# A Novel Frechet-Type Probability Distribution: Its Properties and Applications

Authors: Muhammad Ali; Alamgir Khalil; Wali Khan Mashwani; Sharifah Alrajhi; Sanaa Al-Marzouki; Kamal Shah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this article, a new lifetime model, referred to as modified Frechet–Rayleigh distribution (MFRD), is developed by accommodating an additional parameter in Rayleigh distribution on the basis of the modified Frechet method. Numerous statistical properties of the suggested model are derived and discussed. The technique of maximum likelihood (ML) estimation is adopted to get estimates of the parameters. The suggested model is very flexible and has the capability to model datasets having both monotonic and nonmonotonic failure rates. The proposed model is applied on two real datasets for checking its performance in comparison with available well-known models. The suggested model has shown outclass performance in comparison with the available versions of the Rayleigh distribution used in the literature.


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## Body

## 1. Introduction

The distribution has many applications. It can be defined as follows.LetX∼N0,δ2 and Y∼N0,δ2 be independent normally distributed random variables. Then, Z = square root of (X2 + Y2) has a Rayleigh distribution.Rayleigh distribution is widely used in various fields, such as life sciences, agriculture, and biological sciences, and in the analysis of metrological characteristics, such as wind speed. The Rayleigh distribution has applications in situations where sizes of normally distributed variables are vital. Rayleigh distribution is also used in the investigation of wind velocity into its orthogonal XY coordinates. Furthermore, assuming that these components are independently and normally distributed with equal variances and means equal to zero. Thus, the variable of interest is wind speed and can be modeled by a Rayleigh distribution. According to Fernandez et al. [1] and Akhter et al. [2], it has useful applications in the field of medical sciences. In the field of engineering, it is used to find the expected lifespan of material objects, reliability investigation, service periods, and theory of communication. The density function (PDF) and distribution function (CDF) of RD are specified, respectively, as(1)fx=2λ2xe−λx2,x≥0,λ>0,(2)Fx=1−e−λx2,x≥0,λ>0.In the literature, there are many variants of the Rayleigh model which are used to provide a better fit of data. Voda [3] developed generalized Rayleigh and left-truncated model, Gomes et al. [4] suggested the Kumaraswamy generalized Rayleigh model, Merovci [5] suggested the transmuted Rayleigh model, Cordeiro et al. [6] suggested the beta generalized Rayleigh model. In [7], the inverse Rayleigh (IR) model was proposed to fit survival and reliability data. Voda [8] obtained estimates of the parameters of the IR model by ML estimation, Mukarjee and Maitim [9] discussed certain important statistical properties of the IR model, and Gharraph [10] developed expressions of descriptive statistics of the IR model. Moreover, Soliman et al. [11] and Gharraph [10] used maximum likelihood and Bayesian estimation methods to obtain estimates of parameters.In recent times, the literature of probability models has been updated by inserting additional parameter(s) in the existing models using existing techniques available in literature or by proposing a new method to produce a better model as compared to the baseline model. Rehman and Sajjad [12] developed the IR model; Ahmad et al. [13] suggested transmuted the IR model, beta IR model [14], modified inverse Rayleigh model [15], generalized inverted scale family models [16], alpha power exponentiated inverse Rayleigh model [17], and alpha power Rayleigh distribution [18]. These newly proposed distributions were obtained and comparatively provided a better fit of the complex data.The primary rationale behind producing modified Frechet–Rayleigh distribution (MFRD) is to overcome the inadequacy present in the Rayleigh model. The proposed distribution provides various shapes for hazard rate functions and shows more flexibility as compared to the Rayleigh distribution.


### 1.1. Research Problem

In practice, we might deal with data that exhibit monotonic or nonmonotonic hazard rate shapes. The available models may fail to be applied in such cases as in the case of life distributions. Therefore, researchers are trying to derive models that incorporate the deficiencies of these distributions. So, in this study, our aim is to propose a model that might efficiently fit such kind of data. This purpose is achieved by adding an extra parameter using generators or by the existing models; for detail, we refer to see [19]. Therefore, the modified Frechet technique is used to derive the proposed model.


### 1.2. Modified Frechet (MF) Technique

Alamgir et al. [20] suggested a novel technique, called modified Frechet technique (MFT) for developing new efficient probability models. The CDF and PDF of the newly developed technique are specified by the expressions given below:(3)GMFx=e−Fxα−1e−1−1,x>0.(4)gMFx=αfxFxα−1e−Fxα1−e−1,x>0.In equations (3) and (4), Fx and fx denote the CDF and PDF of the input model, respectively. This technique is used to introduce modified Frechet–Rayleigh distribution (MFRD). The basic purpose of producing MFRD has a more flexible distribution to model life time data in comparison to other versions of Rayleigh distribution.


### 1.3. Modified Frechet–Rayleigh Distribution (MFRD)

This section of the article introduces modified Frechet–Rayleigh distribution (MFRD).The CDF of RD is specified by(5)Fx=1−e−λx2,x≥0,λ>0.Definition 1.
IfX∼MFRD, then its PDF is specified by(6)fMFRDx=2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,x,λ,α>0.
The PDF and CDF are plotted in Figure1.
The CDF of MFRD is specified by(7)FMFRDx=e−1−e−λx2α−1e−1−1,α,λ,x>0.
The hazard rate (HR) function of MFRD is defined by(8)hMFRD=pdfsurvivalfunction=2αλ2xe−λx2−1−e−λx2α1−e−λx2α−1/1−e−1e−1−e−1−e−λx2α/e−1−1,hAMKRDx=2αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−λx2α−e−1.
The survival function (SF) of MFRD is specified by(9)SMFRDx=e−1−e−1−e−λx2αe−1−1.
Figure2 exhibits graphs of HF and SF of MFRD.Figure 1
Graphs of PDF and CDF of MFRD.Figure 2
Graphs of HF and SF of MFRD.Lemma 1.
Ifα<1 and fx is nonincreasing function, then fMFRDx is also a nonincreasing function.Proof.
Iffx is differentiable and if d/dxlogfx<0 for X in 0,∞, then fx is nonincreasing and vice versa.
Taked/dxlogfMFRDx<0, that is,(10)ddxlogfMFRDx=ddxlog2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,ddxlogfMFRDx=ddxlog2αλ2+logx−λx2−1−e−λx2α+α−1log1−e−λx2−log1−e−1,ddxlogfMFRDx=1x−2λ2x1−αe−λx21−e−λx2α−1−α−1e−λx21−e−λx2.
For nonnegative and less than 1 values ofα and λ>0, (9) shows that d/dxlogfMFRDx<0.
Hence, forα<1, fMFRDx is nonincreasing function.Lemma 2.
Ifα<1, fx is nonincreasing function, and fx is log-convex, then hMFRDx is a nonincreasing function.Proof.
Ifd2/dx2fx exists and d2/dx2logfx>0, then fx referred as log-convex.
By taking second differential of (10), we obtain(11)d2dx2logfMFRDx=ddx1x−2λ2x1−αe−λx21−e−λx2α−1−α−1e−λx21−e−λx2,d2dx2logfMFRDx=−1x2−2λ2−2αλ2e−λx21−e−λx2α−11−2λ2x2+2λ2α−1x2e−λx21−e−λx2−1+2λ2α−1e−λx21−e−λx2−11−2λ2x2−2λ2x2e−λx21−e−λx2−1.
Whenα is nonnegative and less than 1and λ>0, then d2/dx2logfMFRDx>0.
Therefore, for0<α<1, fMFRDx is log-convex [21].


