Useful Generalization of the Inverse Lomax Distribution: Statistical Properties and Application to Lifetime Data

Various distributions have been proposed to serve as models for wide applications on data from different real-life situations through the extension of existing distribution. This has been achieved in various ways. The Lomax distribution also called “Pareto type II” is a special case of the generalized beta distribution of the second kind [1], and can be seen in many application areas, such as actuarial science, economics, biological sciences, engineering, lifetime and reliability modeling and so on [2]. This heavy-duty distribution is considered useful as an alternative distribution to survival problems and life-testing in engineering and survival analysis [3]. Inverse Lomax distribution is a member of the inverted family of distributions and discovered to be very flexible in analyzing situations with a realized non-monotonic failure rate [4]. If a random


Introduction
Various distributions have been proposed to serve as models for wide applications on data from different real-life situations through the extension of existing distribution. This has been achieved in various ways. The Lomax distribution also called "Pareto type II" is a special case of the generalized beta distribution of the second kind [1], and can be seen in many application areas, such as actuarial science, economics, biological sciences, engineering, lifetime and reliability modeling and so on [2]. This heavy-duty distribution is considered useful as an alternative distribution to survival problems and life-testing in engineering and survival analysis [3].
Inverse Lomax distribution is a member of the inverted family of distributions and discovered to be very flexible in analyzing situations with a realized non-monotonic failure rate [4]. If a random variable Χ has a Lomax distribution, then has an inverse Lomax Distribution. Thus, a random variable X is said to have an In verted Lomax distribution if the corresponding probability density function and cumulative density function are given by [5]: In literature exist several families of distribution and the references to them are listed in [6][7]. In the course of this study, the Odd Generalized Exponential family of distribution developed by [8] having probability density and cumulative distribution function given by;

The Odd Generalized Exponentiated Inverse Lomax (OGE-IL) distribution
A four parameter Odd Generalized Exponentiated Inverse Lomax distribution was investigated in this section. Obtained by substituting equation (2) into (4), the cumulative distribution function of the OGE-IL distribution is given by; The corresponding probability density function is given by; The reliability function is given by; The failure rate function/hazard function is given by; The reverse hazard function is given by; The cumulative hazard function is given by; And the odd function is given by;

Structural Properties of the Odd Generalized Exponentiated Inverse Lomax (OGE-IL) distribution
In this section we derive some of the basic theoretical/structural properties of the OGE-IL distribution, such as; its asymptotic behavior, quantile, median, Skewness and kurtosis of the Odd Generalized Exponentiated Inverse Lomax distribution.

Asymptotic Behavior of the OGE-IL Distribution
We examine the behavior of the OGE-IL distribution in equation (6) as x→0 and as x→∞ and in equation (10) as i.e. x→∞

Thus showing that the Odd Generalized Exponentiated Inverse
Lomax distribution is unimodal i.e. it has only one mode We can also deduce that for the OGE-IL Distribution, lim ( ) 1

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Copy@ Obubu Maxwell Thus, indicating that the OGE-IL distribution has a proper probability density function.

Quantile function and Median
The Quantile function of the Odd Generalized Exponentiated Inverse Lomax distribution is; The median of the Odd Generalize Exponentiated Inverse Lomax distribution can be gotten by placing u as 0.5 in equation (12) above, we obtain

Skewness and kurtosis
According to Kenney and Keeping [9], the Bowely Quartile Skewness is given as;

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Using binomial expansion and solving through, we obtained an expression for the moment as;

Order Statistics
Consider random sample denoted by The pdf of the j th order statistics for the OGE-IL distribution is given as; The minimum order statistics for the OGE-IL distribution is given as; And the maximum order statistics for the OGE-IL distribution is given as; Thus the log likelihood function becomes; By taking the derivative with respect to α, β, a, b, and equating to zero, we have; The parameters of the OGE-IL distribution is obtained by solving equation 20-24.

Simulation
In simulation studies using R software, the behavior of the OGE-IL distribution parameters was investigated. Data sets from the OGE-IL distribution were generated with replication number  Tables 1-4 below.    Thus, the estimates tend approaches the true parameter value, as the sample size increases.

Application
The Odd Generalized Exponentiated Inverse Lomax distribution was applied to a real-life dataset and its performance was compared to the Odd Generalized Exponentiated Exponential Distribution (OGE-E) [10][11][12], and the Exponential Exponentiated Distribution. The most suitable selection criteria were based on the values of the Log-likelihood, and, Akaike Information Criterion (AIC),

Data I: Strengths of Glass Fibres Dataset
The dataset obtained from Smith and Naylor [13][14][15][16] represents the strengths of 1.5 cm glass fibres, measured at the National Physi-