A New Three-Parameter Inverse Weibull Distribution with Medical and Engineering Applications

1  Introduction

Recently, many statistical distributions have been proposed by statisticians. The necessity to develop new distributions appear either due to practical investigations or theoretical concerns or both. Many applications in domains including dependability analysis, finance and risk modelling, insurance, and biological sciences, among others, have indicated in recent years that data sets that follow standard distributions are more often the exception than the usual. Because modified distributions are necessary, significant progress has been made in the modification of several classic distributions and their efficient utilization to challenges in these fields. Lately, specifically since 1980, the direction of statisticians to develop new distributions is on adding parameter(s) to some existing distributions or integrating classical distributions, see Lee et al. [1]. This addition of parameter(s) has been demonstrated valuable in analysing tail properties and also for enhancing the goodness of fit of the family under consideration. For more details about the methods of adding parameters, one can refer to Mudholkar et al. [2], Marshall et al. [3] and Mahdavi et al. [4].

The inverse-Weibull (IW) distribution has many vital applications in life testing and reliability studies. The IW distribution bears some other names like Fréchet distribution. The IW distribution is viewed as reciprocal to the usual Weibull distribution, see Drapella [5] and Mudholkar et al. [6], it is used to describe the degradation of mechanical components in diesel engines, such as the crankshaft and pistons, see Keller et al. [7]. Several academics proposed several modified variations of the IW distribution to increase its flexibility. Khan [8] looked at the beta IW distribution, for example. The generalized IW distribution was proposed by de Gusmao et al. [9], the modified IW distribution was proposed by Khan et al. [10], the Kumaraswamy generalized IW distribution was proposed by Oluyede et al. [11], and the Kumaraswamy modified IW distribution was proposed by Aryal et al. [12]. In addition, Okasha et al. [13] suggested the Marshall-Olkin IW distribution, Basheer [14] proposed the alpha power IW distribution, Dey et al. [15] discussed the logarithmic transformed IW distribution, and Afify et al. [16] suggested the logarithmic transformed IW distribution.

Assume that X is a random variable follows IW distribution with δ as the scale parameter and θ as the shape parameter, where δ,θ>0. Then, the probability density function (PDF) of X can be written as follows:

g(x;δ,θ)=δθx−θ−1e−δx−θ,(1)

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

G(x;δ,θ)=e−δx−θ,x>0.(2)

See Johnson et al. [17] for further information on the IW distribution. In this study, we offer a new three-parameter IW distribution utilizing the same idea proposed by Alotaibi et al. [18]. The modified alpha power transformed IW (MAPTIW) distribution is the name given to the new distribution. Two shape parameters and one scale parameter are included. It includes some lifetime distributions, such as IW, inverted exponential and inverse Rayleigh distributions. Recently, The alpha power transformation approach was used by Alotaibi et al. [18] to suggest a new method for adding a new shape parameter to a baseline distribution. Modified alpha power transformation (MAPT) was the name given to the approach they proposed. The CDF and PDF of MAPT method are expressed as

F(x)=αG(x)−1(α−1)(1+α−αG(x)),α>0,α≠1(3)

and

f(x)=α1+G(x)log⁡(α)g(x)(α−1)(1+α−αG(x))2,α>0,α≠1,(4)

where α is a new shape parameter. This article seeks to furnish a new lifetime model with the lowest number of parameters, yet satisfactory to depict more complicated data patterns; one scale and two shape parameters are involved to accommodate for decreasing and upside-down bathtub-shaped failure rates. The MAPTIW distribution is motivated by the following causes: (1) It is a comprehensive distribution that holds some well-known lifetime distributions. (2) It is established in Section 2 that the MAPTIW distribution can be considered as a mixture of the traditional IW distribution. (3) The MAPTIW distribution can deliver decreasing and unimodal hazard rates which makes it outstanding from some other distributions. (4) The MAPTIW can be viewed as a practical model for modelling skewed data which can not be adequately modelled by other distributions. (5) The MAPTIW distribution furnishes more significant flexibility when compared with many well-known distributions in two real data applications. We seek that it will attract wider applications in medicine, engineering and other fields of research.