### 1.4. Quantile Function (QF)

IfX ∼ MFRDα,λ, then MFRD has the following QF: Fx=u. This implies x=F−1u, where the distribution of u is uniform. The QF of MFRD is(12)Xp=1λ−log1−−logue−1−1+11/α1/2.


#### 1.4.1. Median

By puttingu=1/2 in (12), we get the median of MFRD as follows:(13)Median=1λ2−log1−−loge−1+121/α1/2.


#### 1.4.2. Mode

To find mode of MFRD, solve the following forx:(14)ddxfMFRDx=0⇒ddx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1=0,1−2λ2x21+e−λx21−e−λx2α−1α−α−1e−λx21−e−λx2α=0.Mode of the distribution satisfies (13).


#### 1.4.3.Rth Moment

LetX ∼ MFRDα,λ; then, by definition,(15)μr/=EXr=∫0∞xr2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Pute−λx2=y in (15) and simplify; we obtain the expression below: μ/r=EXr=α/1−e−1λr−1r/2∫01logyr/2e−1−yα1−yα−1dy.Insert1−yα=z in the above expression; after simplification, it will take the form given below:(16)μ/r=11−e−1λr−1r/2∫01log1−z1/αr/2e−zdz.Insert seriese−z=∑k=0∞−zk/k! and log1−z1/α=−∑m=1∞−1m−z1/αm/m for z1/α<1; from equation (16), we obtain the final expression as given below:(17)μr/=11−e−1λr−1r/2∫01−∑m=1∞−1m−z1/αmmr/2∑k=0∞−zkk!dz.The expression ofRth moment is incomplete integral. It can be solved numerically using some numerical integration technique.


#### 1.4.4. Moment Generating Function (MGF)

SupposeX : ∼MFRDα,λ; then, the MGF is specified by(18)Mxt=Eetx=∫0∞etx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Insertetx=∑r=0∞trxr/r! in (18), and after some calculation, we obtain(19)Mxt=Eetx=∑r=0∞trr!∫0∞xr2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Using result (15) in (19), we obtain(20)Mxt=11−e−1∑r=0∞trr!λr−1r2∫01−∑m=1∞−1m−z1/αmmr/2∑k=0∞−zkk!dz.The expression of MGF is an incomplete integral. It can be solved numerically by adopting any iterative method.


### 1.5. Order Statistics

Consider an order statisticsX1,X2,…,Xn taken from MFRD. Then, the PDF of ith order statistics Xi:n is specified by(21)fi:nx=n!i−1!n−i!fxFxi−11−Fxn−i.Insert (5) and (6) of MFRD in (21); we obtain ith order statistic is as follows:(22)fi:nx=n!i−1!n−i!2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1e−1−e−λx2α−1e−1−1i−11−e−1−e−λx2α−1e−1−1n−i,fi:nx=n!i−1!n−i!11−e−1e−1−1n−1e−1−e−λx2α−1i−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−1−e−λx2αn−i.By puttingi=1 in (22), we have expression for first-order statistic as given below:(23)f1:nx=n1−e−1e−1−1n−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−1−e−λx2αn−1.By substitutingi=n in (22), the expression for the nth order statistic will take the form given below:(24)fn:nx=n1−e−1e−1−1n−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−λx2α−1n−1.For median’s distribution inserti=n/2 in (22), we have(25)fn2:nx=n!n/2−1!n−n/2!11−e−1e−1−1n−1e−1−e−λx2α−1n/2−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−1−e−λx2αn−n/2.Lemma 3.
The Shannon entropy (SE) of MFRD is given by(26)S.Ex=−log21/α−1α2λ1−e−12∫02−∑k=1∞−1k−2−1/αz1/αkk1/2∑m=0∞−zmm!21/α−z1/αz2−2/αdz.Proof.
The SE of MFRD is given as follows:(27)S.Ex=E−logfx=E−log2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,S.Ex=−logE2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,S.Ex=−log4α2λ41−e−12∫0∞x.xe−2λx2e−21−e−λx2α1−e−λx22α−2dx.
Substituting1−e−λx2=y in (27), the expression will be like(28)S.Ex=−log2α2λ1−e−12∫01−log1−y1/21−yy2α−2e−2yαdy.
Putting2yα=z in (28), after simplification, it will take the following form:(29)S.Ex=−log22/α−1α2λ−11/21−e−12∫02log1−12z1/α1/2e−zz2−2α1−z21/αdz.
Using series representatione−z=∑m=0∞−zm/m! and log1−z/21/α=−∑k=1∞−1k−2−1/αz1/αk/k in (27), we get the final expression given below:(30)S.Ex=−log21/α−1α2λ1−e−12∫02−∑k=1∞−1k−2−1/αz1/αkk1/2∑m=0∞−zmm!21/α−z1/αz2−2/αdz.
The final expression for SE is an incomplete integral. It can be solved numerically using numerical integration methods or some iterative procedures.


