Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.014407

ARTICLE

A New Modified Inverse Lomax Distribution: Properties, Estimation and Applications to Engineering and Medical Data

Abdullah M. Almarashi*

Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
*Corresponding Author: Abdullah M. Almarashi. Email: aalmarashi@kau.edu.sa
Received: 24 September 2020; Accepted: 09 February 2021

1  Introduction

Inverse Lomax (IL) distribution is a very important lifetime distribution which can be used as a good alternative to the well known distributions such as gamma, inverse Weibull, Weibull and Lomax distributions. It can be considered as a member of generalized beta family of distributions. It has different applications in modelling various types of data including economics and actuarial sciences data because its hazard rate can be decreasing and upside down bathtub shaped. The random variable X is said to have an IL distribution if its probability density function (PDF) is

h(x;α,θ)=αθx-2(1+θx)-(α+1),x>0,α,θ>0.(1)

The cumulative distribution function (CDF) of (1) is

H(x;α,θ)=(1+θx)-α,(2)

where α is the shape parameter and θ is the scale parameter. For more details about the inverse distributions one can refers to [1–4]. The traditional IL distribution does not give a good fit for modeling many real life data such as in reliability and biological studies. Recently, many generalizations of IL distribution have been developed. Hassan et al. [5] introduced Weibull IL distribution and studied some of its statistical properties. Maxwell et al. [6] proposed the Marshall-Olkin IL distribution by adding a new shape parameter for more flexibility. ZeinEldin et al. [7] used the alpha power method to add a new shape parameter to the IL distribution and proposed a new distribution which called alpha power IL (APIL) distribution. In spite of these generalizations, still a new generalization of IL is needed because these distributions are not able to model various types of data.

The main objective of this paper is to propose a new form of the IL distribution by adding a new shape parameter to the CDF in (2) using the same approach of [8]. They introduced a new method to add an extra shape parameter to an existing distributions which called the logarithmic transformed (LT) method. The LT method has the following CDF and PDF

F(x;λ)=1-log[1-(1-λ)H̄(x)]log(λ)(3)

and

f(x;λ)=(λ-1)h(x)log(λ)[1-(1-λ)H̄(x)].(4)

respectively, with λ>0, λ≠1. By taking H̄(x)=1-H(x) in (3) of the IL distribution CDF in (2) we obtain a modified logarithmic transformed IL (MLTIL) distribution. Many researchers used the LT method to propose a new continuous distributions by taking the baseline H(x) of any known distribution. [9,10] introduced a generalizations for modified Weibull extension and generalized exponential distributions, respectively. [11] introduced a logarithmic inverse Lindley distribution by taking the H(x) in (3) of the inverse Lindley distribution. Also, the logarithm transformed Fréchet distribution by [12]. The overall motivations to propose the MLTIL distribution are

1.    To develop different shapes for the PDF and hazard rate function.

2.    To increase the flexibility of the traditional IL distribution in modelling different phenomenons.

3.    To model skewed data which can not be modeled by other traditional models.

4.    To increase the flexibility of the traditional IL distribution properties like mean, variance, skewness and kurtosis.

5.    Two applications showed that the MLTIL distribution provides a better fit than the traditional IL distribution and some of its generalizations.

6.    Another motivation to this article is to use six classical estimation methods to estimate the parameters in order to recommend which method provide the best estimates based on mean square error criteria and via a simulation study.

The hazard rate function of the MLTIL distribution can has decreasing or upside-down shapes depending on its shape parameters which makes the distribution is quite effectively in modelling lifetime data. It can be used as an alternative to IL and inverse Weibull distributions. Simulation results reveal that the Anderson-Darling (AD) estimators perform better than other estimators in terms of minimum mean-squared errors. Finally, the analysis of engineering and medical data sets show the ability of the MLTIL distribution to provide a better fit than some other competitive models. The rest of the paper is organized as follows: In the next Section we describe the MLTIL distribution and a mixture representation of its density. Some of its statistical properties are discussed in Section 3. Six classical estimation methods are considered in Section 4. A simulation study is conducted in Section 5. Two applications are considered in Section 6. In Section 7, the paper is concluded.