The remainder of this article is structured as follows. The MAPTIW distribution is introduced and its mixed representation is derived in Section 2. In Section 3, we will look at some of the MAPTIW’s most important characteristics. Section 4 shows how to estimate unknown parameters using maximum likelihood estimation. We present the simulation results in Section 5 to show how well the estimations performed. Three real-life data sets are used to assess the importance of the MAPTIW distribution in Section 6. Section 7 presents the conclusion.

2  MAPTIW Distribution

We may define the CDF of the MAPTIW distribution in the following form by inserting the CDF of the IW distribution obtained by Eq. (2) into the CDF of the MAPT generator given by Eq. (3)

F(x;α,δ,θ)=αe−δx−θ−1(α−1)(1+α−αe−δx−θ),x,α,δ>0,θ>0,α≠1.(5)

The PDF that corresponds to the CDF in Eq. (5) can be expressed as follows:

f(x;α,δ,θ)=δθlog⁡(α)x−θ−1e−δx−θα1+e−δx−θ(α−1)(1+α−αe−δx−θ)2,x,α,δ>0,θ>0,θ>0,α≠1.(6)

MAPTIW distribution has the following special sub-models. If α→1, it reduces to the IW distribution. If θ=1 and α→1, it reduces to the inverted exponential distribution. If θ=2 and α→1, it reduces to the inverse Rayleigh distribution. Also, when θ=1, it reduces to the MAPT-inverted exponential distribution. Finally, when θ=2, it simplifies to the MAPT-Rayleigh distribution. The MAPTIW distribution’s reliability function (RF) and hazard rate function (HRF) are

R(x;α,δ,θ)=1−αe−δx−θ−1(α−1)(1+α−αe−δx−θ),(7)

and

h(x;α,δ,θ)=θδlog⁡(α)x−θ−1e−δx−θαe−δx−θ(1+α−αe−δx−θ)(α−αe−δx−θ),(8)

respectively. Henceforth, we will use X to refer to the random variable that possesses the PDF in Eq. (6).

The various plots of the PDF of the MAPTIW distribution with δ=1 in all settings and different values for α and θ are shown in Fig. 1. The newly introduced parameter α offered more extra flexibility to the PDF of the MAPTIW distribution, as seen in Fig. 1. The MAPTIW distribution is right-skewed, and because of this, it is more suited to modelling right-skewed data than other competing distributions. The many forms of the HRF of the MAPTIW distribution are shown in Fig. 2. It shows that the MAPTIW distribution’s HRF can be decreasing or unimodal in form.

Figure 1: MAPTIW PDF plots

Figure 2: HRF plots for the MAPTIW distribution

The PDF in Eq. (6) might be used to obtain various statistical characteristics of the MAPTIW distribution. Finding the PDF’s mixture structure in Eq. (5) is a simple method. Observe the power series below:

αy=∑m=0∞(log⁡α)mm!ym,(9)

where y is a variable and log⁡(α) is a constant, as well as the generalized binomial expansion, which has the form

(1−y)−a=∑k=0∞Γ(j+a)Γ(a)j!yk,|y|<1,k>0,(10)

where y is a variable, a is a constant and Γ(a) is the gamma function defined as Γ(a)=∫0∞za−1e−zdz.

When Eqs. (9) and (10) are applied to the PDF in Eq. (6), the following is a suitable linear representation for the PDF

f(x;α,δ,θ)=∑m=0∞ϖmg(x;(m+1)δ,θ),(11)

where g(x;(m+1)δ,θ) is the PDF of the IW distribution with scale parameter (m+1)δ and shape parameter θ, and

ϖm=∑k=0∞α(k+1)m+1(log⁡α)m+1(α−1)(α+1)k+2(m+1)!.

On the other hand, the expansion of the CDF in Eq. (5) may be provided as by integrating Eq. (11) as

F(x;α,δ,θ)=∑m=0∞ϖmG(x;(m+1)δ,θ).(12)

Hence G(x;(m+1)δ,θ) is the CDF corresponding to g(x;(m+1)δ,θ).

3  MAPTIW Distribution’s Properties

The quantile function, moments and stress-strength parameter are some of the properties of the MAPTIW distribution that we derive in this part.