### 1.6. Mean Residual Life Function

SupposeX follows MFRD. Then, μt of MFRD has the expression given below:(31)μt=1StEt−∫0txfxdx−t,t≥0.where(32)∫0txfxdx=∫0tx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Putting,1−e−λx2=y, the above expression will take the form given below:(33)∫0txfxdx=α−11/2λ1−e−1∫01−e−λt2log1−y1/2e−yαyα−1dy.Substituteyα=z in (33); we obtain(34)∫0txfxdx=−11/2λ1−e−1∫01−e−λt2αlog1−z1/α1/2e−zdz.Insert seriese−z=∑k=0∞−zk/k! and log1−z1/α=−∑m=1∞−1m−z1/αm/m for z1/α<1 in (31), we obtain(35)∫0txfxdx=−11/2λ1−e−1∫01−e−λt2α−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz,(36)Et=∫0∞tft=∫0∞t2αλ2te−λt2−1−e−λt2α1−e−λt2α−11−e−1dt.Substitute1−e−λt2=y in (33) and simplify; we have(37)Et=α−11/2λ1−e−1∫01log1−y1/2e−yαyα−1dy.Puttingyα=z in (37) and simplifying, we obtain(38)Et=−11/2λ1−e−1∫01log1−z1/α1/2e−zdz.Substitute seriese−z=∑k=0∞−zk/k! and log1−z1/α=−∑m=1∞−1m−z1/αm/m for z1/α<1 in (35), to have(39)Et=−11/2λ1−e−1∫01−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz.Put (8), (32), and (36) in (29); we obtain(40)μt=−11/2e−1−1λ1−e−1e−1−e−1−e−λx2α∫01−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz−∫01−e−λt2α−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz−t.The final expression ofμt is an incomplete integral. The expression in (40) can be solved numerically using any iterative method.


### 1.7. Stress-Strength Parameter (SSP)

LetX1 and X2 be two random variables, which are independent and follow the same distribution such that X1∼MFRDα1,λ and X2∼MFRDα2,λ; then, the expression for the SSP is specified as(41)R=∫−∞+∞f1xF2xdx.Utilizing (5) and (6) in (41), then SSP is given as(42)R=∫0∞2α1λ2xe−λx2−1−e−λx2α11−e−λx2α1−11−e−1e−1−e−λx2α2−1e−1−1dx,R=1e−1−11−e−1∫0∞2α1λ2xe−λx2e−1−e−λx2α11−e−λx2α1−1e−1−e−λx2α2dx−1e−1−1.Substitute1−e−λx2=y in (42) and simplify; we obtain(43)R=α1e−1−11−e−1∫01e−yα1yα1−1e−yα2dy−1e−1−1.Now, puttingyα1=z in (43), after simplification, we get the expression given below:(44)R=1e−1−11−e−1∫01e−ze−zα2α1dz−1e−1−1.Put seriese−z=∑k=0∞−zk/k! and e−zα2/α1=∑m=0∞−zα2/α1m/m! in (44); we get the following expression for SSP:(45)R=1e−1−11−e−1∑k=0∞∑m=0∞−1k+mk!m!1k+cm+1−1e−1−1,where c=α2/α1.This is the required result.


### 1.8. Parameters’ Estimation

#### 1.8.1. MLE

The joint density function has the following expression:(46)lα,λ=2αλ21−e−1n∏i=1nxie−∑i=1nλxi2e−∑i=1n1−e−λxi2α∏i=1n1−e−λxi2α−1.Taking natural logarithm of (46), we obtain(47)loglα,λ=nlog2λ2α−nlog1−e−1+∑i=1nlogxi−∑i=1nλxi2−∑i=1n1−e−λxi2α+α−1∑i=1nlog1−e−λxi2.Differentiate (44) with respect to α and λ, respectively, to have the likelihood equations as(48)∂loglα,λ∂α=nα−∑i=1n1−e−λxi2αlog1−e−λxi2+∑i=1nlog1−e−λxi2=0,(49)∂loglα,λ∂λ=2nλ−2λ∑i=1nxi2−2αλ∑i=1nxi2e−λxi21−e−λxi2α−1+2λα−1∑i=1nxi2e−λxi21−e−λxi2α−1=0.The solution of above equations is obtained using the Newton–Raphson method. For large sample size, the ML Estimators are normally distributed, that is,nα^−α,λ^−λ∼N20,Σ, where Σ denotes variance covariance matrix acquired by inverting Fisher information matrix F as follows:(50)F=∂2logl∂α2∂2logl∂α∂λ∂2logl∂α∂λ∂2logl∂λ2.Again differentiating (45) and (46) with respect to α and λ, respectively, we obtain(51)∂2logl∂α2=−nα2−∑i=1n1−e−λxi2αlog1−e−λxi22,(52)∂2logl∂λ2=−2nλ2+2∑i=1nxi2+2α∑i=1nxi2e−λxi21−e−λxi2α−1−4αλ2∑i=1nxi4e−λxi21−e−λxi2α−1+4αα−1λ2∑i=1nxi4e−2λxi21−e−λxi2α−2+2α−1∑i=1nxi2e−λxi21−e−λxi2−α+1−4λ2α−1∑i=1nxi4e−λxi21−e−λxi2−α+1−4α−12λ2∑i=1nxi4e−2λxi21−e−λxi2−α.Again differentiating (45) w.r.t λ, we obtain(53)∂2logl∂α∂λ=−2λ2α∑i=1nxie−λxi21−e−λxi2α−1log1−e−λxi2+2λ∑i=1nxie−λxi21−e−λxi2.Asymptotic1−ζ100% confidence intervals for parameters of MFRD are obtained by α^±Zζ/2Σ11 and λ^±Zζ/2Σ22, where Zζ denote the upper ζth percentile of the standard normal distribution.


