Inference of Truncated Lomax Inverse Lomax Distribution with Applications

This paper introduces a modified form of the inverse Lomax distribution which offers more flexibility for modeling lifetime data. The new three-parameter model is provided as a member of the truncated Lomax-G procedure. The new modified distribution is called the truncated Lomax inverse Lomax distribution. The density of the new model can be represented as a linear combination of the inverse Lomax distribution. Expansions for quantile function, moment generating function, probability weighted moments, ordinary moments, incomplete moments, inverse moments, conditional moments, and Rényi entropy measure are investigated. The new distribution is capable of monotonically increasing, decreasing, reversed J-shaped and upside-down shaped hazard rates. Maximum likelihood estimators of the population parameters are derived. Also, the approximate confidence interval of parameters is constructed. A simulation study framework is established to assess the accuracy of estimates through some measures. Simulation outcomes show that there is a great agreement between theoretical and empirical studies. The applicability of the truncated Lomax inverse Lomax model is illustrated through two real lifetime data sets and its goodness-of-fit is compared with that of the recent models. In fact, it provides a better fit to these data than the other competitive models.


Introduction
The inverse Lomax (IL) distribution is very flexible in analyzing situations of different types of hazard rate function. It is a member of the inverted family of distributions, that is, the IL distribution is the reciprocal of the well-known Lomax distribution. Kleiber et al. [1] reported that the IL distribution can be used in economics and actuarial sciences. Singh et al. [2] the reliability estimators of the IL distribution were investigated under type-II censoring. Yadav et al. [3] discussed parameter estimators of the IL model in the case of hybrid censored samples. Reyad et al. [4] regarded the Bayesian estimation of a two-component mixture of IL distribution based on a type-I censoring scheme. Bantan et al. [5] provided entropy estimators based on multiple censored scheme. The cumulative distribution function (cdf) of the IL distribution is given by: where, b and d are the scale and shape parameters respectively. The probability density function (pdf) of the IL distribution is as follows: Recent studies about the generalization and extensions of the IL distribution have been proposed by several authors. Hassan et al. [6] provided the inverse power Lomax distribution. Hassan et al. [7] introduced the Weibull IL distribution. ZeinEldin et al. [8] provided the alpha power transformed IL distribution. Maxwell et al. [9] introduced the Marshall-Olkin IL distribution. ZeinEldin et al. [10] introduced odd Fréchet IL distribution. Hassan et al. [11] proposed Topp-Leone IL distribution.
A classical strategy to generate families of probability distributions consists of adding parameter(s) to baseline distributions. These families have the ability to improve the desirable properties of the probability distributions as well as to extract more information from the several data applied in many areas like engineering, economics, biological studies and environmental sciences. Another useful generator that works with the truncated random variable. The notable studies about the truncated-G families in this regard, are the truncated Fréchet-G [12], truncated inverted Kumaraswamy-G [13], truncated Lomax-G [14], truncated power Lomax-G [15] and truncated Cauchy power-G [16].
Hassan et al. [14] suggested the newly truncated Lomax-G (TL-G) family with the following cdf, where a > 0; A ¼ ð1 À 2 Àa Þ À1 ; & is the parameter vector and Gðx; &Þ is the cdf of any distribution and the truncated Lomax pdf is the generator. The pdf associated with Eq. (3) is given by f ðy; a; &Þ ¼ aAgðy; &Þ 1 þ Gðy; &Þ ð Þ ÀaÀ1 : simulation study is considered in Section 4. Applications are given in Section 5 followed by comments and conclusions.

Description of the Model
In this section, we define and provide the density and distribution function of the TLIL distribution. This model is yielded by taking the base-line Eq. (3) to be the cdf of the IL model. A random variable Y is said to have the TLIL, if its cdf is represented as: where, Â ðb; a; dÞ is the set of parameters. The pdf associated with Eq. (5) is given by Expression of hazard rate function (hrf) is given by: Graphic features of the pdf and hrf plots of Y ∼ TLIL ða; d; bÞ are represented in Fig. 1 Fig . 1 gives density and hazard functions plots for specified values of parameters. It is observed that the pdf of the TLIL can be right-skewed, uni-modal and reversed J-shaped. The hazard function can be increasing, decreasing, reversed J-shaped and upside-down bathtub.

