Alpha Power Transformed Inverse Lomax Distribution with Different Methods of Estimation and Applications

Deanship of Scientific Research, King Abdulaziz University, Jeddah, Saudi Arabia Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt Quality Enhancement Cell, National College of Arts, Lahore, Pakistan College of Statistical & Actuarial Sciences, University of the Punjab, Lahore, Pakistan Lahore College for Women University (LCWU), Lahore, Pakistan King Abdulaziz University, Jeddah, Saudi Arabia


Introduction
e inverse Lomax (IL) is originally developed as a lifetime distribution. e inverse Lomax is member of family of generalized beta distribution. Kleiber and Kotz [1] showed that IL distribution can be used in economics.
e IL distribution has many applications in modeling of different trends of hazard rate function (hrf ), i.e., decreasing or upside-down bathtub failure rate of life testing of components. e IL distribution is used in [2] to get Lorenz ordering relationship among ordered statistics. McKenzie et al. [3] used the IL model and applied it to geophysical databases. Singh et al. [4] investigated the reliability estimators of IL distribution under Type II censoring. Bayesian estimation of mixture of the IL model under the censoring scheme was studied in [5]. Inverse power Lomax distribution was studied in [6] and Weibull IL distribution in [7]. e probability density (pdf ) and cumulative distribution function (cdf ) of the IL distribution are as follows: g(x; a, b) � ab and the corresponding pdf is In the literature, many probability distributions are generalized using this approach; for example, alpha power transformed Weibull (APTW) distribution in [9], APT generalized exponential distribution in [10], APT Lindley distribution in [11], APT extended exponential distribution in [12], alpha power inverted exponential distribution in [13], alpha power Inverse-Weibull distribution in [14], APT inverse-Lindley distribution in [15], APT power Lindley studied in [16], and APT Pareto distribution proposed in [17].
e main goal of this research article is to introduce a simpler and more flexible model called APT inverse Lomax (APTIL) distribution. Furthermore, the key motivations of using APTIL distribution in the practice are follows: (i) To improve the flexibility of the existing distributions by using APT-G (ii) To introduce the extended version of the IL distribution whose closed form of cdf exist. (iii) To provide better fits than the competing modified models e rest of the paper is arranged as follows. In Section 2, we define the new model called AP inverse Lomax (APTIL) distribution. Various statistical properties of the APTIL distribution are derived in Section 3 along with more attractive expressions for quantile function, median, mode, moments, order statistics, and stress strength parameter. Lemmas 1 and 2 contain expressions for stochastic ordering and characterization related to hazard function, and the asymptotic behavior of density is derived. In Section 4, we estimate the parameters using eight methods of parameters estimation as follows: maximum likelihood (ML), least squares (LS), weighted least squares (WLS), percentile (PC), Cramer-von Mises (CV), maximum product of spacing (MPS), Anderson-Darling (AD), and right-tail Anderson-Darling (RTAD). Section 5 deals with three applications to show the efficiency of the proposed model. Final remarks are mentioned in Section 6.

APTIL Distribution
e random variable (rv) X is said to have APTIL distribution denoted by APTIL(α, a, and b) with two shape parameters and one scale as α, a, and b, respectively. e pdf of X for x ≥ 0 is e survival function (sf ) and the hrf for APTIL distribution for x > 0 are in the following forms: Figure 1 demonstrates the graphs of pdf and hazard function of APTIL distribution for different values of α, a, and b. Clearly, the pdf of APTIL distribution is the function for α ≠ 1 and a < 1 and unimodal for α ≠ 1 and a > 1. e hrf of the APTIL model can be decreasing or upside-down bathtub for α ≠ 1 and a < 1 and a > 1, respectively.

Useful Expansions.
Here, an explicit expression for APTIL pdf is given. By using the series representation of exponential function, equation (5) can be written as 2 Complexity where g a(i+1),b (x) denotes the pdf of IL distribution given in equation (1) with parameters a(i + 1)and b.

Quantile Function and
Median. e generation from APTIL distribution is obtained by inverting equation (6): If q uniform (0, 1), then X APTIL(α, a, b), the qth quantile function of APTIL(α, a, b) is given by and the median can be obtained as and when α � 1, the median of the APTIL distribution is equal to the median of the IL distribution. e analysis of shape of distribution can be performed by study of skewness and kurtosis. Using Bowley's coefficient of skewness as follows: Moor's coefficient of kurtosis is given as where x (.) is the quantile function.