### 1.9. Simulation Study

To assess the consistency of MLE, the MSE and bias are measured based onW = 1000 simulations. Samples having sizes 50, 100, and 200 are generated from MFRD using the following expression of the quantile function:(54)Xp=1λ−log1−−logue−1−1+11/α1/2,where U follows uniform distribution over [0, 1]. Bias and MSE of MFRD are computed by the following expressions:(55)Bias=1W∑l=1wbi^−b,MSE=1W∑l=1wbi^−b2,where b=α,λ. Simulation results were obtained for various values of α and λ. The results are shown in Table 1. The consistency pattern can easily be observed as estimates’ approach to their true values by increasing the sample size. Furthermore, the values of error measures of the parameter estimates’ decrease for all combinations of parameter values. Hence, it can be concluded that MLE procedure produces consistent estimates in estimating the parameters of MFRD. The same consistency of parameters has been observed for all choices of parameter values. It is also evident from Table 1 that bias for the estimators are decreasing with increase in the sample sizes and parameter α. Therefore, we conclude that these estimators are asymptotically unbiased.Table 1
Simulation results.
ParameterNMSE (α^)MSE (λ^)Bias (α^)Bias (λ^)α = 0.3635288λ = 0.4765585500.0025780260.29812270.01997809−0.29818741000.0013160360.19594890.003959582−0.1953882000.0006791100.057038540.003634784−0.05379642α = 0.3735288λ = 0.3735288500.00299770.21091660.008424304−0.27124791000.0017796350.14419120.004146879−0.19164792000.0008198250.095600070.003931851−0.1200957α = 0.4048975λ = 0.3565585500.0042890240.20295630.02249881−0.26223891000.002105580.14425190.01043012−0.19012632000.0008846410.077661410.004018672−0.10063α = 0.4335288λ = 0.3865585500.0055709320.27229050.01798329−0.33140391000.0015806990.19304870.006078905−0.2364862000.0011439010.096536060.001407309−0.1170932α = 0.4535288λ = 0.4065585500.0055649720.25720820.0184824−0.30219721000.0025640850.21137780.01034615−0.24782112000.0014945820.11279260.007373421−0.1322202α = 0.4648975λ = 0.3565585500.0040060280.22102960.02249881−0.29469361000.0021537030.17374560.00791557−0.2319122000.0011775390.07731780.002197945−0.1055793α = 0.4735288λ = 0.3865585500.0055967820.24607490.02127052−0.30023371000.002031430.19743260.00842539−0.2455152000.0015126190.079871720.005135786−0.09911058


## 1.1. Research Problem

In practice, we might deal with data that exhibit monotonic or nonmonotonic hazard rate shapes. The available models may fail to be applied in such cases as in the case of life distributions. Therefore, researchers are trying to derive models that incorporate the deficiencies of these distributions. So, in this study, our aim is to propose a model that might efficiently fit such kind of data. This purpose is achieved by adding an extra parameter using generators or by the existing models; for detail, we refer to see [19]. Therefore, the modified Frechet technique is used to derive the proposed model.


## 1.2. Modified Frechet (MF) Technique

Alamgir et al. [20] suggested a novel technique, called modified Frechet technique (MFT) for developing new efficient probability models. The CDF and PDF of the newly developed technique are specified by the expressions given below:(3)GMFx=e−Fxα−1e−1−1,x>0.(4)gMFx=αfxFxα−1e−Fxα1−e−1,x>0.In equations (3) and (4), Fx and fx denote the CDF and PDF of the input model, respectively. This technique is used to introduce modified Frechet–Rayleigh distribution (MFRD). The basic purpose of producing MFRD has a more flexible distribution to model life time data in comparison to other versions of Rayleigh distribution.


## 1.3. Modified Frechet–Rayleigh Distribution (MFRD)

This section of the article introduces modified Frechet–Rayleigh distribution (MFRD).The CDF of RD is specified by(5)Fx=1−e−λx2,x≥0,λ>0.Definition 1.
IfX∼MFRD, then its PDF is specified by(6)fMFRDx=2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,x,λ,α>0.
The PDF and CDF are plotted in Figure1.
The CDF of MFRD is specified by(7)FMFRDx=e−1−e−λx2α−1e−1−1,α,λ,x>0.
The hazard rate (HR) function of MFRD is defined by(8)hMFRD=pdfsurvivalfunction=2αλ2xe−λx2−1−e−λx2α1−e−λx2α−1/1−e−1e−1−e−1−e−λx2α/e−1−1,hAMKRDx=2αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−λx2α−e−1.
The survival function (SF) of MFRD is specified by(9)SMFRDx=e−1−e−1−e−λx2αe−1−1.
Figure2 exhibits graphs of HF and SF of MFRD.Figure 1
Graphs of PDF and CDF of MFRD.Figure 2
Graphs of HF and SF of MFRD.Lemma 1.
Ifα<1 and fx is nonincreasing function, then fMFRDx is also a nonincreasing function.Proof.
Iffx is differentiable and if d/dxlogfx<0 for X in 0,∞, then fx is nonincreasing and vice versa.
Taked/dxlogfMFRDx<0, that is,(10)ddxlogfMFRDx=ddxlog2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,ddxlogfMFRDx=ddxlog2αλ2+logx−λx2−1−e−λx2α+α−1log1−e−λx2−log1−e−1,ddxlogfMFRDx=1x−2λ2x1−αe−λx21−e−λx2α−1−α−1e−λx21−e−λx2.
For nonnegative and less than 1 values ofα and λ>0, (9) shows that d/dxlogfMFRDx<0.
Hence, forα<1, fMFRDx is nonincreasing function.Lemma 2.
Ifα<1, fx is nonincreasing function, and fx is log-convex, then hMFRDx is a nonincreasing function.Proof.
Ifd2/dx2fx exists and d2/dx2logfx>0, then fx referred as log-convex.
By taking second differential of (10), we obtain(11)d2dx2logfMFRDx=ddx1x−2λ2x1−αe−λx21−e−λx2α−1−α−1e−λx21−e−λx2,d2dx2logfMFRDx=−1x2−2λ2−2αλ2e−λx21−e−λx2α−11−2λ2x2+2λ2α−1x2e−λx21−e−λx2−1+2λ2α−1e−λx21−e−λx2−11−2λ2x2−2λ2x2e−λx21−e−λx2−1.
Whenα is nonnegative and less than 1and λ>0, then d2/dx2logfMFRDx>0.
Therefore, for0<α<1, fMFRDx is log-convex [21].


## 1.4. Quantile Function (QF)

IfX ∼ MFRDα,λ, then MFRD has the following QF: Fx=u. This implies x=F−1u, where the distribution of u is uniform. The QF of MFRD is(12)Xp=1λ−log1−−logue−1−1+11/α1/2.


