A Novel Extended Power-Lomax Distribution for Modeling Real-Life Data: Properties and Inference

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Rady et al. [15] proposed power-Lomax (PLx) model as a new extension of the Lx distribution with an extra shape parameter and applied it to medical data.Bayesian estimation by Markov chain Monte Carlo (MCMC), maximum likelihood, maximum product spacing, ordinary least squares, and Cramér-von Mises of the PLx model is studied in [16].Furthermore, a truncated version of the PLx distribution is introduced in [17].
Mahdavi and Kundu [18] introduced the alpha-power-G (AP-G) family for adding an additional parameter to a baseline distribution to increase its fexibility.
In this research article, we introduced a simpler and more fexible model called the alpha-power power-Lomax (APPLx) distribution.Te PLx and Lx models follow as special cases of the new APPLx distribution.Te APPLx distribution is constructed based on the PLx and AP-G family by inserting the baseline PLx model in the AP-G family.Te failure rate (FR) of the APPLx distribution is very fexible to exhibit increasing, unimodal, J-shape, reversed Jshape, and decreasing FR shapes.Moreover, the APPLx density can provide left-skewed, unimodal, right-skewed, symmetric, J-shape, and reversed J-shaped densities.Te skewness of the APPLx distribution can range in the interval (− 0.8,41.7),whereas the skewness of the PLx distribution varies only in the interval (− 0.5, 35.4) for the scale parameter λ � 1 and at the same values of the shape parameters.Furthermore, the spread for the APPLx kurtosis is much larger ranging from 3.3 to 4116.9, whereas the spread for the PLx kurtosis only varies from 3.46 to 2939.67 with the same parameter values.
We are also motivated to illustrate how diferent classical estimators of the APPLx parameters can perform for several parameter combinations and sample sizes.Hence, the APPLx parameters are estimated using some estimation methods along with extensive simulation studies to explore the performance of the diferent estimators.Additionally, these estimators are ordered by using partial and overall ranks to discover the best estimation method for estimating the APPLx parameters.
Te paper is outlined in seven sections.Te APPLx distribution is presented in Section 2. Te key distribution properties are introduced in Section 3. Diferent estimators of the APPLx parameters are provided in Section 4. Section 5 provides extensive simulation studies for exploring the introduced estimators.Tree real-life data sets are ftted using the APPLx distribution in Section 6.Some fnal conclusions are ofered in Section 7.

The APPLx Distribution
In this section, we introduced the APPLx distribution using the AP-G family.Te APPLx distribution follows by inserting the baseline PLx model in the AP-G family.
Te PLx distribution is specifed by the following cumulative distribution function (CDF): Its probability density function (PDF) reduces to the following equation: ( Te CDF of the APT-G family takes the following form: Te corresponding PDF is as follows: Te survival function (SF) and hazard rate function (HRF) of the AP-G family are as follows: Te random variable (rv) X has a four-parameter APPLx distribution if its CDF is given (for x > 0) by the following equation: Te corresponding PDF of equation ( 6) takes the following form: Te SF and HRF of the APPLx distribution have the following forms: Journal of Mathematics Te rv X with PDF ( 8) is denoted by APPLx (α, c, β, λ).Figures 1 and 2 display possible shapes for the PDF and HRF of the APPLx distribution.Tese plots reveal that the APPLx PDF provides left-skewed, unimodal, right-skewed, symmetric, J-shape, and reversed J-shaped densities.Also, the APPLx HRF can be increasing, unimodal, decreasing, Jshape, and reversed J-shape.

Properties of APPLx Distribution
Tis section presents some statistical characteristics of the APPLx distribution.
3.1.Useful Expansions.Tis section provides an explicit expression for the APPLx density.By applying the series representation of exponential function, equation (7) can be written as follows: where denotes the PDF of PLx distribution in equation ( 2) with parameters c(i + 1), β, and λ.
3.2.Quantile Function and Mode.Te quantile function (QF) of the APPLx distribution is, by inverting equation ( 6), as follows: where q is uniform (0,1) and the median follows directly from the following equation: Te mode of APPLx distribution is derived by solving f ′ (x) � 0, where

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Tere is no closed form expression for the mode of APPLx distribution; hence, statistical software such as R and Mathematica can be used to calculate the APPLx mode values for diferent values of parameters.Te mode of the PLx distribution is, from equation (13) for α � 1, as follows: 4 Journal of Mathematics 3.3.Moments.Te r th moment of the APPLx distribution takes the following form: Setting r � 1 in equation ( 15), the mean of x is as follows: Te r th central moment μ r of X is derived as follows: Te variance of the APPLx distribution is as follows: Te coefcients of skewness (CS) and kurtosis (CK) are generally defned by the following equation: Table 1 provides numerical values of the mean (μ), variance (σ 2 ), CS, and kurtosis CK of the APPLx distribution for diferent values of α, β, λ, and c.Te numerical values in Table 1 show that CS of the APPLx distribution can range in the interval (− 0.8029, 41.6885).Te spread of its CK is much larger ranging from 3.3021 to 4116.9250.Table 1 illustrates that the APPLx distribution can be right-skewed or left-skewed.Furthermore, it can be leptokurtic (CK > 3).Hence, the APPLx distribution is a fexible distribution and it can be used to model real-life skewed data.
is given by the following equation: Proof.By using the series representation of the exponential function and Taylor's series expansion of the function e δx , equation (20) can be expressed as follows Te integral in the last equation is determined by the following equation:

