# Sine Half-Logistic Inverse Rayleigh Distribution: Properties, Estimation, and Applications in Biomedical Data

Authors: M. Shrahili; I. Elbatal; Mohammed Elgarhy
Journal: Journal of Mathematics
 (2021)

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## Abstract

A new lifetime distribution with two parameters, known as the sine half-logistic inverse Rayleigh distribution, is proposed and studied as an extension of the half-logistic inverse Rayleigh model. The sine half-logistic inverse Rayleigh model is a new inverse Rayleigh distribution extension. In the application section, we show that the sine half-logistic inverse Rayleigh distribution is more flexible than the half-logistic inverse Rayleigh and inverse Rayleigh distributions. The statistical properties of the half-logistic inverse Rayleigh model are calculated, including the quantile function, moments, moment generating function, incomplete moment, and Lorenz and Bonferroni curves. Entropy measures such as Rényi entropy, Havrda and Charvat entropy, Arimoto entropy, and Tsallis entropy are proposed for the sine half-logistic inverse Rayleigh distribution. To estimate the sine half-logistic inverse Rayleigh distribution parameters, statistical inference using the maximum likelihood method is used. Applications of the sine half-logistic inverse Rayleigh model to real datasets demonstrate the flexibility of the sine half-logistic inverse Rayleigh distribution by comparing it to well-known models such as half-logistic inverse Rayleigh, type II Topp–Leone inverse Rayleigh, transmuted inverse Rayleigh, and inverse Rayleigh distributions.


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## Body

## 1. Introduction

In recent years, inverse and half-inverse problems are studied in general operator theory [1–3], and many statisticians are focusing on generated families of distributions such as Kumaraswamy-G [4], T-X family [5], sine-G [6], type II half logistic-G [7], Weibull-G [8], the Burr type X-G [9], a new power Topp–Leone-G [10], truncated Cauchy power-G [11], beta generalized Marshall–Olkin–Kumaraswamy-G [12], transmuted odd Fréchet-G [13], new Kumaraswamy-G [14], Kumaraswamy Kumaraswamy-G [15], generalized Kumaraswamy-G [16], sine Topp–Leone-G [17], generalized transmuted exponentiated G [18], and Kumaraswamy transmuted-G [19].A new generated family of distributions which is called the Sine-G (SG) family was introduced in [6]. The distribution function (CDF) of SG is(1)Fx;ε=sinπ2Gx;ε,x∈R,where Gx;ε is the CDF of baseline model with parameter vector ε and Fx;ε is the CDF derived by the T-X generator proposed in [3]. The probability density function (PDF) of the SG is(2)fx;ε=π2gx;εcosπ2Gx;ε,x∈R,respectively.The inverse Rayleigh (IR) distribution is a useful model for calculating lifetimes. Several authors have developed a number of extensions for the IR distribution in recent years, using various methods of generalization (see, for example, beta IR in [20], transmuted IR (TIR) in [21], modified IR in [22], transmuted modified IR in [23], Kumaraswamy exponentiated IR in [24], weighted IR in [25], odd Fréchet IR in [26], and half-logistic IR (HLIR) in [27]).The CDF and PDF of HLIR distribution are given by(3)Gx;λ,α=1−1−e−α/x2λ1+1−e−α/x2λ,x,λ,α>0,and(4)gx;λ,α=4λα2x−3e−α/x21−e−α/x2λ−11+1−e−α/x2λ2,x,λ,α>0,where α is a scale parameter and λ is a shape parameter.We now present the sine half-logistic IR (SHLIR) distribution, a new lifetime model with two parameters. Inserting (3) into (1) yields the cdf of the SHLIR distribution as(5)Fx;λ,α=sinπ21−1−e−α/x2λ1+1−e−α/x2λ,x,λ,α>0.The corresponding PDF to (5) is(6)fx;λ,α=2πλα2x−3e−α/x21−e−α/x2λ−11+1−e−α/x2λ2cosπ21−1−e−α/x2λ1+1−e−α/x2λ,x,λ,α>0.The SHLIR distribution’s survival function (SF), hazard rate function (HRF), reversed HRF, and cumulative HRF are as follows:(7)Rx;λ,α=1−sinπ21−1−e−α/x2λ1+1−e−α/x2λ,(8)hx;λ,α=2πλα2x−3e−α/x21−e−α/x2λ−1cosπ/21−1−e−α/x2λ/1+1−e−α/x2λ1+1−e−α/x2λ21−sinπ/21−1−e−α/x2λ/1+1−e−α/x2λ,(9)τx;λ,α=2πλα2x−3e−α/x21−e−α/x2λ−11+1−e−α/x2λ2cotπ21−1−e−α/x2λ1+1−e−α/x2λ,(10)Hx;λ,α=−ln1−sinπ21−1−e−α/x2λ1+1−e−α/x2λ.Figures1 and 2 show plots of the SHLIR PDF and HRF for various parameter values.Figure 1
PDF plots for SHLIR distribution.Figure 2
HRF plots for the SHLIR distribution.We can conclude from Figures1 and 2 that the PDF of the SHLIR distribution can be unimodel and right skewed. SHLIR distribution HRF can be J-shaped and increasing.The remainder of this paper is structured as follows. Section2 discusses some of the structural characteristics of the SHLIR distribution, such as the quantile function, moments, incomplete moments, Lorenz and Bonferroni curves, and various measures of entropy. Section 3 discusses maximum likelihood (ML) parameter estimators for the SHLIR distribution. Section 4 implements simulation schemes. In Section 5, two sets of real-world data applications are used to demonstrate the potential of the SHLIR distribution in comparison to other distributions. The article concludes with some closing remarks.


