Intelligent Automation & Soft Computing

Inference of Truncated Lomax Inverse Lomax Distribution with Applications

Abdullah Ali H. Ahmadini1, Amal Hassan2, M. Elgarhy3,*, Mahmoud Elsehetry4, Shokrya S. Alshqaq5 and Said G. Nassr6

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

1  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:

G(y;b,δ)=(1+by)δ,y,b,δ>0, (1)

where, b and δ are the scale and shape parameters respectively. The probability density function (pdf) of the IL distribution is as follows:

g(y;b,δ)=bδy2(1+by)δ1,y,b,δ>0. (2)

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,

F(y;α,ς)=0G(y;ς)α(1+t)(α+1)12αdt=A(1(1+G(y;ς))α), (3)

where α>0,A=(12α)1, ς is the parameter vector and G(x;ς) is the cdf of any distribution and the truncated Lomax pdf is the generator. The pdf associated with Eq. (3) is given by

f(y;α,ς)=αAg(y;ς)(1+G(y;ς))α1. (4)

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.

2  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:

F(y;Θ)=A[1(1+(1+by)δ)α],y,b,δ,α>0, (5)

where, Θ(b,α,δ) is the set of parameters. The pdf associated with Eq. (5) is given by

f(y;Θ)=Aαδby2(1+by)δ1(1+(1+by)δ)α1,y,b,δ,α>0. (6)

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

h(y;Θ)=Aαδby2(1+by)δ1(1+(1+by)δ)α1{1A[1(1+(1+by)δ)α]}1. (7)

Graphic features of the pdf and hrf plots of Y ∼ TLIL (α,δ,b) are represented in Fig. 1


Figure 1: Plots of the pdf and hrf for TLIL model

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

Q(u)=b{[(1uA)1α1]1δ1}1, (8)

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).

2.2 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:

f(y;α,ς)=j=0τjg(y;ς)G(y;ς)j, (9)

where τj=Aα(1)j(α+1j). Hence, a useful expression of the TLIL distribution is produced by inserting pdf Eq. (1) and cdf Eq. (2) in Eq. (9) as follows:

f(y;Θ)=j=0τjbδy2(1+by)δ(j+1)1=j=0τj(j+1)g(y;b,δ(j+1)), (10)

where, g(y;b,δ(j+1)) is the pdf of the IL distribution with parameters b and δ(j+1). 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 [F(y;ς)]h, where h is an integer, as follows:

[F(y;ς)]h=l=0SlG(y;ς)l, (11)

where Sl=Ahi=0h(1)i+l(hi)(αi+l1l). Hence, as mentioned in Eq. (11), the cdf [F(y;Θ)]h, of the TLIL takes the following form:

[F(y;Θ)]h=l=0Sl(1+by)δl. (12)

3  Statistical Properties

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 Υk,h, is defined as:

Υk,h=E(Yk(F(y)h)=0yk(F(y)hf(y)dy. (13)

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

Υk,h=l,j=0Slτjbδ0yk2(1+by)δ(j+l+1)1dy=l,j=0SlτjδΓ(1k)Γ(δ(j+l+1)+k)Γ(δ(j+l+1)+1), (14)

where; Γ(0)=γ , Γ(k)=(1)kk!φ(k)(1)kk!γfork=1,2,..., γ denotes Euler's constant and φ(k)=i=1k1i (see [17]).

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:

μn=j=0τjbδ0yn2(1+by)δ(j+1)1dy=j=0τjδΓ(1n)Γ(δ(j+1)+n)Γ(δ(j+1)+1), (15)

The first four moments are obtained by setting n =1, 2, 3 and 4 in Eq. (15). The n-th central moment (μn ) of Y is given by

μn=E(Yμ1)n=i=0n(1)i(ni)(μ1)iμni (16)

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

Mx(t)=j=0tnτjδΓ(1n)Γ(δ(j+1)+n)n!Γ(δ(j+1)+1),n=1,2,..., (17)

The m–th inverse moment, for the TLIL distribution is derived by using pdf Eq. (10) as follows:

E(Ym)=j=0τjδΓ(1+m)Γ(δ(j+1)m)Γ(δ(j+1)+1);m=1,2,... (18)

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:

ωn(t)=j=0τjbδ0tyn2(1+by)δ(j+1)1dy=j=0τjδbnB(1n,δ(j+1)+n,tt+b), (19)

where, B(.,.,t) is the incomplete beta function.

3.3 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:

Λr(t)=1F¯(t)t(yt)rf(y)dy=i=0r(1)ri(in)triF¯(t)tyrf(y)dy. (20)

The r-th moment of the residual life of the TLIL distribution is obtained by substituting Eq. (10) in Eq. (20) as follows

Λr(t)=δF¯(t;Θ)j=0i=0r(1)ri(ir)triτjbrB(1r,r+δ(j+1),bt+b) (21)

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