A new generalization of the inverse Lomax distribution with statistical properties and applications

In this paper, we introduce a new generalization of the inverse Lomax distribution with one extra shape parameter, the so-called power inverse Lomax (PIL) distribution, derived by using the power transformation method. We provide a more flexible density function with right-skewed, uni-modal, and reversed J-shapes. The new three-parameter lifetime distribution capable of modeling decreasing, Reversed-J and upside-down hazard rates shapes. Some statistical properties of the PIL distribution are explored, such as quantile measure, moments, moment generating function, incomplete moments, residual life function, and entropy measure. The estimation of the model parameters is discussed using maximum likelihood, least squares, and weighted least squares methods. A simulation study is carried out to compare the efficiencies of different methods of estimation. This study indicated that the maximum likelihood estimates are more efficient than the corresponding least squares and weighted least squares estimates in approximately most of the situations Also, the mean square errors for all estimates are decreasing as the sample size increases. Further, two real data applications are provided in order to examine the flexibility of the PIL model by comparing it with some known distributions. The PIL model offers a more flexible distribution for modeling lifetime data and provides better fits than other models such as inverse Lomax, inverse Weibull, and generalized inverse Weibull.


Introduction
or Pareto type II distribution has been suggested by Lomax (1954) as an important model for lifetime analysis. The Lomax distribution is widely applied in some areas, such as analysis of income and wealth data, modeling business failure data, biological sciences, model firm size and queuing problems, reliability modeling, and life testing (Harris, 1968;Atkinson and Harrison, 1978;Holland et al., 2006;Corbellini et al., 2010;Hassan and Al-Ghamdi, 2009;Hassan et al., 2016). In the literature, some extensions of the Lomax distribution are available such as the Marshall-Olkin Extended-Lomax (Ghitany et al., 2007), gamma-Lomax (Cordeiro et al., 2015), power Lomax (Rady et al., 2016), exponentiated Lomax geometric (Hassan and Abdelghafar, 2017), power Lomax Poisson (Hassan and Nassr, 2018), exponentiated Weibull-Lomax (Hassan and Abd-Allah, 2018), and Type II half logistic Lomax (Hassan et al., 2020) distributions among others.
The inverse Lomax (IL) distribution is a special case of the generalized beta distribution of the second kind. It is one of the significant lifetime models in statistical applications. Also, it has an application in various fields like stochastic modeling, economics, actuarial sciences, and life testing as discussed by Kleiber and Kotz (2003). Besides this, it has been used to obtain the Lorenz ordering relationship among ordered statistics (Kleiber, 2004). The IL distribution has been used on geophysical databases especially on the sizes of land fibers in California State of United States (McKenzie et al., 2011). The IL distribution can be derived from Lomax distribution using the transformation, distribution has a Lomax Z where Y=1/Z.
The probability density function (pdf) of twoparameter IL distribution is specified by, ( ; , ) = −1 (1 + ) −( +1) ; , , > 0, here, and are the scale and shape parameters respectively. The associated cumulative distribution function (CDF) is: Rahman and Aslam (2014) used a twocomponent mixture IL model for the prediction of future ordered observations in the Bayesian framework using predictive models. Singh et al. (2016) obtained reliability estimates of the IL model under Type II censoring. In addition to this, the IL distribution from hybrid censored was applied to the survival data by Yadav et al. (2016).
In this paper, we provide a more flexible model by inducing just one extra shape parameter to inverse Lomax model for improving its goodness-offit to real data. We discuss some of its statistical properties. Estimation of the unknown parameters for the subject model using the maximum likelihood (ML), least squares (LS), and weighted least squares (WLS) methods is considered. Finally, simulation issues, as well as application to real data, are provided. The layout of the paper contains the following sections. In Section 2, we introduce the three-parameter PIL distribution. Some statistical properties of PIL distribution are presented in Section 3. In Section 4, ML, LS, and WLS estimators for the model parameters are obtained. Numerical study and analysis of real data sets are presented in Section 5. The article ends with concluding remarks.

Power inverse Lomax distribution
The power inverse Lomax distribution is developed by considering the power transformation = 1 , where the random variable follows IL distribution with parameters and . The cdf of the PIL distribution is specified by, The pdf of PIL distribution corresponding to 3 can be written as follows: , , , > 0.
A random variable X has PIL distribution will be denoted by ~PIL( , , ). Plots of the pdf and hrf for some selected parameters values are displayed in Fig. 1 and Fig. 2. As seen from these figures that the pdf and hrf take different shapes according to different values of parameters.

Properties of PIL distribution
In this section, some important mathematical characteristics of the PIL distribution are developed, specifically; the r th moment, the moment-generating function, incomplete moments, moments of residual life function, and Rényi entropy.

Moments
An explicit expression of the r th moment for the PIL can be obtained from pdf 4 as follows: Hence, after simplification, the r th moment of PIL distribution is obtained as follows: Setting r=1, 2, 3, 4 in 5, we can obtain the first four moments about zero. Generally, the moment generating function of PIL distribution is obtained as follows: The r th central moment ( ) of is given by, The mean ) ( 1 ) and variance ( 2 ) of the PIL distribution for some selected values of the parameters which can be calculated numerically in Table 1. Also, the skewness (SK) and (KU)kurtosis of the PIL distribution for various values of parameters and can be calculated numerically in Table 2.  From Table 1, it can be observed that both values of the mean and variance of the PIL distribution increase as the values of and increase. Also, the values of the variance decrease as the values of increase. From Table 2, it can be noticed that both the skewness and the kurtosis are decreasing functions of and .
Next, we derive a simple formula for the s th incomplete moment of X defined by ( ) = ( | < ). So, the quantity ( )comes from 4 as follows where ( + ; + , 1 − ) is the incomplete beta function. The first incomplete moment of X is important to determine the mean deviations, which can be used to measure the amount of scattering in a population, and the Bonferroni and Lorenz curves. It can be obtained by substituting s=1in (6). Additionally, the Bonferroni and Lorenz curves of PIL distribution are, respectively, given by, and,

Moments of residual life function
The n th moment of the residual life of X is given by, Hence, n th moment of residual life of PIL distribution can be obtained as follows: where ( + ; 1 − , + ) is the incomplete beta function.