Quantile Function and Median
The quantile function of Y, denoted by Q(u), is defined by inverting Eq. (5) as follows where u~U(0,1). Further, the TLIL distribution can be simply simulated from Eq. (8). The median of Y is obtained by putting u = 0.5 in Eq. (8).

Linear Representation
Here, we provide the representation of cdf Eq. (5) and pdf Eq. (6) of the TLIL distribution. As mentioned in Hassan et al. [14], the pdf of the TL-G family is expressed as follows: s j gðy; &ÞGðy; &Þ j ; where s j ¼ AaðÀ1Þ j a þ 1 j : Hence, a useful expression of the TLIL distribution is produced by inserting pdf Eq. (1) and cdf Eq. (2) in Eq. (9) as follows: where, gðy; b; dðj þ 1ÞÞ is the pdf of the IL distribution with parameters b and dðj þ 1Þ: That is the pdf Eq. (10) of TLIL is represented as a linear combination of the IL distribution.
Furthermore, Hassan et al. [14] provided a useful expression of cdf ½F y; & ð Þ h ; where h is an integer, as follows: where Hence, as mentioned in Eq. (11), the cdf ½F y; Â ð Þ h ; of the TLIL takes the following form: 3 Statistical Properties

Probability-Weighted Moments
The probability-weighted moments (PWMs) are less sensitive to outliers. They are utilized to study characteristics of the probability distributions. They are sometimes used when maximum likelihood estimates are unavailable or difficult to compute. The PWMs, denoted by Ç k;h ; is defined as: where, k and h are positive integers. The class of the PWMs of the TLIL is obtained by substituting Eq. (10) and Eq. (12) in Eq. (13) as follows where; Àð0Þ ¼ Àc, ÀðÀkÞ ¼ ðÀ1Þ k k! 'ðkÞ À ðÀ1Þ k k! c for k ¼ 1; 2; …;c denotes Euler's constant and [17]).

Moments and Related Measures
Here, we provide an infinite sum representation for the n-th moment about the origin, inverse moments and incomplete moments of the TLIL model, since it has a pivotal role in the study of the distribution and real data applications. The n-th moment for the TLIL is obtained as follows: The first four moments are obtained by setting n =1, 2, 3 and 4 in Eq. (15). The n-th central moment (l n ) of Y is given by Based on Eq. (16), we can obtain the skewness and kurtosis measures using the well-known relationships. Further, the moment generating function of the TLIL distribution for |t| < 1, is given by The m-th inverse moment, for the TLIL distribution is derived by using pdf Eq. (10) as follows: The harmonic mean of the TLIL distribution can be obtained by using the first inverse moment.
Additionally, the n-th incomplete moment of Y can be obtained from Eq. (10) as follows: where, B(.,.,t) is the incomplete beta function.

Conditional Moments
The conditional moments and the mean residual lifetime function are of interest for lifetime models to be obtained. The r-th conditional moment is defined as: The r-th moment of the residual life of the TLIL distribution is obtained by substituting Eq. (10) in Eq. (20) as follows where B(.,.,t) is the incomplete beta function. In particular, the mean residual life of the TLIL model is obtained by substituting r =1 in Eq. (21).
Moreover, the reversed residual life (RRL) is defined as the conditional random variable ð¡−Y|Y≤ð¡ which denotes the time elapsed from the failure of a component given that its life is less than or equal to ð¡. The r-th moment of the RRL is defined by: The r-th moment of the RRL of TLIL distribution is obtained by substituting Eq. (10) in Eq. (22) as follows The mean of RRL (mean waiting time) represents the waiting time elapsed since the failure of an item on condition that this failure had occurred. For, r = 1 in Eq. (23) we obtain the mean of RRL of the TLIL distribution.