## 2. Statistical Characteristics

Some statistical properties of the SHLIR distribution are obtained in this section.


### 2.1. Linear Representation

In this section, we will go over the most important linear PDF combinations for SHLIR distribution.The sine function’s series:(11)sinZ=∑i=0∞−1i2i!Z2i.By inserting (11) in (6), we get(12)fx=∑i=0∞−1iπ2i+12i!22i−1λα2x−3e−α/x21−e−α/x2λ−11−1−e−α/x2λ2i1+1−e−α/x2λ2i+1,where fx=fx;λ,α.Consider the following well-known binomial expansions (for 0 <a < 1):(13)1+a−n=∑j=0∞−1in+j−1jaj.Thus, inserting (13) in (12), we get(14)fx=∑i,j=0∞λα2−1i+jπ2i+1x32i!22i−12i+j+1je−α/x21−e−α/x2λj+1−11−1−e−α/x2λ2i.Again, we can using the binomial expansion in the following term:(15)1−1−e−α/x2λ2i=∑k=0∞−1k2ik1−e−α/x2kλ.Therefore, by inserting (15) in (14),(16)fx=∑i,j,k=0∞−1i+j+kπ2i+12i!22i−12i+j+1j2ikλα2x−3e−α/x21−e−α/x2λk+j+1−1.Again, we can using the binomial expansion in the following term:(17)1−e−α/x2λk+j+1−1=∑m=0∞−1mλk+j+1−1me−mα/x2.Therefore, by inserting (17) in (16), we can write the PDF of SHLIR as(18)fx=2α2∑i=0∞Δmx−3e−m+1α/x2,where Δm=∑i,j,k=0∞−1i+j+k+mπ2i+1/2i!22i2i+j+1j2ikλk+j+1−1mλ.


### 2.2. Quantile Function

By inverting (5), we can obtain the quantile function of the SHLIR distribution, say Qu=F−1u of X, as follows:(19)Qu=α−ln1−1−2/πsin−1u/1+2/πsin−1u1/λ,where u is thought of as a uniform random variable on (0, 1).


### 2.3. Moments

IfX has PDF (6), then its moment can be calculated using the following relation.(20)μr\=EXr=∫0∞xrfxdx.Substituting (18) into (20) yields(21)μr\=2α2∑m=0∞Δm∫0∞xr−3e−m+1α/x2dx.Lety=α/x2; then,(22)μr\=αr∑m=0∞Δm∫0∞y−r/2e−m+1ydy.Then,μr\ becomes(23)μr\=αr∑m=0∞ΔmΓ1−r/2m+11−r/2,r2<1.The SHLIR distribution's moment generating function is given by(24)MXt=∑r=0∞trr!EXr=αr∑r,m=0∞Δmtrr!Γ1−r/2m+11−r/2,r2<1.The incomplete moments of SHLIR are defined by(25)ϕst=∫0txsfxdx.Using (18), φst will be(26)φst=2α2∑m=0∞Δm∫0txs−3e−m+1α/x2dx=αs∑m=0∞ΔmΓ1−s/2,m+1α/t2m+11−s/2,where Γs,t=∫0txs−1e−xdx is the lower incomplete gamma function.The Lorenz and Bonferroni curves are obtained as follows:(27)LFx=φ1xEX=α∑m=0∞ΔmΓ1/2,m+1α/t2/m+11/2∑m=0∞Δmαπ/m+11/2=∑m=0∞ΔmΓ1/2,m+1α/t2/m+11/2∑m=0∞Δmπ/m+11/2,and(28)BFx=LFxFx=∑m=0∞ΔmΓ1/2,m+1α/t2/i+11/2sinπ/21−1−e−α/x2λ/1+1−e−α/x2λ∑m=0∞Δmπ/m+11/2.