2  Model Description

In this section we introduce the MLTIL distribution. Let the random variable X follows the IL distribution with PDF and CDF, respectively, given by (1), (2), then from (1)–(3) the PDF of the MLTIL distribution is given by

f(x;λ,α,θ)=(λ-1)αθx-2(1+θx)-(α+1) log(λ)[λ-(λ-1)(1+θx)-α],x>0,λ,α,θ>0,λ≠1,(5)

and its CDF is

F(x;λ,α,θ)=1-log[λ-(λ-1)(1+θx)-α] log(λ).(6)

The survival function (SF) is given by

S(x;λ,α,θ)=log[λ-(λ-1)(1+θx)-α] log(λ)(7)

and the hazard rate functions is

h(x;λ,α,θ)=(λ-1)αθx-2(1+θx)-(α+1)[λ-(λ-1)(1+θx)-α] log[λ-(λ-1)(1+θx)-α].(8)

For some selected values of the parameters λ, α and θ, Fig. 1 indicates the various shapes of the PDF of the MLTIL distribution, while Fig. 2 shows the different shapes of the hazard rate function. These Figure indicate the flexibility of the MLTIL distribution to model right skewed data as well as the data with decreasing and upside down bathtub shaped.

Figure 1: Plots of the MLTIL density for different values of λ, α and θ

Figure 2: Plots of the MLTIL hazard rate function for different values of λ, α and θ

Now, we can obtain an useful representation for the PDF and CDF of the MLTIL distribution. Using the series representation in the form

(1-q)n-1=∑j=0∞(-1)j(n-1j)qj,if|q|<1,n>0positive real non-integer.(9)

Applying (9) for the PDF in (1), we can obtain

f(x)=∑k=0∞ϑkh(x;α(k+1),θ),(10)

where h(x;α(k+1),θ) denotes the PDF of the IL distribution with scale parameter θ and shape parameter α(k+1) and

ϑk=λ(k+1)log(λ)(λ-1λ)k+1.

Different structural properties of the MLTIL distribution can be determined using this representation. By integrating (10), the CDF of X is given by

F(x)=∑k=0∞ϑkH(x;α(k+1),θ),(11)

where H(.) is the CDF of the IL distribution with scale and shape parameters θ and shape parameter α(k+1), respectively.

3  Properties of the MLTIL Distribution

In this section some statistical properties of the MLTIL distribution are obtained including quintile function, moments, incomplete moments, conditional moments and entropies.

3.1 Quantile Function

For the MLTIL distribution the quantile function, say x = QMLTIL(q), can be obtained by inverting (6) as

F-1(q)=QMLTIL(q)=θ(λ-1λ(1-λ-q))1α-1.(12)

We can easily generate X by taking q as a uniform random variable in (0,1). Based on (12), one can compute the measures of skewness (Sk) and kurtosis (Ku) as follows:

Sk=Q(34)+Q(14)-2Q(12)Q(34)-Q(14) and Ku=Q(78)-Q(58)+Q(38)-Q(18)Q(68)-Q(28), respectively, where Q(⋅) refers to the quantile function.

3.2 Moments

The nth moment of MLTIL distribution can be derives as μn′=E(Xn)=∫-∞∞xnf(x)dx=∑k=0∞ϑk∫0∞xnh(x;α(k+1),θ)dx=∑k=0∞ϑkα(k+1)θnB(1-n,α(k+1)+n),n=1,2,3,… where B(a,b) is the beta function, Γ(0)=-γ and Γ(-n)=(-1)nn!ϕ(n)-(-1)nn!γ for n=1,2,3,…,γ denotes Euler’s constant and ϕ(n)=∑j=1n1j. For more details see [13]. Similarly, for the MLTIL distribution we can obtain the nth inverse moment as follows: E(1Xn)=∑k=0∞ϑkB(1+n,α(k+1)-n)θn.

In particular,

μ1′=E(X)=∑k=0∞ϑkα(k+1)θΓ(0), where γ denotes Euler’s constant. μ2′=E(X2)=∑k=0∞ϑkα(k+1)(α(k+1)+1)θ2Γ(-1) and var(X)=μ2′-(μ1′)2.

The rth central moment μr of X is derived as

μn=E(X-μ1′)n=∑k=0n(-1)r-k(nk)μn-k′(μ1′)k.

The Sk and Ku measures can be computed using the following expressions:

Sk=μ3′−3μ2′μ+2μ3(μ2′−μ2)32,Ku=μ4′−4μ3′μ+6μ2′μ2−3μ4(μ2′−μ2)2.

