Intelligent Automation & Soft Computing DOI:10.32604/iasc.2021.017890 | |

Article |

Inference of Truncated Lomax Inverse Lomax Distribution with Applications

1Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

2Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, 12613, Egypt

3The Higher Institute of Commercial Sciences, Al Mahalla Al Kubra, 31951, Egypt

4Deanship of Information Technology, King Abdulaziz University, Jeddah, 21589, Kingdom of Saudi Arabia

5Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

6Faculty of Business Administration, Sinai University, Cairo, Egypt

*Corresponding Author: M. Elgarhy. Email: m_elgarhy85@sva.edu.eg

Received: 12 February 2021; Accepted: 16 March 2021

Abstract: 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.

Keywords: Truncated Lomax-G family; inverse Lomax distribution; maximum likelihood method; moments

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

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

The main purpose of this work is to suggest a more flexible and enhance model called the truncated Lomax IL (TLIL) distribution. The key motivations of the TLIL distribution in the practice are (i) to enhance the flexibility of the IL distribution by using TL-G, (ii) to introduce the modified form of the IL distribution whose density can be expressed as a linear combination of the IL distributions, (iii) to develop a new model with increasing, decreasing, reversed J-shaped and upside-down shaped hazard functions and (iv) to provide better fits than the competing models. Further, we study various statistical properties and estimate the model parameters by using the maximum likelihood (ML) method. Finally, simulation studies as well as applications to real data are given.

This work can be arranged as follows. In Section 2, we give the model description of the TLIL distribution and provide a linear representation of its density and distribution functions. Section 3 provides some structural properties of the TLIL distribution. Estimation of parameters along with simulation study is considered in Section 4. Applications are given in Section 5 followed by comments and conclusions.

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,

Expression of hazard rate function (hrf) is given by:

Graphic features of the pdf and hrf plots of Y ∼ TLIL

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.

2.1 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).

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:

where

where,

Furthermore, Hassan et al. [14] provided a useful expression of cdf

where

3.1 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

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;

3.2 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 (

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.

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