Rényi Entropy
The entropy affords a great tool to evaluate the amount of information (or uncertainty) exists in a random observation relating to its parent distribution. A small value of entropy provides the smaller uncertainty in the data. Hassan et al. [14] mentioned that the Rényi entropy of the TL-G distributions is defined as: gðy; nÞ t Gðy; nÞ i dy; where g i Ã ¼ ðÀ1Þ i ðAaÞ t tða þ 1Þ þ i À So, after some manipulation, the Rényi entropy of the TLIL distribution is given by where, B(.,.) is the beta function.

Estimation and Numerical Study
In this section, the ML estimators of the model parameters are derived in the case of complete samples. Also, approximate confidence intervals (CIs) are obtained. Furthermore, a numerical study is conducted.

ML Estimators
Suppose that x 1 , x 2 ,…,x n be the observed values from TLIL distribution whose lifetimes have the pdf Eq. (6). The log-likelihood based on complete sample, is given by where, W i ¼ 1 þ b y i : Then, the first partial derivatives of the log-likelihood are given by For interval estimation of the parameters, it is known that under regularity conditions, the asymptotic distribution of ML estimators of elements of unknown parameters for a; b and d is given where, I À1 a; b; d ð Þis the variance-covariance matrix of the unknown parameters a; b and d: The elements of the Fisher information matrix are obtained. Therefore, the two-sided approximate 2 100 percent limits for the ML estimators of the population parameters for a; b and d can be obtained, respectively, as follows: where z 2=2 is the 100 1 À z 2=2 À Á % standard normal percentile and var(.)'s denotes the diagonal elements of the variance covariance matrix corresponding to the model parameters.

Simulation Study
In this sub-section, we perform a simulation study to evaluate the behavior of the ML estimates (MLEs). The attitude of the different estimates is checked in terms of their mean square errors (MSEs), standard errors (SEs) and average lengths (ALs) of the CIs. 10000 random samples of sizes 30, 50, 75 and 100 are generated from TLIL distribution. Four sets of parameter values are chosen as; ðiÞ a ¼ 2:

Applications to Real Data
In this section, we provide the applicability of TLIL distribution by using two real data sets. These data have been used by several authors to show the superiority of other competing models. We also provide a formative evaluation of the goodness of-fit of the models and make comparisons with other distributions. The measures of goodness of fit including; the Akaike information criterion (AIC), Consistent AIC (CAIC), Hannan-Quinn information criterion (HQIC), Anderson-Darling(A * ) and Cramér-von Mises (W * ) are calculated to compare the fitted models. Generally, the best fit to the data that is correspond to the lowest values of these statistics.
It is clear from Tab. 3, Tab. 4 and Fig. 2 that the TLIL provides a better fit to the data and therefore could be chosen as the best model.

Cancer Patient Data
This data set describes there mission times (in months) of a random sample of 128 bladder cancer patients studied by Lee et al. [28]. For these data, we compare the fit of the TLIL with some other distributions like; the ZL, transmuted modified Weibull (TMW) distribution [29], transmuted additive Weibull (TAW) distribution [30], generalized inverse gamma distribution distribion [31], beta exponentiated Burr XII (BEBXII) distribution [32], beta Fréchet (BFr) distribution [33] and ETGR  distribution The estimated parameters along with their SEs are provided in Tab. 5. The statistics of the fitted models are presented in Tab. 6. We note from Tab. 6 that the TLIL gives the lowest values of AIC, CAIC, HQIC, A * and W * as compared to the other competitive models. Therefore, TLIL distribution provides the best fit for the cancer patient data. More information can be found in Fig. 3.   As seen from Fig. 3 that the TLIL provides a better fit to cancer data and so could be preferred than the other competitive model.

Concluding Remarks
We provide a new three-parameter lifetime distribution depends on the recent truncated Lomax-G family. The new truncated Lomax inverse Lomax model offers flexibility for modeling lifetime data. Expressions of density and distribution functions are obtained as a linear combination of the inverse Lomax distribution. Several mathematical properties of the new model are derived like; probability weighted moments, quantile function, moment generating function, ordinary and incomplete moments, inverse moments, conditional moments, and Rényi entropy. The maximum likelihood method of estimation is employed to obtain the point and approximate confidence interval of population parameters. We assess the accuracy of estimates viz simulation study. Applications to two real data are utilized to illustrate the importance and usefulness of the new model compared to some models.