The following propositions are a description of three different types of moments such as incomplete moments, moment generating function (mgf) and conditional moment.

prop 3.1. If X~MLTIL, then the incomplete moments of X is given as follows:

IX(t)=∑k=0∞ϑk∫−∞txnh(x;α(k+1),θ)dx            =∑k=0∞ϑk∫0txn(αθ(k+1)x−2(1+θx)−α(k+1)−1)dx            =αθ∑k=0∞ϑk(k+1)∫0txα(k+1)+n−1(θ+x)−α(k+1)−1dx            =∑k=0∞ϑkα(k+1)tn(tθ)α(k+1)α(k+1)+nF2:1(α(k+1)+1;α(k+1)+n;α(k+1)+n+1;−tθ),(13)

where, F2:1(u;v;c;y)=1B(u,c-v)∫01zv-1(1-z)c-v-1(1-zy)udz is the hypergeometric function see [14]. See also [15–17].

The first incomplete moment, I1(t), follows from Eq. (13) with n = 1.

prop 3.2. If X~MLTIL, then the mgf of X is

MX(t)=∑j=0∞tjj!E(Xj)=∑j=0∞∑k=0∞ϑk(θt)jj!α(k+1)B(1-j,α(k+1)+j).

prop 3.3. If X~MLTIL, then the characteristic generating function (cgf) of X is

MX(t)=∑j=0∞(it)jj!E(Xj)=∑j=0∞∑k=0∞ϑk(θit)jj!α(k+1)B(1-j,α(k+1)+j).

prop 3.4. If X~MLTIL, then the conditional moment of X is

CX(t)=E(E(Xn∣X>x)=ψn(x)1−F(x), where ψn(x)=∑k=0∞ϖk∫x∞ynh(x;α(k+1),θ)dy=∑k=0∞ϑkα(k+1)θ(-θ)n-1B-θx(1-n,-α(k+1)), where, Bz(a,b)=∫0zya-1(1-y)b-1dy is the incomplete beta function.

3.3 Entropies

Entropy has been used in areas like in physics (sparse kernel density estimation), medicin (molecular imaging of tumors) and engineering (measure the randomness of systems). The entropy is a measure of variation of the uncertainty. The Rényi entropy (RE) of order ρ is defined as

IR(ρ)=11-ρlog(∫fρ(x)dx).(14)

For the MLTIL distribution in (1), we can write

IR(ρ)=11−ρlog{ ((λ−1)αθlog(λ))ρ∫0∞(x−2(1+θx)−(α+1)λ−(λ−1)(1+θx)−α)ρdx }, ρ≥0, ρ≠1           =11−ρlog{ ((λ−1)αλθlog(λ))ρ∑k=0∞((λ−1)λ)kΓ(ρ+k)k!B(2ρ−1,α(k+ρ)−ρ+1). }.

The ρ-entropy, Hρ-R(x), can be written as

Hρ-R(ρ)=11-ρlog(1-∫0∞fρ(x)dx).(15)

Shannon’s entropy (SE) is defined as E[-logf(X)] and it is a special case of (15) for ρff1. Limiting ρff1 in (15) and using L´Hospital´s rule, one obtains

E[−logf(X)]=E[ −log((λ−1)αθx−2(1+θx)−(α+1)log(λ)[ λ−(λ−1)(1+θx)−α ]) ]      =−log[ (λ−1)αθlog(λ) ]−E[ log(x−2(1+θx)−(α+1)λ−(λ−1)(1+θx)−α) ]      =log[ log(λ)αθ(λ−1) ]+2E(log(x))+(α+1)E(log(1+θx))      +E(log[ λ−(λ−1)(1+θx)−α ]).

3.4 Distribution of Order Statistics

Let X(1)<X(2),…,<X(n) be the order statistics of a random sample of size n taken from the MLTIL distribution. In this case, we can write the PDF of the sth order statistic as follows

fX(s)(x)=f(x)B(s,n-s+1)∑k=0n-s(-1)k(n-sk)Fk+s-1(x),(16)

where B(., .) is the beta function. Using (1), (6) and a power series expansion, we obtain

f(x)Fk+s-1(x)=αθ(λ-1)(log(λ))k+sx-2(1+θx)-(α+1)λ-(λ-1)(1+θx)-α[log(λλ-(λ-1)(1+θx)-α)]k+s-1.(17)