Rényi entropy
Entropy is a measure of uncertainty of a random variable X. The Rényi entropy of random variable X for a continuous random variable with range R is defined as follows: The pdf; ( ( ; , , )) of the PIL distribution can be expressed as follows: Therefore, the Rényi entropy of PIL distribution is given by, Table 3 gives ( ) of the PIL distribution for different choices of parameters , and . It's seems that the entropy increases with increasing values of and , while decreases with increasing values of .

Parameter estimation
This section deals with the parameter estimation for PIL distribution based on ML, LS, and WLS methods.

Maximum likelihood estimator
Let 1 , 2 , 3 , … , be the observed values follow PIL distribution with pdf 4, then the log-likelihood function, denoted by lnL, based on a complete sample for the unknown parameters can be expressed as: The partial derivatives of the log-likelihood function with respect to , and can be obtained as follows: , and, Then ML estimators of the parameters , , and denoted by ̂ ,̂ ̂ are determined by solving numerically the non-linear equations ⁄ = 0, ⁄ = 0,and ⁄ = 0 simultaneously.

Least squares estimator
The LS estimators were originally proposed by Swain et al. (1988) to estimate the parameters of the beta distribution. The method of LS is about estimating parameters by minimizing the sum of square errors between the observed data and their expected values with respect to the unknown parameters. That is; , also, the WLS can be obtained by minimizing the following with respect to the unknown parameters: .
Let 1 , 2 , 3 , … , be a random sample of size n from PIL distribution. Suppose that 1: < 2: < 3: < ⋯ < : denotes the corresponding ordered sample. Therefore, the LS estimators of , and say,̃,̃ and ̃ respectively, can be obtained by minimizing the following function with respect to , , and , Also, the WLS estimators of , and say, ̅ , ̅ and ̅ respectively, can be obtained by minimizing the following quantity with respect to , , and , .

Numerical illustration
In this section, a numerical study is presented to compare the performance of the estimates for different parameter values. The performance of the estimates of unknown parameters has been measured in terms of their absolute bias (AB), standard error (SE), and mean square error (MSE) for different sample sizes and for different parameter values. The numerical procedures are defined through the following algorithm.
Step 4: Steps from 1 to 3 are repeated 1000 times for each sample size and for selected sets of parameters. Then, the ABs, SEs, and MSEs of the estimates of the unknown parameters are calculated.

Numerical results
Numerical results are reported in Tables 4 to 6 and represented through some Figs. 3-6. From these tables and figures, the following observations can be detected on the properties of estimated parameters from the PIL distribution.
1. The MSEs and SEs of the ML, LS, and WLS' estimates decrease as the sample sizes increase for a different selected set of parameters ( Fig. 3 and Fig.  4).   (Table 6). 6. The SEs for the LS estimates, ̃ take the smallest value among the corresponding SEs for the other methods in almost all of the cases.

Data analysis
In this subsection, two real data sets are provided to illustrate the importance of the PIL distribution by comparing it with some other distributions (IL, inverse Weibull (IW), and generalized inverse Weibull (GIW). Two real data sets are used to show that PIL distribution can be applied in practice and can be a better model than some others. The first data set is corresponding to remission times (in months) of a random sample of 128 bladder cancer patients given in Lee and Wang (2003). The data are given as follows: The maximum likelihood method is performed to obtain the point estimates of the model parameters.
To compare the fitted models, some selected measures are applied. The selected measures include; -2log-likelihood function evaluated at the parameter estimates, Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and Hannan-Quinn information criterion (HQIC). The better model is corresponding to the lower values of, AIC, CAIC, BIC, and HQIC. The results for the previous measures to the mentioned models are listed in Table 7. The results, in Table 7, show that the PIL distribution has the smallest values of AIC, CAIC, BIC, and HQIC. Then PIL distribution provides a significantly better fit than other distributions considered here IL, IW, and GIW. The second data set refers to Murthy et al. (2004) about the time between failures for the repairable item. The data are listed as the following: 1. 43, 0.11, 0.71, 0.77, 2.63, 1.49, 3.46, 2.46, 0.59, 0.74, 1.23, 0.94, 4.36, 0.40, 1.74, 4.73, 2.23, 0.45, 0.70, 1.06, 1.46, 0.30, 1.82, 2.37, 0.63, 1.23, 1.24, 1.97, 1.86, 1.17. The results in Table 8 indicate that the PIL model is suitable for this data set based on the selected criteria. The PIL model has the lowest AIC, BIC, CAIC, and HQIC values. Therefore, the PIL is a preferable model to the other model for this data set.

Concluding remarks
In this paper, we introduce a new model, the socalled, power inverse Lomax distribution. Several properties of the PIL distribution are investigated, including the moments, incomplete moments, moments of residual life, and Rényi entropy. The estimation of population parameters is discussed through the method of the maximum likelihood, least squares, and weighted least squares. The simulation study is presented to compare the performance of estimates. Applications of the power inverse Lomax distribution to real data show that the new distribution can be used quite effectively to provide better fits than the inverse Lomax, inverse Weibull, and generalized inverse Weibull models. The power inverse Lomax distribution parameters can be investigated using ranked set sampling methods (Al-Saleh and Al-Omari, 2002;Al-Omari, 2010;2011;Haq et al., 2014a;2014b).