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where B(a, b) is a beta function.By substituting equation ( 23) into (22), we get equation (21), which completes the proof.
Te moment generating function (MGF) of X is obtained as follows:

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Proof.Te MGF is defned by the following equation: Using Lemma 1, the MGF is as follows:  (7) and let X 1:n ≤ X 2:n ≤ . . .≤ X n:n be the order statistics of a random sample from the APPLx (7).
Ten, the PDF and CDF of the r-th order statistics are defned (for r � 1, 2, . . ., n) by the following equations: By inserting equations ( 6) and (7) in equations ( 27) and (28), we have the PDF and CDF of the r-th order statistics as follows: Particularly, the PDFs and CDFs of the 1st and nth order statistics are derived from equations ( 29) and (32), respectively, as follows: 3.5.Stress-Strength Model.Consider the two independent random variables X 1 and X 2 , where represents stress and X 2 represents strength, then the reliability R can be defned by the following equation: Hence, we have the following equation: Using Lemma 1, we can write the following equation:

ML Estimation.
Te MLEs of the parameters α, c, β, and, λ of the APPLx distribution are presented in this section.Hence, the log-likelihood function of ξ � (α, c, β, λ) ⊺ reduces to the following equation: On diferentiating the log-likelihood equation with respect to the parameters and equating them to zero, we obtained the following equation: 8

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Solving these equations give the MLEs of the APPLx parameters.

Ordinary and Weighted Least-Squares Estimators.
Te LSEs [19] of the APPLx parameters can be obtained by minimizing the following equation: Also, these estimators can be obtained by solving the following equation: where where and Ω k for k � 1, 2, 3, 4 which can be solved numerically.
Te WLSEs (see [19]) of the APPLx parameters are derived by minimizing the following: Te WLSEs can also be determined by solving the following function: where Ω k are given in equations ( 41)-(44).

Anderson-Darling and Right-Tail Anderson-Darling
Estimators.Te ADEs (see [20]) of the APPLx parameters are obtained by minimizing as follows: Te ADEs are also determined by solving the nonlinear equations given by the following equation: where Ω k are given in equations ( 41)-(44).Te RADEs of the APPLx parameters can be determined by minimizing as follows: Furthermore, the RADEs can be calculated by solving the following equation: where Ω k are defned by equations ( 41)-(44).

Maximum Product of Spacing and Cramér-Von Mises
Estimators.Te uniform spacings, say D i (ξ), of a random sample from the APPLx distribution are given by Te MPSEs of the APPLx parameters follow by maximizing the geometric mean (GM) of spacings as follows: Tey are also obtained by maximizing the logarithm of GM of sample spacings which is given by the following equation: Te CRVMEs (see [21]) are derived as the diference between estimated CDF and empirical CDF.Te CRVMEs of the APPLx parameters follow by minimizing the following: Hence, the PCEs of the APPLx parameters can be calculated by minimizing the following: Also, the PCEs can be derived by solving the following equation: where where

Simulation Analysis
Te behavior of eight estimators of the APPLx parameters is explored by using numerical results with respect to their mean square errors (MSEs), MSEs � 1/N N i�1 (  ξ − ξ) 2   2-7.
For each combination of the parameters and each sample, the APPLx parameters α, β, c, and λ are estimated using the eight methods called the MLEs, LSEs, MPSEs, WLSEs, ADEs, CRVMEs, PCEs, and RADEs.Tables 2-7 report the simulation results for all estimators.Te tables

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Figure 1 : 4 Figure 2 :
Figure 1: Some shapes of the PDF of the APPLx distribution for several values of parameters.

Figure 3 :
Figure 3: Te ftted APPLx PDF, CDF, SF, and PP plots of the APPLx distribution for cancer data.

Table 8 :
All ranks of the eight estimation methods for diferent parametric values of the APPLx distribution.

Table 9 :
Findings of the ftted APPLx distribution and its competing distributions for cancer data.

Table 10 :
Findings of the ftted APPLx distribution and its competing distributions for carbon fber data.

Table 11 :
Findings of the ftted APPLx distribution and its competing distributions for Guinea pigs data.

Table 12 :
Findings of the APPLx model under diferent estimation methods for cancer data.

Table 13 :
Findings of the APPLx model under diferent estimation methods for carbon fber data.

Table 14 :
Findings of the APPLx model under diferent estimation methods for Guinea pigs data.