i-1^{i-1}} \\& \left(\frac{2^\alpha-\left(2-e^{-\left(\frac{v}{\lambda}\right)^k}\right)^\alpha}{2^\alpha-1}\right)^{n-j} \frac{\alpha k\left(2-e^{-\left(\frac{u}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{u}{\lambda}\right)^k} u^{k-1}}{\lambda^k\left(2^\alpha-1\right)} \\& \frac{\alpha k\left(2-e^{-\left(\frac{v}{\lambda}\right)^k}\right)^{\alpha-1} e^{-\left(\frac{v}{\lambda}\right)^k} v^{k-1}}{\lambda^k\left(2^\alpha-1\right)}, 0<u<v<\infty\end{aligned}$  (57)

The j.p.d.f of n order random variables $Y_1, Y_2, \ldots, Y_n$ can extract more than one ranking statistics through our use of similar functions. Thus, the joint pdf for all order statistics $f_{Y_1, Y_2, \ldots, Y_n}\left(x_1, \ldots, x_n\right)$, taken from the APIIW distribution, as the following

$\begin{gathered}f_{Y_1, Y_2 \ldots, Y_n}\left(x_{1, \ldots,}, x_n\right)= \\ \left\{\begin{array}{c}\alpha^n k^n \prod_{=1}^n\left(2-e^{-\left(\frac{x_i}{\lambda}\right)^k}\right)^{\alpha-1} \prod_{i=1}^n x_i^{k-1} e^{-\sum_{i=1}^n\left(\frac{x_i}{\lambda}\right)^k} \\ n!\lambda^{n k}\left(2^\alpha-1\right)^n \\ 0 \text { otherwise } \\ 0<x_1<\cdots<x_n<\infty\end{array}\right.\end{gathered}$      (58)

4.5 Simulation studies of the APIIW distribution

In order to understand and interpret experimentally the adopted estimation method MLE that was studied in this section of the study, we will employ the simulation approach to examine the MLEs of the all parameters of the APIIW distribution. This part includes a description of the Monte Carlo simulation experiment for the research in terms of the sizes of samples which generated when the number of iterations of the simulation 1000 . We also present the simulation test results obtained where we used statistical R software to apply the simulation method. The stages of building simulation experiments contain some important stages, which are the stage of choosing the sample size, where n was chosen: $(\mathrm{n}=30,50,75,100,150,250,500)$, the stage of choosing values for the parameters for three experiments, the default values for the parameters $(\alpha, \lambda, k)=(0.5,4,0.5)$, $(\alpha, \lambda, k)=(1.5,0.5,1.5)$ and $(\alpha, \lambda, k)=(2.5,1.5,4)$. The stage of generating appropriate data for the APII-W distribution and calculating the MLE for the three parameters $(\alpha, \lambda, k)$ and following the calculation of the MSE and the bias. Following the simulation, the outcomes are shown in Figure 6.

4.6 Application APIIW distribution with real data

In this section, as in the Section 3.6, our focus will be directed towards the examination and interpretation of an authentic data collection that serves to show the advantages of the application of the aforementioned APIIW distribution methodology on real datasets. To gauge the relevance of the model, multiple information criteria were determined. These requirements featured the AIC, which weighs the trade-off between model fit and complexity. Additionally, the CAIC was computed to address potential issues with small sample sizes, the BIC underwent scrutiny to penalize models with a significant number of parameters, demonstrating a preference for simpler models. Finally, the HQIC was integrated in the assessment, which is a modification of the AIC that considers the number of observations in the dataset.

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Figure 6. The MSE and the bias of the APIIW distribution for different parameters values

The first authentic dataset illustrates an uncensored dataset derived from Nichols and Padgett's research on the breaking stress of the carbon fibers (measured in Gba) [20]. This dataset is presented as shown in Table 7.

These data were recently used by Hassan and Hemeda [21]. By comparing the distributions, they chose with the APIIW distribution, we found that the new APIIW distribution is better than the other distributions, as shown in Table 8.

The second dataset showcases the durations of remission (expressed in months) for a specific group of 128 patients diagnosed with bladder cancer, as detailed by Lee and Wang in 2003. This dataset is shown in Table 9.

Table 7. Data on the stress of the carbon fibres

Table 8. Goodness for first dataset

Table 9. Data on duration of remission of bladder cancer

These data were recently used by Almetwally [22]. So, when we studied this evidence on the new APII distribution, it turned out to be better than the other two distributions, as shown in Table 10.

The third dataset comprises 76 observations pertaining to the endurance limits of fatigue fracture in Kevlar 373/epxy under constant pressure at a stress level of 90%, until all specimens experienced failure. This dataset is presented as shown in Table 11.

These data were recently used by Selim [23]. The above-mentioned dataset was utilized by him for fitting to the inv. generated power Weibull (IGPWD), inv. Naderajah-Haghighi (INHD), inv. Weibull (IWD), and inv. exponential (IED). So when we studied this evidence on the new APIIW distribution, it turned out to be better than the other two distributions, as shown in Table 12.

Based on the results in Tables 4-6, we see that the APIIW model is the best according to the criteria of fit to the data that was used compared to the other distributions with which it was compared in the tables.

Table 10. Goodness for the second dataset

Table 11. Data on endurance limits of fatigue fracture

Table 12. Goodness for the third dataset