3.1 Quantile Function

The quantile function for the MAPTIW distribution may be calculated as follows using Eq. (5):

xu=QMAPIW(u)=[−1δlog⁡{[log⁡(1+u(α2−1)1+u(α−1))log⁡α]}]−1θ,0<u<1.(13)

One can employ Eq. (13) to simulate the random variable X by taking u∼Uniform(0,1).

3.2 Moments

Moments play an essential part in statistics. Numerous vital aspects of any statistical distribution can be explored via moments. For the MAPTIW distribution, the nth moment can be obtained as

μn′=E(Xn)=∫−∞∞xnf(x)dx=∑m=0∞ϖm∫0∞xng(x;(m+1)δ,θ)dx=∑m=0∞ϖm[(m+1)δ]nθΓ(1−nθ),nθ≤1,n=1,2,3,…,n,

where Γ(.) is the gamma function. In particular, the first two moments can be obtained as

μ1′=E(X)=∑m=0∞ϖm[(m+1)δ]1θΓ(1−1θ),θ≥1,

and

μ2′=E(X2)=∑m=0∞ϖm[(m+1)δ]2θΓ(1−2θ),θ≥2,

The following formulae can also be used to obtain the variance (var), skewness (Sk), and kurtosis (Ku):

var(X)=μ2′−(μ1′)2,

Sk=[μ3′−3μ2′μ+2μ3]/[μ2′−μ2]3/2,

and

Ku=[μ4′−4μ3′μ+6μ2′μ2−3μ4]/[μ2′−μ2]2.

where μ=μ1′. The propositions that follow are an explanation of three different types of moments, namely, incomplete moments (IMs), moment generating function (MGF), characteristic generating function (CGF) and conditional moments (CMs).

Prop 3.1. If X∼MAPTIW, then the IMs of X is given as follows:

ψr(t)=∑m=0∞ϖm[(m+1)δ]rθγ(1−rθ,(m+1)δtθ),r<θ,(14)

where γ(.,.) is the incomplete gamma function.

Prop 3.2. If X∼MAPTIW, then the MGF, denoted by M(t), and the cumulant generating function, denoted by K(t), of X are, respectively, given by

MX(t)=∑k=0∞∑m=0∞ϖmtkk![(m+1)δ]kθΓ(1−kθ),

and

KX(t)=log⁡[∑k=0∞∑m=0∞ϖmtkk!((m+1)δ)kθΓ(1−kθ)].

Prop 3.3. If X∼MAPTIW, then the CGF of X is

MX(t)=∑k=0∞∑m=0∞ϖm(it)kk![(m+1)δ]kθΓ(1−kθ),k<θ.

Prop 3.4. If X∼MAPTIW, then the CMs of X is

CX(t)=E(Xn|X>x)=ψn(x)1−F(x),

where ψn(t) is the nth IM of X.

3.3 Entropies of MAPTIW Distribution

Entropy is used to measure uncertainty. It recreates a vital role in the area of engineering, probability, statistics, information theory and financial analysis. For example, Gençay et al. [19] delivered a relative analysis of the stock market during the 1987 and 2008 financial crises. Rashidi et al. [20] examined and simulated the heat transfer flow employing entropy generation in solar still. The Rényi entropy (RE) for the MAPTIW distribution may be calculated using Eq. (6) as follows:

Iρ(x)=11−ρlog⁡{(δθlog⁡αα−1)ρ∫0∞x−ρ(θ+1)α(1+e−δx−θ)ρe−δρx−θ(1+α−αe−δx−θ)2ρdx}=11−ρlog⁡{(δθlog⁡αα−1)ρ∑s=0∞∑r=0∞(2ρ+s−1s)(ρlog⁡α)rr!(1+α)2ρ+s∫0∞x−ρ(θ+1)e−δρx−θ}×[1+s+ρρe−δx−θ]rdx=11−ρlog⁡{(δθlog⁡αα−1)ρ∑s=0∞∑r=0∞∑u=0r(ru)(2ρ+s−1s)(ρlog⁡α)rr!(1+α)2ρ+s(s+ρρ)u×∫0∞x−ρ(θ+1)e−(u+1)δx−θdx}=11−ρlog⁡{(δθlog⁡αα−1)ρ∑s=0∞∑r=0∞∑u=0r∑v=0∞(−1)vv!(ru)(2ρ+s−1s)(ρlog⁡α)rr!(1+α)2ρ+s(s+ρρ)u×Γ(ρ(θ+1)−1θ)[(u+1)δ]ρ(θ+1)−1θ}.(15)