Properties of APTIL Distribution
is section deals with some statistical properties of APTIL distribution.

Mode.
e mode of APTIL distribution is derived by f ′ (x) � 0: Complexity e mode of APTIL distribution cannot be expressed in the closed form. Computer software, e.g., Mathematica or R, can be used to compute the mode of APTIL distribution for specific values of parameters.
For α � 1, the mode of IL distribution can be easily calculated from the following equation:

Asymptotic
Behavior. e behavior of APTIL distribution is investigated here as x ⟶ 0 and x ⟶ ∞: Proof.

Moments
Theorem 1. Let X be a rv from APTIL distribution, then its r th moment is Proof. Let X be an r.v. with pdf given in equation (5). For any real number a > 0, b > 0, α > 0, and r ≥ 0, the r th moments of APTIL distribution are obtained as Using expression from equation (10), we get where . . , c denotes Euler's constant and φ(r) � r k�1 1/k [18]. e mean of X can be obtained using equation (20) by putting r � 1: where c denotes Euler's constant. e r th central moment μ r of X is derived as e variance of APTIL distribution is easily obtained as □ Lemma 2. Let X be a r. v. with pdf (equation (5)). For any real number a > 0, b > 0, α > 0, r ≥ 0 and δ ≥ 0, the integral, 4 Complexity is calculated as Proof of Lemma 2 is given in Appendix.
Proof. e moment generating function can be derived by Using Lemma 2, the moments, moment generating function, characteristic generating function, and raw moment can be easily obtained by □ 3.5. Order Statistics. Let X 1 , X 2 , . . . , X n be r.sample from the APTIL(α a, b) distribution with order statistics e pdf of X (r) can be expressed as where Particularly, pdf of the first and n th order statistics can be easily derived from equation (32) as respectively.

Stress-Strength Model.
Let X 1 and X 2 be two independent random variables with APTIL(α 1 , a 1 , b 1 ) and APTIL(α 2 , a 2 , b 2 ) distributions, respectively. If X 1 represents "stress" and X 2 represents "strength," the reliability is defined by Using Lemma 2 from equation (27), we have e effect of parameters a, b, and α on mean, variance, skewness, and kurtosis is displayed in Figures 2 and 3, respectively.

Remark 1
(i) e mean and variance of APTIL distribution increase as "a" or "b" increase for fixed value of α (ii) For increasing α, the mean of distribution decreases as "b" decreases for higher value of "a" and increases for lower value of "a" (iii) For increasing α, the variance of distribution increases as "b" increases for lower value of "a" (iv) e skewness and kurtosis of APTIL distribution decrease as "a" increases or "b" decreases for fixed value of α 3.7. Characterization Based on Hazard Function. In this section, characterizations of APTIL distribution based on the hazard function are presented. It is known that hazard function h(x) satisfies the following differential equation: under the boundary conditions h(0) ≥ 0. Proof. If rv X has the hrf given in (8), then which implies C � 0.

ML Estimation.
Let X 1 , . . ., X n have the observed values from APTIL distribution. e MLEs of the proposed model parameters α, a, and b are derived using the log-likelihood function say ℓ which is given by e ML equations of the APTIL distribution are given by Equating zℓ/zα, zℓ/za, and zℓ/zb with zeros and solving simultaneously, we obtain the ML estimators of α, a, and b.

Ordinary and Weighted LS Estimators.
Suppose X 1 , X 2 , . . . , X n is a random sample from APTIL distribution with corresponding ordered sample of X (1) , X (2) , . . . , X (n) . e mean and variance of APTIL are independent of unknown parameter and are as follows: where F(X (i) ) is the cdf of APTIL distribution withX (i) being the i th order statistic. en, LS estimators ( [19]) are obtained by minimizing the SSE: with respect to α, a, and b. So, the LS estimators (LSEs) of the parameters α, a, and b of the APTIL are obtained by minimizing the following: with respect to α, a, and b. e WLS estimators [19] of α, a, and b can be obtained by minimizing the following expression: with respect to α, a, and b.

e Cramer-von Mises Minimum Distance Estimators.
e CV method is based on the difference between the estimated cdf and the empirical cdf ( [22,23]). e CV estimators are obtained by minimizing

Complexity 7
Macdonald [24] stated about the CV method that it depends on minimum distance estimators providing empirical evidence that the bias of the estimator is smaller than the other minimum distance estimators.