From (17) and (16), we can write

fX(s)(x)=αθ(λ−1)(B(s,n−s+1)∑k=0n−s(−1)k(log(λ))k+s(n−sk)x−2(1+θx)−(α+1)(λ−(λ−1)(1+θx)−α)                   ×[ log(λλ−(λ−1)(1+θx)−α) ]k+s−1.(18)

Particularly, we can obtain the PDF of the first and last order statistics from (18), respectively, as

fX(1)(x)=αθ(λ−1)n∑k=0n−1(−1)k(log(λ))k+1(n−1[3pt]k)x−2(1+θx)−(α+1)(λ−(λ−1)(1+θx)−α)   ×[ log(λλ−(λ−1)(1+θx)−α) ]k,(19)

and

fX(n)(x)=αθ(λ-1)n(log(λ))nx-2(1+θx)-(α+1)(λ-(λ-1)(1+θx)-α)[log(λλ-(λ-1)(1+θx)-α)]n-1.(20)

3.5 Probability Weighted Moments

The (u,v)th probability weighted moments of the random variable X is defined by

ρu,v=E{XuF(X)v}=∫-∞∞xuf(x)Fv(X)dx.(21)

Based on Eq. (21), we can obtain the probability weighted moments of the MLTIL distribution as

ρu,v=αθ(λ-1)ρ(log(λ))v+1,(22)

where

ρ=∫0∞xu-2(1+θx)-(α+1)λ-(λ-1)(1+θx)-α[log(λλ-(λ-1)(1+θx)-α)]vdx.(23)

3.6 Residual Life and Reversed Failure Rate Function

The nth moment of the residual life (RL) of the random variable X is defined as follows

Rn(t)=E((X-t)n∣X>t)=11-F(t)∫t∞(x-t)nf(x)dx,n≥1

Based on Eq. (1) and applying the binomial expansion of (x − t)n, we have

Rn(t)=11−F(t)∑m=0n(−t)n−m(nm)∫t∞xnf(x)dx            =αθ(λ−1)λ∑m=0n∑k=0∞(−t)n−m(−θ)n−1(n[3pt]m)(λ−1λ)kB−θt(1−n,−(k+1)α)log(λ−(λ−1)(1+θx)−α).

The nth moment of the reversed RL can be derived using the general formula

mn(t)=E((t-X)n∣X≤t)=1F(t)∫0t(t-x)nf(x)dx,n≥1.

Based on Eq. (1) and applying the binomial expansion of (t − x)n, we can write

mn(t)=αθ(λ-1)λlog(θ)F(t)∑m=0n(-t)n-m(nm)∫0txnf(x)dx=α(λ-1)λlog(θ)F(t)∑m=0n∑k=0∞(-1)n-m(nm)(λ-1λ)kt2n-m+α(k+1)θα(k+1)(n+α(k+1))×F2:1(a(k+1)+1;n+a(k+1);n+a(k+1)+1;-tθ).

3.7 Stress-Strength Model

Let Y1~MLTIL(λ1,α1,θ1) and Y2~MLTIL(λ2,α2,θ2). If Y1 represents stress and Y2 represents strength, then the stress-strength parameter, denoted by R, for the MLTIL distribution is given by

R=P(Y2>Y1)=∫0∞F1(y)dF2(y)=1−∫0∞F¯1(y)dF2(y) =1−(λ2−1)α2θ2log(λ1)log(λ2)∫0∞y−2(1+θ2y)−(α2+1)log[ λ1−(λ1−1)(1+ θ 1x)−α1 ]λ2−(λ2−1)(1+θ2x)−α2dy =1−(λ2−1)α2θ2log(λ1)log(λ2)∫0∞y−2(1+θ2y)−(α2+1)[ log(λ1)+log(1−(1−λ1)(1+ θ 1 x)−α1) ]λ2−(λ2−1)(1+θ2x)−α2dy =1−(λ2−1)α2θ2log(λ2)∫0∞y−2(1+θ2y)−(α2+1)λ2−(λ2−1)(1+θ2x)−α2dy −(λ2−1)α2θ2log(λ1)log(λ2)∫0∞y−2(1+θ2y)−(α2+1)log(1−(1−λ1)(1+ θ 1x)−α1)λ2−(λ2−1)(1+θ2x)−α2dy.