The Shannon entropy can also be calculated as

E[−log⁡f(X)]=−log⁡[δθlog⁡αα−1]+(θ+1)E[log⁡x]+δE[x−θ]−(log⁡α){1+E[e−δx−θ]}+2E[log⁡(1+α−αe−δx−θ)]=−log⁡[δθlog⁡αα−1]−(θ+1)θ∑m=0∞ϖm(log⁡(δ(m+1))+γ)−δ∑m=0∞ϖmm+1−(log⁡α){1−∑m=0∞ϖm(m+1)m+2}+2E[log⁡(1+α−αe−δx−θ)].

3.4 Order Statistics

Let X1,…,Xn  be a MAPTIW random sample of size n. Assume that X(1),…,X(n)  are the corresponding order statistics. The ith order statistic’s PDF is then presented as follows:

fX(i)(x)=f(x)B(i,n−i+1)∑a=0n−i(−1)a(n−ia)Fj+i−1(x),(16)

The beta function is represented as B(.,.). Using Eqs. (5), (6) and after some simplifications, we can obtain

fX(i)(x)=∑l=0∞ωlg(x;(l+1)δ,θ),(17)

where g(x;(l+1)δ,θ) is the PDF of the IW distribution with scale parameter (l+1)δ and shape parameter θ, and

ωl=αlog⁡α(α2−1)i+jB(i,n−s+1)∑a=0n−i∑k,m,n=0∞(−1)i+j+m+n+a−2(α+1)k+1(n+1)((k+m+1)log⁡α)l(l+1)!×Γ(i+j+k+1)Γ(i+j+1)(i+j−1m)(n−ia)(ln).(18)

Using Eq. (17), we can derive the properties of X(i) directly from the properties of Z(l+1), where Z(l+1) follows IW distribution with scale parameter (l+1)δ and shape parameter θ. Also, the qth moments of X(i) can be derive as follows:

E(X(i)q)=∑l=0∞ωlE(Z(l+1)q).(19)

3.5 Moments of Residual Life and Reversed Failure Rate Function

The mean residual lifetime has been studied by engineers and survival analysts. Given a feature or a system is of age t, the remaining lifetime after t is a random variable. The expected value of this random residual lifetime is named the mean residual life. The mean residual lifetime is often a significant measure for discovering an optimal burn-in time for a feature. The rth moment of the residual life of X is given by

Rr(t)=E((X−t)r|X>t)=11−F(t)∫t∞(x−t)rf(x)dx,r≥1.

Using Eqs. (5) and (6), and the binomial expansion of (x−t)r, we get

Rr(t)=11−F(t)∑k=0r(−t)r−k(rk)∫t∞xrf(x)dx=11−F(t)∑k=0r∑m,j=0∞ϖm,j(−t)r−k(rk)∫t∞xrg(x;(m+1)δ,θ)dx=(α−1)α{1+(α−αe−δt−θ)−1}∑k=0r∑m,j=0∞ϖm,j(−t)r−k(rk)∫t∞xrg(x;(m+1)δ,θ)dx.

The mean residual lifetime can be obtained from the last equation by setting r=1. The rth moment of the reversed residual lifetime can be obtained as follows:

mr(t)=E((t−X)r|X≤t)=1F(t)∫0t(t−x)rf(x)dx,r≥1.

Based on Eqs. (5), (6) and using the binomial expansion of (t−x)r, we can write

mr(t)=(α−1)(1+α−αe−δx−θ)αe−δx−θ−1∑k=0r∑m,j=0∞ϖm,j(−t)r−k(rk)∫0txrg(x;(m+1)δ,θ)dx.