### 1.4.1. Median

By puttingu=1/2 in (12), we get the median of MFRD as follows:(13)Median=1λ2−log1−−loge−1+121/α1/2.


### 1.4.2. Mode

To find mode of MFRD, solve the following forx:(14)ddxfMFRDx=0⇒ddx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1=0,1−2λ2x21+e−λx21−e−λx2α−1α−α−1e−λx21−e−λx2α=0.Mode of the distribution satisfies (13).


### 1.4.3.Rth Moment

LetX ∼ MFRDα,λ; then, by definition,(15)μr/=EXr=∫0∞xr2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Pute−λx2=y in (15) and simplify; we obtain the expression below: μ/r=EXr=α/1−e−1λr−1r/2∫01logyr/2e−1−yα1−yα−1dy.Insert1−yα=z in the above expression; after simplification, it will take the form given below:(16)μ/r=11−e−1λr−1r/2∫01log1−z1/αr/2e−zdz.Insert seriese−z=∑k=0∞−zk/k! and log1−z1/α=−∑m=1∞−1m−z1/αm/m for z1/α<1; from equation (16), we obtain the final expression as given below:(17)μr/=11−e−1λr−1r/2∫01−∑m=1∞−1m−z1/αmmr/2∑k=0∞−zkk!dz.The expression ofRth moment is incomplete integral. It can be solved numerically using some numerical integration technique.


### 1.4.4. Moment Generating Function (MGF)

SupposeX : ∼MFRDα,λ; then, the MGF is specified by(18)Mxt=Eetx=∫0∞etx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Insertetx=∑r=0∞trxr/r! in (18), and after some calculation, we obtain(19)Mxt=Eetx=∑r=0∞trr!∫0∞xr2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Using result (15) in (19), we obtain(20)Mxt=11−e−1∑r=0∞trr!λr−1r2∫01−∑m=1∞−1m−z1/αmmr/2∑k=0∞−zkk!dz.The expression of MGF is an incomplete integral. It can be solved numerically by adopting any iterative method.


## 1.4.1. Median

By puttingu=1/2 in (12), we get the median of MFRD as follows:(13)Median=1λ2−log1−−loge−1+121/α1/2.


## 1.4.2. Mode

To find mode of MFRD, solve the following forx:(14)ddxfMFRDx=0⇒ddx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1=0,1−2λ2x21+e−λx21−e−λx2α−1α−α−1e−λx21−e−λx2α=0.Mode of the distribution satisfies (13).


## 1.4.3.Rth Moment

LetX ∼ MFRDα,λ; then, by definition,(15)μr/=EXr=∫0∞xr2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Pute−λx2=y in (15) and simplify; we obtain the expression below: μ/r=EXr=α/1−e−1λr−1r/2∫01logyr/2e−1−yα1−yα−1dy.Insert1−yα=z in the above expression; after simplification, it will take the form given below:(16)μ/r=11−e−1λr−1r/2∫01log1−z1/αr/2e−zdz.Insert seriese−z=∑k=0∞−zk/k! and log1−z1/α=−∑m=1∞−1m−z1/αm/m for z1/α<1; from equation (16), we obtain the final expression as given below:(17)μr/=11−e−1λr−1r/2∫01−∑m=1∞−1m−z1/αmmr/2∑k=0∞−zkk!dz.The expression ofRth moment is incomplete integral. It can be solved numerically using some numerical integration technique.


## 1.4.4. Moment Generating Function (MGF)

SupposeX : ∼MFRDα,λ; then, the MGF is specified by(18)Mxt=Eetx=∫0∞etx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Insertetx=∑r=0∞trxr/r! in (18), and after some calculation, we obtain(19)Mxt=Eetx=∑r=0∞trr!∫0∞xr2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Using result (15) in (19), we obtain(20)Mxt=11−e−1∑r=0∞trr!λr−1r2∫01−∑m=1∞−1m−z1/αmmr/2∑k=0∞−zkk!dz.The expression of MGF is an incomplete integral. It can be solved numerically by adopting any iterative method.


## 1.5. Order Statistics

Consider an order statisticsX1,X2,…,Xn taken from MFRD. Then, the PDF of ith order statistics Xi:n is specified by(21)fi:nx=n!i−1!n−i!fxFxi−11−Fxn−i.Insert (5) and (6) of MFRD in (21); we obtain ith order statistic is as follows:(22)fi:nx=n!i−1!n−i!2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1e−1−e−λx2α−1e−1−1i−11−e−1−e−λx2α−1e−1−1n−i,fi:nx=n!i−1!n−i!11−e−1e−1−1n−1e−1−e−λx2α−1i−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−1−e−λx2αn−i.By puttingi=1 in (22), we have expression for first-order statistic as given below:(23)f1:nx=n1−e−1e−1−1n−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−1−e−λx2αn−1.By substitutingi=n in (22), the expression for the nth order statistic will take the form given below:(24)fn:nx=n1−e−1e−1−1n−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−λx2α−1n−1.For median’s distribution inserti=n/2 in (22), we have(25)fn2:nx=n!n/2−1!n−n/2!11−e−1e−1−1n−1e−1−e−λx2α−1n/2−12αλ2xe−λx2−1−e−λx2α1−e−λx2α−1e−1−e−1−e−λx2αn−n/2.Lemma 3.
The Shannon entropy (SE) of MFRD is given by(26)S.Ex=−log21/α−1α2λ1−e−12∫02−∑k=1∞−1k−2−1/αz1/αkk1/2∑m=0∞−zmm!21/α−z1/αz2−2/αdz.Proof.
The SE of MFRD is given as follows:(27)S.Ex=E−logfx=E−log2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,S.Ex=−logE2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1,S.Ex=−log4α2λ41−e−12∫0∞x.xe−2λx2e−21−e−λx2α1−e−λx22α−2dx.
Substituting1−e−λx2=y in (27), the expression will be like(28)S.Ex=−log2α2λ1−e−12∫01−log1−y1/21−yy2α−2e−2yαdy.
Putting2yα=z in (28), after simplification, it will take the following form:(29)S.Ex=−log22/α−1α2λ−11/21−e−12∫02log1−12z1/α1/2e−zz2−2α1−z21/αdz.
Using series representatione−z=∑m=0∞−zm/m! and log1−z/21/α=−∑k=1∞−1k−2−1/αz1/αk/k in (27), we get the final expression given below:(30)S.Ex=−log21/α−1α2λ1−e−12∫02−∑k=1∞−1k−2−1/αz1/αkk1/2∑m=0∞−zmm!21/α−z1/αz2−2/αdz.
The final expression for SE is an incomplete integral. It can be solved numerically using numerical integration methods or some iterative procedures.


