key: cord-0070372-4vew6iop
authors: Falgore, Jamilu Yunusa; Isah, Muhammad Nazir; Abdulsalam, Hussein Ahmad
title: Inverse Lomax-Rayleigh distribution with application
date: 2021-11-18
journal: Heliyon
DOI: 10.1016/j.heliyon.2021.e08383
sha: b90b8b1f7332c13cfc406f10325ea7fab9123e67
doc_id: 70372
cord_uid: 4vew6iop

In this paper, an extension of Rayleigh distribution called Inverse Lomax Rayleigh (ILR) is proposed by using the Inverse Lomax generator of [13]. Properties of ILR were derived. This includes the complete and incomplete moments, entropy, distribution of order statistics, and quantile function. A simulation study was presented to explore the properties of the estimates. This shows that they are unbiased, consistent, and efficient. An application to fatigue data shows the flexibility of ILR distribution, as it outperforms all the comparators with minimum values of all the measures.

Using probability distributions to represent real-life situations is one of the most important tasks of a statistician. Modeling and interpreting lifetime data is essential in many practical situations, such as medical, actuarial science, engineering, and finance. In recent decades, this has prompted academics to focus on developing families of probability distributions.

Some of the recent families of distributions in the literature include Kumaraswamy Poisson G by [11] , Zubair G by [3] , Beta Poisson G by [15] , Extended Exp G by [8] , Inverse Lomax G by [13] , Burr X Exponential G by [24] , Odd Log-Logistic Lindley G by [7] , Weibull Marshall-Olkin Lindley G by [2] , Kumaraswamy-Odd Rayleigh-G by [14] , Inverse Lomax Exponentiated G by [12] , as well as Topp-Leone Odd Frechet G by [5] , among others. Extension of probability distributions is a regular practice in the theory of statistics. Different strategies are proposed to generalize probability distributions in the literature. This is necessary so as the addition of parameter(s) will expand the adaptability of the models to catch the multifaceted nature of the data. Several generalized (or G) classes of distributions are available in the literature, but our main focus in this paper is to extend the Rayleigh distribution with the Inverse Lomax G family.

The Cumulative distribution function and probability density function of Rayleigh distribution are given by * Corresponding author.

E-mail address: jamiluyf@gmail.com (J.Y. Falgore).

and

where > 0. The Rayleigh distribution has several applications, including life testing experiments, communication theory, technology, reliability analysis, applied statistics, medical testing, and clinical studies. With regard to this significance and the desire to give this distribution greater versatility, several researchers have proposed extensions to the Rayleigh distribution. This includes Odd Lindley-Rayleigh distribution by [16] , Lomax-Rayleigh distribution by [4] , Rayleigh-Rayleigh distribution by [6] , an extension of Rayleigh distribution by [10] , Weibull Rayleigh by [21] , Transmuted Rayleigh by [19] , New generalized Rayleigh distribution by [25] , Generalized Rayleigh distribution by [17] , among others.

Falgore and Doguwa [13] , introduced the Inverse Lomax family of distribution by adopting T-X methodology by [9] . The and of the family have the following form

and where , > 0, ( ; ) is the pdf of the baseline distribution and ( ; ) is the of the baseline distribution, and ̄( ; ) = 1 − ( ; ), and is a vector of parameter(s).

which is the same as

we arrived at

The quantile function of ILD can be derived by inverting equation (5) as follows Let

by taking log of both sides and some simplifications, we have the quantile function as

The pdf in equation (6) can be re-written in closed form. This form can be used in deriving basic properties such as moments, entropies, and distribution of order statistics.

after simplifications and replacement, we have

another identity is

after replacing the above quantity back, it becomes

where

6. Moment and moment generating function

The moments of the ILR distribution can be given in terms of the mixture representations discussed in section 5.

substituting = 2 2 2 (1 + + ), we arrived at

The moment generating function of ILR distribution can be given in terms of the moments as shown below

The entropy considered here is the Renyi entropy by [23] . The Renyi entropy for the ILR random variable is given by

by replacing back, we have

Order statistics are important in statistical theory especially in the theory of extreme value. The pdf of the ℎ order statistics of the ILR is derived here

for simplicity, we use

Here = , ( ) = 2 Γ( + 1)

Here = 1 1, ( ) = ! 2 Γ( + 1) ∑ − 2 2 2 (1+ + ) × (−1) ( − 1 − )! ! ⎡ ⎢ ⎢ ⎣ 1 + ⎛ ⎜ ⎜ ⎝ − 2 2 2 1 − − 2 2 2 ⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ −(26)

Incomplete moment of X Incomplete moments play a significance role in computing measures of statistical theory. The ℎ incomplete moments ( ) of the ILR distribution are (1 + + ) 2 +1

( 2 (1 + + )

Lorenz and Bonferroni curves are used in various areas like reliability studies, econometrics, and insurance.