3.6 Stress-Strength Model

Let Y1∼MAPTIW(α1,θ1,δ1) and Y2∼MAPTIW(α2,θ2,δ2). If Y1 describes the stress and Y2 describes the strength, then the MAPTIW distribution’s stress-strength parameter, indicated by R. It has a wide range of applications in engineering and research. We will concentrate on two different cases.

Case one: α1≠α2, δ1=δ2=δ and θ1=θ2=θ

R=(Y2>Y1)=∫0∞F1(y)dF2(y)dy=δθlog⁡(α2)(α1−1)(α1+1)(α2−1)(α2+1)2∫0∞y−θ−1e−δy−θα21+e−δy−θ(α1e−δy−θ−1)×(1−α1e−δy−θα1+1)−1(1−α2e−δy−θα2+1)−2dy.

Using the series expansion in Eq. (10), it follows:

R=δθlog⁡(α2)(α1−1)(α2−1)∑a=0∞∑b=0∞Ψ1(a,b)∫0∞y−θ−1e−δy−θ(α1e−δy−θ−1)α1ae−δy−θα2(b+1)e−δy−θdy.

Using the series expansion in Eq. (9), we can obtain

R=δθlog⁡(α2)(α1−1)(α2−1)∑a=0∞∑b=0∞Ψ1(a,b)∫0∞y−θ−1e−δy−θ(α1e−δy−θ−1)α1ae−δy−θα2(b+1)e−δy−θdy=δθlog⁡(α2)(α1−1)(α2−1)∑a,b=0∞∑c,d=0∞Ψ1(a,b)T2(c,d)∫0∞y−θ−1e−(c+d+1))δy−θdy=δθlog⁡(α2)(α1−1)(α2−1)∑a,b=0∞∑c,d=0∞Ψ1(a,b)Ψ2(c,d)(c+d+1))δ,

where

Ψ1(a,b)=(b+1)α2(α1+1)a+1(α2+1)b+2

and

Ψ2(c,d)=(b+1)d(log⁡(α1))c(log⁡(α2))dc!d!((a+1)c−ac).

Case two: α1≠α2, δ1≠δ2 and θ1≠θ2

R=P(Y2>Y1)=∫0∞F1(y)dF2(y)dy=δ2θ2log⁡(α2)(α1−1)(α1+1)(α2−1)(α2+1)2∫0∞y−θ2−1e−δ2y−θ2α21+e−δ2y−θ2(α1e−δ1y−θ1−1)×(1−α1e−δ1y−θ1α1+1)−1(1−α2e−δ2y−θ2α2+1)−2dy.

Using the series expansion in Eq. (10), we have

R=δ2θ2log⁡(α2)(α1−1)(α2−1)∑j=0∞∑k=0∞Ψ1(a,b)∫0∞y−θ2−1e−δ2y−θ2(α1e−δ1y−θ1−1)α1ae−δ1y−θ1α2(b+1)e−δ2y−θ2dy.

Using the series expansion in Eq. (9), we can obtain

R=δ2θ2log⁡(α2)(α1−1)(α2−1)∑a=0∞∑b=0∞Ψ1(a,b)∫0∞y−θ2−1e−δ2y−θ2(α1e−δ1y−θ1−1)α1ae−δ1y−θ1α2(b+1)e−δ2y−θ2dy=δ2θ2log⁡(α2)(α1−1)(α2−1)∑a,b=0∞∑c,d=0∞Ψ1(a,b)Ψ2(c,d)∫0∞y−θ2−1e−(cδ1y−θ1+(d+1)δ2y−θ2)dy.

4  Parameter Estimation

In this section, we consider the maximum likelihood approach to estimate the unknown parameters of the MAPTIW distribution. Furthermore, the approximate confidence intervals (ACIs) of the unknown parameters are acquired.