Maximum Product of Spacing Estimators.
e MPS method is a powerful alternative to the ML method for estimating the population parameters of continuous distributions ( [25]). Let be the uniform spacings of a random sample from the APTL distribution, where e MPS estimator is obtained by maximizing the geometric mean (GM) of the spacings: w.r.t. α, a, and b. e MPS estimator of α, a, and b can be obtained by maximizing the logarithm of the GM of sample spacing's equation (51). No closed solution exists, so the numerical method is used to find the estimates.

Anderson-Darling and Right-Tail Anderson-Darling
Estimators. e method of Anderson-Darling estimation was introduced by [26] in the context of statistical tests. By adapting it to the APTIL model, the Anderson-Darling estimates (ADEs) of α, a, and b can be obtained by minimizing, with respect to α, a, and b, the function given by        69 0.081 0.990 0.394 1.026 0.399 0.881 0.380 0.885 0.276 0.696 0.168 0.872 0.482 0.780 0.288  2.667 0.238 2.412 1.160 2.388 1.406 2.766 1.290 2.500 0.441 2.813 0.477 2.332 1.489 2.781 0.908 Complexity 9 Similarly, the right-tail Anderson-Darling estimates (RTADEs) of α, a, and b can be obtained by minimizing, with respect to α, a, and b, the function given by

Simulation Study
Here, we come up with a numerical study to compare the behavior of different estimates. We generate 1000 random samples of size n � 50, 100, and 200 from the APTIL distribution. Four sets of the parameters are assigned as follows:    Tables 1-4. Form Table 5, for the parameter combinations, we can conclude that the ML estimation method outperforms all the other estimation methods (overall score of 20.5). erefore, depending on our study, we can consider the ML estimation method is the best for APTIL distribution.

Applications
In this section, we utilized three data sets to show that APTIL can be a better life testing distribution compared with some known probability distributions such as APT Weibull (APW) distribution [9], alpha power transformed inverse exponential (APTIE) distribution [10],
For the selection of best fit model, we used the following criteria: Akaike information criterion (AIC), Bayesian information criterion (BIC), Anderson-darling (A * ), and Cramer-von Mises (W * ) test. e maximum likelihood estimates are presented in Table 6, and the goodness of fit measures are presented in Table 7.
We can use the likelihood ratio (LR) test to compare the fit of the ALTIL distribution with other models for given data sets. e form of the test is suggested its name LRT � 2 log L s (θ) where the LR is the ratio of two likelihood functions; the simpler model (s) has fewer parameters than the general (g) model. e LR test rejects the null hypothesis if χ 2 > χ 2 d , where χ 2 d denotes the upper 100% point of the χ 2 distribution. e shape hazard function for modeling can be analyzed using graphical technique called total time in the test (TTT) plot (for more details, see [31]). From Figure 4, for the first and second data, the TTT plot is concave and provides evidence of the monotonic hazard rate. For the third data set, e TTT plot is convex and according to [31], it provides evidence that the hazard rate is decreasing. e APTIL distribution gives the lowest values of AIC, BIC, A * , and W * tests among all the fitted models to these data sets. So, it could be selected as the best model among them. e fitted pdf and estimated cumulative distribution function of the APTIL are displayed in Figures 5-7 for the three data sets, respectively. e empirical data and estimated density plot show closeness which depict that the APTIL model fits all three data sets well. e APTIL model is compared with other competitive models. e estimated cdf curve of APTIL model also confirms the above results. e likelihood ratio test is performed to compare APTIL distribution with other fitted models to test H o against H 1 discussed above, and results are shown in Table 8. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared with the alternative. We conclude that APTIL distribution provides better fit than all other competitive models at the level of significance ≤0.05.

Conclusions
In this research, we proposed and studied the APTIL distribution. Some structural characteristics of the APTIL distribution are derived. e asymptotic behavior of its density function is studied. Characterization related to hazard rate function is also obtained. Estimation of the population parameters is achieved using eight various procedures. Simulation results are carried out to assess the performance of estimators. Real data sets are used for the applications to show the flexibility of the APTIL model.