Lorenz curve for ILR distribution can be given as

where ∫ 0 ( ) , is the 1 incomplete moment,

∴ ( ) = Γ( + 1) (1 + + ) 3 2 ( 2 (1 + + )

The Bonferoni curve for the ILR distribution can be given as (1 + + ) 3 2 ( 2 (1 + + )

be the observed values of n observations independently drawn from the ILR distribution with parameter vector , = ( , , ) . Then,

The log-likelihood (ll) function for denoted by ( ) can be expressed as

haven taken the partial derivatives of Equation (34) with respect to , , and we derived ( ) i.e. the Score Vector components are as follows

Setting Equations (35), (36), and (37) to zero and also solving simultaneously yields the MLE (̂) = (̂, ̂, ̂) of . However, these equations can not be solved analytically. Therefore, statistical software can be employed to solve the equations numerically through iterative methods.

In this section, a Monte Carlo simulation analysis is performed and the findings are presented to demonstrate the performance of the estimates at different true parameter values. We set the true parameter values as ( = 0.6, = 0.5, = 0.3). The numerical study is described as follows: (i). For true parameter values i.e. = ( , , ) , we simulated a random sample of size n from the ILR distribution using the quantile function defined in Equation (11) . (ii). We then Estimate the parameters of the ILR distribution from the sample using method of maximum likelihood. (iii). We conduct N=1,000 replications of steps (i) and (ii). (iv). For each of the three (3) estimated parameters of the ILR, from the N replicates, we compute the mean estimate, Bias, and MSE. The statistics are given bŷ

where the vector of estimated parameters ̂i s the maximum likelihood estimate for each iteration ( = 30, 75, 150, 300, 500, 1, 000) .

The simulation results are presented in Table 1 . The simulation study has shown that irrespective of the parameter values chosen, the Bias and MSE of the parameter estimate decay as the sample size n increases. Thus, the larger the sample size, the more consistent are the estimates of the parameters. The estimates are good as they approach the true parameter values as the sample size increases.

A demonstration of the applicability of the ILR was demonstrated using Fatigue data as in [1] . The summary of the data is as follows: n=76, minimum = 0.0251, maximum = 9.9096, mean = 1.9592, mode = 1.5, median = 1.7362, variance = 2.4774, skewness = 1.9796, as well as kurtosis = 5.1608. Table 2 summarizes the comparators with their [22] references. They are: Transmuted Generalized Rayleigh (TGR) distribution, Weibull-Rayleigh (WR) distribution, Type II Topp-Leone Generalized Inverse Rayleigh (TLR) distribution, and Rayleigh distribution, respectively. Table 4 presents the criteria for selecting fitted models. A package in R software called Adequacy Model by [18] was used in the analysis. The Maximum Likelihood estimates with the standard errors in (parentheses) for the ILR distribution are presented in Table 3 . These are the estimated values of the parameters of the ILR distribution based on the Fatigue data set. The standard errors can be used to compute the confidence interval for drawing inferences. In Table 4 , ILR seems to be the best with large P-Value and smaller AIC, CAIC, BIC, HQIC, -ll, and KS, respectively. This shows that the proposed ILR distribution fits the Fatigue data better than the compared distributions.

In this paper, a new sub-model of the Inverse Lomax G family of distributions was proposed. It is called Inverse Lomax Rayleigh (ILR) distribution. Some of the properties of ILR distribution were presented. Figs. 1, 2, and 3 show the pdfs, hazards, cdf, and survival functions, respectively. This indicates that the ILR distribution can take symmetric and asymmetric shapes depending on the values of the parameters. To test the proposed distribution, a simulation study was conducted by setting the initial values of the parameters as ( = 0.6, = 0.5, = 0.3) for 1,000 iterations. Furthermore, the ILR distribution was fitted to a Fatigue data set alongside some other distributions in the literature. Based on the results in Table 4 , the proposed ILR distribution seems to be the best. Fig. 4 also indicated that the ILR distribution fitted the data best than the other comparators.

Author contribution statement M. I. Nazir: Conceived and designed the experiments. H. A. Abdulsalam: Performed the experiments; Wrote the paper. J. Y. Falgore: Analyzed and interpreted the data; Wrote the paper.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data included in article/supplementary material/referenced in article.

The authors declare no conflict of interest.

No additional information is available for this paper.

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The authors appreciate the efforts of the editor and the reviewers.