## 1.6. Mean Residual Life Function

SupposeX follows MFRD. Then, μt of MFRD has the expression given below:(31)μt=1StEt−∫0txfxdx−t,t≥0.where(32)∫0txfxdx=∫0tx2αλ2xe−λx2−1−e−λx2α1−e−λx2α−11−e−1dx.Putting,1−e−λx2=y, the above expression will take the form given below:(33)∫0txfxdx=α−11/2λ1−e−1∫01−e−λt2log1−y1/2e−yαyα−1dy.Substituteyα=z in (33); we obtain(34)∫0txfxdx=−11/2λ1−e−1∫01−e−λt2αlog1−z1/α1/2e−zdz.Insert seriese−z=∑k=0∞−zk/k! and log1−z1/α=−∑m=1∞−1m−z1/αm/m for z1/α<1 in (31), we obtain(35)∫0txfxdx=−11/2λ1−e−1∫01−e−λt2α−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz,(36)Et=∫0∞tft=∫0∞t2αλ2te−λt2−1−e−λt2α1−e−λt2α−11−e−1dt.Substitute1−e−λt2=y in (33) and simplify; we have(37)Et=α−11/2λ1−e−1∫01log1−y1/2e−yαyα−1dy.Puttingyα=z in (37) and simplifying, we obtain(38)Et=−11/2λ1−e−1∫01log1−z1/α1/2e−zdz.Substitute seriese−z=∑k=0∞−zk/k! and log1−z1/α=−∑m=1∞−1m−z1/αm/m for z1/α<1 in (35), to have(39)Et=−11/2λ1−e−1∫01−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz.Put (8), (32), and (36) in (29); we obtain(40)μt=−11/2e−1−1λ1−e−1e−1−e−1−e−λx2α∫01−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz−∫01−e−λt2α−∑m=1∞−1m−z1/αmm1/2∑k=0∞−zkk!dz−t.The final expression ofμt is an incomplete integral. The expression in (40) can be solved numerically using any iterative method.


## 1.7. Stress-Strength Parameter (SSP)

LetX1 and X2 be two random variables, which are independent and follow the same distribution such that X1∼MFRDα1,λ and X2∼MFRDα2,λ; then, the expression for the SSP is specified as(41)R=∫−∞+∞f1xF2xdx.Utilizing (5) and (6) in (41), then SSP is given as(42)R=∫0∞2α1λ2xe−λx2−1−e−λx2α11−e−λx2α1−11−e−1e−1−e−λx2α2−1e−1−1dx,R=1e−1−11−e−1∫0∞2α1λ2xe−λx2e−1−e−λx2α11−e−λx2α1−1e−1−e−λx2α2dx−1e−1−1.Substitute1−e−λx2=y in (42) and simplify; we obtain(43)R=α1e−1−11−e−1∫01e−yα1yα1−1e−yα2dy−1e−1−1.Now, puttingyα1=z in (43), after simplification, we get the expression given below:(44)R=1e−1−11−e−1∫01e−ze−zα2α1dz−1e−1−1.Put seriese−z=∑k=0∞−zk/k! and e−zα2/α1=∑m=0∞−zα2/α1m/m! in (44); we get the following expression for SSP:(45)R=1e−1−11−e−1∑k=0∞∑m=0∞−1k+mk!m!1k+cm+1−1e−1−1,where c=α2/α1.This is the required result.


## 1.8. Parameters’ Estimation

### 1.8.1. MLE

The joint density function has the following expression:(46)lα,λ=2αλ21−e−1n∏i=1nxie−∑i=1nλxi2e−∑i=1n1−e−λxi2α∏i=1n1−e−λxi2α−1.Taking natural logarithm of (46), we obtain(47)loglα,λ=nlog2λ2α−nlog1−e−1+∑i=1nlogxi−∑i=1nλxi2−∑i=1n1−e−λxi2α+α−1∑i=1nlog1−e−λxi2.Differentiate (44) with respect to α and λ, respectively, to have the likelihood equations as(48)∂loglα,λ∂α=nα−∑i=1n1−e−λxi2αlog1−e−λxi2+∑i=1nlog1−e−λxi2=0,(49)∂loglα,λ∂λ=2nλ−2λ∑i=1nxi2−2αλ∑i=1nxi2e−λxi21−e−λxi2α−1+2λα−1∑i=1nxi2e−λxi21−e−λxi2α−1=0.The solution of above equations is obtained using the Newton–Raphson method. For large sample size, the ML Estimators are normally distributed, that is,nα^−α,λ^−λ∼N20,Σ, where Σ denotes variance covariance matrix acquired by inverting Fisher information matrix F as follows:(50)F=∂2logl∂α2∂2logl∂α∂λ∂2logl∂α∂λ∂2logl∂λ2.Again differentiating (45) and (46) with respect to α and λ, respectively, we obtain(51)∂2logl∂α2=−nα2−∑i=1n1−e−λxi2αlog1−e−λxi22,(52)∂2logl∂λ2=−2nλ2+2∑i=1nxi2+2α∑i=1nxi2e−λxi21−e−λxi2α−1−4αλ2∑i=1nxi4e−λxi21−e−λxi2α−1+4αα−1λ2∑i=1nxi4e−2λxi21−e−λxi2α−2+2α−1∑i=1nxi2e−λxi21−e−λxi2−α+1−4λ2α−1∑i=1nxi4e−λxi21−e−λxi2−α+1−4α−12λ2∑i=1nxi4e−2λxi21−e−λxi2−α.Again differentiating (45) w.r.t λ, we obtain(53)∂2logl∂α∂λ=−2λ2α∑i=1nxie−λxi21−e−λxi2α−1log1−e−λxi2+2λ∑i=1nxie−λxi21−e−λxi2.Asymptotic1−ζ100% confidence intervals for parameters of MFRD are obtained by α^±Zζ/2Σ11 and λ^±Zζ/2Σ22, where Zζ denote the upper ζth percentile of the standard normal distribution.