4.1 Maximum Likelihood Estimation

Given a random sample of size n taken from the MAPTIW distribution with PDF given by Eq. (6), we can express the log-likelihood function as

ℓ(α,δ)=nlog⁡(δθlog⁡αα−1)−(θ+1)∑i=1nlog⁡(xi)−δ∑i=1nxi−θ+(log⁡α)∑i=1n(1+e−δxi−θ)−2∑i=1nlog⁡(1+α−αe−δxi−θ).(20)

The maximum likelihood estimates (MLEs) denoted by α^, δ^ and θ^ can be got by solving the following normal equations simultaneously

∂ℓ∂α=−nα−1+nαlog⁡(α)+∑i=1n(e−δxi−θ+1)α−2∑i=1nΨα=0,(21)

∂ℓ∂δ=nδ−∑i=1nxi−θ−log⁡(α)∑i=1nxi−θe−δxi−θ−2∑i=1nΨδ=0,(22)

and

∂ℓ∂θ=nθ−∑i=1nlog⁡(xi)+δ∑i=1nxi−θlog⁡(xi)+δlog⁡(α)∑i=1nxi−θlog⁡(xi)e−δxi−θ−2∑i=1nΨθ=0,(23)

where

Ψα=1−e−δxi−θαe−δxi−θ−1α−αe−δxi−θ+1,

Ψδ=log⁡(α)xi−θe−δxi−θαe−δxi−θα−αe−δxi−θ+1,

and

Ψθ=−δlog⁡(α)xi−θlog⁡(xi)e−δxi−θαe−δxi−θα−αe−δxi−θ+1.

To get the MLEs of α, δ and θ, any numerical technique can be used to solve Eqs. (21)–(23). Using the asymptotic properties of the MLEs, we can construct the ACIs of α, δ and θ. It is known that (α,δ,θ)∼N((α^,δ^,θ^),I0−1(α,δ,θ)), where I0−1(α,δ,θ) is the asymptotic variance-covariance of the MLEs. Here, we employ the approximate asymptotic variance-covariance matrix of the MLEs denoted by I0−1(α^,δ^,θ^) which obtained based on the observed Fisher information matrix to compute the ACIs as follows:

I0−1(α^,δ^,θ^)=[−Iαα−Iαδ−Iαθ−Iαδ−Iδδ−Iδθ−Iαθ−Iδθ−Iθθ](α,δ,θ)=(α^,δ^,θ^)−1=[var^(α^)cov^(α^,δ^)cov^(α^,θ^)cov^(δ^,α^)var^(δ^)cov^(δ^,θ^)cov^(θ^,α^)cov^(θ^,δ^)var^(θ^)].(24)

where Iαα,Iδα=Iαδ,Iαθ=Iθα,Iδθ=Iθδ,Iδδ and Iθθ are derived from Eq. (20) as

Iαα=n(1(α−1)2−log⁡(α)+1α2log2⁡(α))−∑i=1n(e−δxi−θ+1)α2−2∑i=1nΨαα,

Iδδ=−nδ2+log⁡(α)∑i=1nxi−2θe−δxi−θ−2∑i=1nΨδδ,

Iθθ=−nθ2−δ∑i=1nxi−θlog2⁡(xi)+δlog⁡(α)∑i=1n(δ−xiθ)xi−2θlog2⁡(xi)e−δxi−θ−2∑i=1nΨθθ,

Iαδ=−1α∑i=1nxi−θe−δxi−θ−2∑i=1nΨαδ,

Iαθ=δ∑i=1nxi−θlog⁡(xi)e−δxi−θα−2∑i=1nΨαθ,

and

Iδθ=∑i=1nxi−θlog⁡(xi)+log⁡(α)∑i=1n(xiθ−δ)xi−2θlog⁡(xi)e−δxi−θ−2∑i=1nΨδθ,

where

Ψ≡Ψ(α,δ,θ)=log⁡(α−αe−δxi−θ+1),

Ψαα≡−e−δxi−θ(e−δxi−θ−1)αe−δxi−θ−2α−αe−δxi−θ+1−(1−e−δxi−θαe−δxi−θ−1)2(α−αe−δxi−θ+1)2,

Ψθθ=δlog⁡(α)xi−2θlog2⁡(xi)e−2δxi−θαe−δxi−θ(δ(log⁡(α)+eδxi−θ)−xiθeδxi−θ)−α+αe−δxi−θ−1−δ2log2⁡(α)xi−2θlog2⁡(xi)e−2δxi−θα2e−δxi−θ(α−αe−δxi−θ+1)2,