Using series expansion in the last equation, we obtain

R=1−(λ2−1)α2θ2log(λ2)∑m=0∞(λ2−1λ2)m∫0∞y−2(1+θ2y)−α2(m+1)−1dy−(λ2−1)α2θ2log(λ1)log(λ2) ×∑m=0∞∑k=0∞(−1)k−1k(λ2−1λ2)m(1−λ1λ1)k∫0∞y−2(1+θ1y)−α1k(1+θ2y)−α2(m+1)−1dy =1−(λ2−1)α2θ2log(λ2)∑m=0∞(λ2−1λ2)m1λ2θ2(m+1)−(λ2−1)α2θ2log(λ1)log(λ2)∑m=0∞∑k=0∞∑j=0∞(−1)k−1(−θ1)jk ×(λ2−1λ2)m(1−λ1λ1)k(α1k+j−1j)∫0∞y−2−j(1+θ2y)−α2(m+1)−1dy =1−(λ2−1)α2θ2log(λ2)∑m=0∞(λ2−1λ2)m1λ2θ2(m+1)−(λ2−1)α2log(λ1)log(λ2)∑m=0∞∑k=0∞∑j=0∞(−1)k−1(−θ1θ2)jk ×(λ2−1λ2)m(1−λ1λ1)k(α1k+j−1j)B(j+1,α2(m+1)−j).

4  Estimation

In this section, six estimation methods are considered to estimate the MLIIL distribution parameters.

4.1 Maximum Likelihood Estimation

Using a random sample of size m taken from the MLTIL distribution, then based on (5) the log-likelihood function can be written as

E1(ϖ)=mlog((λ-1)αθlog(λ))-2∑j=1m log(xj)-(α+1)∑j=1m log(1+θxj)-∑j=1m log[λ-(λ-1)(1+θxj)-α],(24)

where ϖ=(λ,α,θ)T. The maximum likelihood estimates (MLEs) of λ, α, and θ, denoted by λ^ML, α^ML, and θ^ML, are the solution of the following three equations

∂E1(ϖ)∂λ=mλ-1-mλlog(λ)-∑j=1m1-(1+θxj)-αλ-(λ-1)(1+θxj)-α=0, ∂E1(ϖ)∂α=mα-∑j=1m log(1+θxj)-∑j=1m(λ-1)(1+θxj)-α log(1+θxj)λ-(λ-1)(1+θxj)-α=0 and ∂E1(ϖ)∂θ=mθ-(α+1)∑j=1m1xj(1+θxj)+α∑j=1m(λ-1)xj-1(1+θxj)-α-1λ-(λ-1)(1+θxj)-α=0.

It is observed that these equations cannot be solved analytically for λ, α, and θ, therefore to obtain λ^ML, αML, and θML one can use any numerical technique for this purpose.

4.2 Percentile Estimation

Kao et al. [18,19] originally proposed the percentile method of estimation and recently used by many authors to obtain the parameters of some proposed distributions, see for example [20,21]. It is very useful when the distribution has a closed form quantile function. Since the MLTIL distribution has an explicit quantile function as in (12), then we can obtain the percentile estimates (PEs) of λ, α, and θ, denoted by λ^P, α^P, and θ^P, by minimizing

E2(ϖ)=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]2,(25)

where x(j) is the ordered observation of xj, j=1,…,n and qj = j/(m + 1). Instead of minimizing (25) with respect to λ, α, and θ, we can obtain λ^P, α^P, and θ^P by solving

∂E2(ϖ)∂λ=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]Φ1j(ϖ)=0, ∂E2(ϖ)∂α=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]Φ2j(ϖ)=0 and ∂E2(ϖ)∂θ=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]Φ3j(ϖ)=0, where Φ1j(ϖ)=θ(1-λλvj)1/α-1[1-λ-1λ+qj(λ-1)λqj+1vj]αλvj[(1-λλvj)1/α-1]2, Φ2j(ϖ)=θ(1-λλvj)1/α log(1-λλvj)α2[(1-λλvj)1/α-1]2 and Φ3j(ϖ)=[(1-λλvj)1/α-1]-1, where vj=1λqj-1.

4.3 Maximum Product of Spacing Estimation

Cheng et al. [22,23] introduced the method of maximum product of spacing (MPS) to obtain the estimates of the continuous univariate distributions parameters as an alternative to the method of maximum likelihood. Let Ψi(ϖ) is the uniform spacing defined as follows: Ψj(ϖ)=F(x(j)∣ϖ)−F(x(j−1)∣ϖ)               =log[ λ−(λ−1)(1+θx(j−1))−α ]−log[ λ−(λ−1)(1+θx(j))−α ]log(λ).