## 1.8.1. MLE

The joint density function has the following expression:(46)lα,λ=2αλ21−e−1n∏i=1nxie−∑i=1nλxi2e−∑i=1n1−e−λxi2α∏i=1n1−e−λxi2α−1.Taking natural logarithm of (46), we obtain(47)loglα,λ=nlog2λ2α−nlog1−e−1+∑i=1nlogxi−∑i=1nλxi2−∑i=1n1−e−λxi2α+α−1∑i=1nlog1−e−λxi2.Differentiate (44) with respect to α and λ, respectively, to have the likelihood equations as(48)∂loglα,λ∂α=nα−∑i=1n1−e−λxi2αlog1−e−λxi2+∑i=1nlog1−e−λxi2=0,(49)∂loglα,λ∂λ=2nλ−2λ∑i=1nxi2−2αλ∑i=1nxi2e−λxi21−e−λxi2α−1+2λα−1∑i=1nxi2e−λxi21−e−λxi2α−1=0.The solution of above equations is obtained using the Newton–Raphson method. For large sample size, the ML Estimators are normally distributed, that is,nα^−α,λ^−λ∼N20,Σ, where Σ denotes variance covariance matrix acquired by inverting Fisher information matrix F as follows:(50)F=∂2logl∂α2∂2logl∂α∂λ∂2logl∂α∂λ∂2logl∂λ2.Again differentiating (45) and (46) with respect to α and λ, respectively, we obtain(51)∂2logl∂α2=−nα2−∑i=1n1−e−λxi2αlog1−e−λxi22,(52)∂2logl∂λ2=−2nλ2+2∑i=1nxi2+2α∑i=1nxi2e−λxi21−e−λxi2α−1−4αλ2∑i=1nxi4e−λxi21−e−λxi2α−1+4αα−1λ2∑i=1nxi4e−2λxi21−e−λxi2α−2+2α−1∑i=1nxi2e−λxi21−e−λxi2−α+1−4λ2α−1∑i=1nxi4e−λxi21−e−λxi2−α+1−4α−12λ2∑i=1nxi4e−2λxi21−e−λxi2−α.Again differentiating (45) w.r.t λ, we obtain(53)∂2logl∂α∂λ=−2λ2α∑i=1nxie−λxi21−e−λxi2α−1log1−e−λxi2+2λ∑i=1nxie−λxi21−e−λxi2.Asymptotic1−ζ100% confidence intervals for parameters of MFRD are obtained by α^±Zζ/2Σ11 and λ^±Zζ/2Σ22, where Zζ denote the upper ζth percentile of the standard normal distribution.


## 1.9. Simulation Study

To assess the consistency of MLE, the MSE and bias are measured based onW = 1000 simulations. Samples having sizes 50, 100, and 200 are generated from MFRD using the following expression of the quantile function:(54)Xp=1λ−log1−−logue−1−1+11/α1/2,where U follows uniform distribution over [0, 1]. Bias and MSE of MFRD are computed by the following expressions:(55)Bias=1W∑l=1wbi^−b,MSE=1W∑l=1wbi^−b2,where b=α,λ. Simulation results were obtained for various values of α and λ. The results are shown in Table 1. The consistency pattern can easily be observed as estimates’ approach to their true values by increasing the sample size. Furthermore, the values of error measures of the parameter estimates’ decrease for all combinations of parameter values. Hence, it can be concluded that MLE procedure produces consistent estimates in estimating the parameters of MFRD. The same consistency of parameters has been observed for all choices of parameter values. It is also evident from Table 1 that bias for the estimators are decreasing with increase in the sample sizes and parameter α. Therefore, we conclude that these estimators are asymptotically unbiased.Table 1
Simulation results.
ParameterNMSE (α^)MSE (λ^)Bias (α^)Bias (λ^)α = 0.3635288λ = 0.4765585500.0025780260.29812270.01997809−0.29818741000.0013160360.19594890.003959582−0.1953882000.0006791100.057038540.003634784−0.05379642α = 0.3735288λ = 0.3735288500.00299770.21091660.008424304−0.27124791000.0017796350.14419120.004146879−0.19164792000.0008198250.095600070.003931851−0.1200957α = 0.4048975λ = 0.3565585500.0042890240.20295630.02249881−0.26223891000.002105580.14425190.01043012−0.19012632000.0008846410.077661410.004018672−0.10063α = 0.4335288λ = 0.3865585500.0055709320.27229050.01798329−0.33140391000.0015806990.19304870.006078905−0.2364862000.0011439010.096536060.001407309−0.1170932α = 0.4535288λ = 0.4065585500.0055649720.25720820.0184824−0.30219721000.0025640850.21137780.01034615−0.24782112000.0014945820.11279260.007373421−0.1322202α = 0.4648975λ = 0.3565585500.0040060280.22102960.02249881−0.29469361000.0021537030.17374560.00791557−0.2319122000.0011775390.07731780.002197945−0.1055793α = 0.4735288λ = 0.3865585500.0055967820.24607490.02127052−0.30023371000.002031430.19743260.00842539−0.2455152000.0015126190.079871720.005135786−0.09911058


## 2. Applications

To assess the performance of the MFRD, two practical datasets are used and the results are compared using some standard model selection criteria’s. The proposed model is compared with Rayleigh distribution (RD) [22], modified inverse Rayleigh distribution (MIRD) [15], generalized inverse Rayleigh distribution (GIRD) [16], transmuted inverse Rayleigh distribution (TIRD) (Ahmad et al., 2014), exponentiated inverse Rayleigh distribution (EIRD) [12], and generalized Rayleigh distribution (GRD) [23].


### 2.1. Dataset 1

Dataset 1 was taken from Bjerkedal [24]. The data points are given below: 0.1, 0.33, 1.08, 1.08, 1.08, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 2.54, 2.78, 2.93, 3.27, 3.42, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 07, 1.09, 1.12, 1.13, 1.15,1.36, 1.39, 1.44, 1.83, 1.95, 1.96, 1.97, 2.02, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 2.13, 1.46,1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53,2.54, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55.


### 2.2. Dataset 2

Dataset 2 is taken from Nassar and Nada [25]. The data points are as follows: 5.9, 20.4, 14.9, 16.2, 17.2, 7.8, 6.1, 9.2, 10.2, 9.6, 13.3, 8.5, 21.6, 18.5, 5.1, 6.7, 17, 8.6, 9.7, 39.2, 35.7, 15.7, 9.7, 10, 4.1, 36, 8.5, 8, 9.2, 26.2, 21.9, 16.7, 21.3, 35.4, 14.3, 8.5, 10.6, 19.1, 20.5, 7.1, 7.7, 18.1, 16.5, 11.9, 7, 8.6, 12.5, 10.3, 11.2, 6.1, 8.4, 11, 11.6, 11.9, 5.2, 6.8, 8.9, 7.1, 10.8.P-value and some well-known standard model selection criteria’s are used for comparison purposes. The results are displayed in Tables 2 and 3 of dataset 1 and 2, respectively. Comparative analysis of different distributions based on two data points are shown in Figures 3 and 4, respectively.Table 2
Goodness of fit result for dataset 1.
ModelMLEAICCAICBICHQICK-Sp-valueMFRD0.96910.3959208.0488208.2227212.6022209.86150.11940.2561RD−0.4542212.2285212.2856214.5051213.13480.14040.1172MIRD1.2641−0.0484247.3044247.4783251.8578249.11710.23470.0007GIRD0.39242.2022290.9948291.1687295.5482292.80750.79772.2e − 16TIRD0.3558−0.9404294.6214294.7953299.1747296.43410.35064.106e − 08EIRD0.22852.0379341.9955342.1694346.5488343.80820.47501.554e − 14Table 3
Goodness-of-fit result for dataset 2.
ModelMLEAICCAICBICHQICK-Sp-valueMFRD1.1859−0.0603394.472394.686398.627396.0940.1600.095RD0.0638397.421397.491399.499398.2320.1720.060MIRD8.484612.2370416.948417.162421.103418.5700.2530.001GRD1.02950.0643399.393399.607403.548401.0150.1770.049Figure 3
Comparison of MFRD with other competitive models for dataset 1.Figure 4
Comparison of MFRD with other competitive models for dataset 2.The results of Tables2 and 3 indicate that MFRD provides improved result in comparison to other versions of the Rayleigh model on the basis of well-known model selection criteria. Figure 5 and 6 provide PP-plots, QQ-plots, plots of PDF, and CDF for the two datasets.Figure 5
Plots of MFRD for dataset 1.Figure 6
Plots of MFRD for dataset 2.


## 2.1. Dataset 1

Dataset 1 was taken from Bjerkedal [24]. The data points are given below: 0.1, 0.33, 1.08, 1.08, 1.08, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 2.54, 2.78, 2.93, 3.27, 3.42, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 07, 1.09, 1.12, 1.13, 1.15,1.36, 1.39, 1.44, 1.83, 1.95, 1.96, 1.97, 2.02, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 2.13, 1.46,1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53,2.54, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55.


## 2.2. Dataset 2

Dataset 2 is taken from Nassar and Nada [25]. The data points are as follows: 5.9, 20.4, 14.9, 16.2, 17.2, 7.8, 6.1, 9.2, 10.2, 9.6, 13.3, 8.5, 21.6, 18.5, 5.1, 6.7, 17, 8.6, 9.7, 39.2, 35.7, 15.7, 9.7, 10, 4.1, 36, 8.5, 8, 9.2, 26.2, 21.9, 16.7, 21.3, 35.4, 14.3, 8.5, 10.6, 19.1, 20.5, 7.1, 7.7, 18.1, 16.5, 11.9, 7, 8.6, 12.5, 10.3, 11.2, 6.1, 8.4, 11, 11.6, 11.9, 5.2, 6.8, 8.9, 7.1, 10.8.P-value and some well-known standard model selection criteria’s are used for comparison purposes. The results are displayed in Tables 2 and 3 of dataset 1 and 2, respectively. Comparative analysis of different distributions based on two data points are shown in Figures 3 and 4, respectively.Table 2
Goodness of fit result for dataset 1.
ModelMLEAICCAICBICHQICK-Sp-valueMFRD0.96910.3959208.0488208.2227212.6022209.86150.11940.2561RD−0.4542212.2285212.2856214.5051213.13480.14040.1172MIRD1.2641−0.0484247.3044247.4783251.8578249.11710.23470.0007GIRD0.39242.2022290.9948291.1687295.5482292.80750.79772.2e − 16TIRD0.3558−0.9404294.6214294.7953299.1747296.43410.35064.106e − 08EIRD0.22852.0379341.9955342.1694346.5488343.80820.47501.554e − 14Table 3
Goodness-of-fit result for dataset 2.
ModelMLEAICCAICBICHQICK-Sp-valueMFRD1.1859−0.0603394.472394.686398.627396.0940.1600.095RD0.0638397.421397.491399.499398.2320.1720.060MIRD8.484612.2370416.948417.162421.103418.5700.2530.001GRD1.02950.0643399.393399.607403.548401.0150.1770.049Figure 3
Comparison of MFRD with other competitive models for dataset 1.Figure 4
Comparison of MFRD with other competitive models for dataset 2.The results of Tables2 and 3 indicate that MFRD provides improved result in comparison to other versions of the Rayleigh model on the basis of well-known model selection criteria. Figure 5 and 6 provide PP-plots, QQ-plots, plots of PDF, and CDF for the two datasets.Figure 5
Plots of MFRD for dataset 1.Figure 6
Plots of MFRD for dataset 2.


## 3. Conclusion

In this article, we suggested a new lifetime model referred to as modified Frechet–Rayleigh distribution (MFRD) using the modified Frechet technique. Several statistical properties of the MFRD were obtained such as moments, MGF, and median in closed form, stress-strength parameter, Quantile function, mean residuals’ life function, order statistics, and the expression for entropies. The parameters of the proposed model were estimated using the MLE approach. The simulation results revealed that the estimates of the parameters of MFRD are consistent and asymptotically unbiased. The performance of MFRD was evaluated on the basis of real datasets. The results indicated better performance of MFRD as compared to various versions of Rayleigh distribution.



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*Source: 2537332-2022-01-10.xml*