### 2.4. Entropies

The entropy of the SHLIR model can be measured by various measures such as Rényi entropy (RE) [28], Havrda and Charvat entropy (HCE) [29], Arimoto entropy (AE) [30], and Tsallis entropy (TE) [31]. These measures of entropy are mentioned in Table 1.Table 1
Various measures of entropy for a distribution with PDFfx at γ.
The measuresFormulaREIRγ=1/1−γLog∫0∞fγxdx,γ≠1,γ>0.HCEHCRγ=1/21−γ−1∫0∞fγxdx−1,γ≠1,γ>0.AEARγ=γ/1−γ∫0∞fγxdx1/γ−1,γ≠1,γ>0.TETRγ=1/γ−11−∫0∞fγxdx,γ≠1,γ>0.∫0∞fγxdx is very complicated to calculate, so it will be solved numerically.


## 2.1. Linear Representation

In this section, we will go over the most important linear PDF combinations for SHLIR distribution.The sine function’s series:(11)sinZ=∑i=0∞−1i2i!Z2i.By inserting (11) in (6), we get(12)fx=∑i=0∞−1iπ2i+12i!22i−1λα2x−3e−α/x21−e−α/x2λ−11−1−e−α/x2λ2i1+1−e−α/x2λ2i+1,where fx=fx;λ,α.Consider the following well-known binomial expansions (for 0 <a < 1):(13)1+a−n=∑j=0∞−1in+j−1jaj.Thus, inserting (13) in (12), we get(14)fx=∑i,j=0∞λα2−1i+jπ2i+1x32i!22i−12i+j+1je−α/x21−e−α/x2λj+1−11−1−e−α/x2λ2i.Again, we can using the binomial expansion in the following term:(15)1−1−e−α/x2λ2i=∑k=0∞−1k2ik1−e−α/x2kλ.Therefore, by inserting (15) in (14),(16)fx=∑i,j,k=0∞−1i+j+kπ2i+12i!22i−12i+j+1j2ikλα2x−3e−α/x21−e−α/x2λk+j+1−1.Again, we can using the binomial expansion in the following term:(17)1−e−α/x2λk+j+1−1=∑m=0∞−1mλk+j+1−1me−mα/x2.Therefore, by inserting (17) in (16), we can write the PDF of SHLIR as(18)fx=2α2∑i=0∞Δmx−3e−m+1α/x2,where Δm=∑i,j,k=0∞−1i+j+k+mπ2i+1/2i!22i2i+j+1j2ikλk+j+1−1mλ.


## 2.2. Quantile Function

By inverting (5), we can obtain the quantile function of the SHLIR distribution, say Qu=F−1u of X, as follows:(19)Qu=α−ln1−1−2/πsin−1u/1+2/πsin−1u1/λ,where u is thought of as a uniform random variable on (0, 1).


## 2.3. Moments

IfX has PDF (6), then its moment can be calculated using the following relation.(20)μr\=EXr=∫0∞xrfxdx.Substituting (18) into (20) yields(21)μr\=2α2∑m=0∞Δm∫0∞xr−3e−m+1α/x2dx.Lety=α/x2; then,(22)μr\=αr∑m=0∞Δm∫0∞y−r/2e−m+1ydy.Then,μr\ becomes(23)μr\=αr∑m=0∞ΔmΓ1−r/2m+11−r/2,r2<1.The SHLIR distribution's moment generating function is given by(24)MXt=∑r=0∞trr!EXr=αr∑r,m=0∞Δmtrr!Γ1−r/2m+11−r/2,r2<1.The incomplete moments of SHLIR are defined by(25)ϕst=∫0txsfxdx.Using (18), φst will be(26)φst=2α2∑m=0∞Δm∫0txs−3e−m+1α/x2dx=αs∑m=0∞ΔmΓ1−s/2,m+1α/t2m+11−s/2,where Γs,t=∫0txs−1e−xdx is the lower incomplete gamma function.The Lorenz and Bonferroni curves are obtained as follows:(27)LFx=φ1xEX=α∑m=0∞ΔmΓ1/2,m+1α/t2/m+11/2∑m=0∞Δmαπ/m+11/2=∑m=0∞ΔmΓ1/2,m+1α/t2/m+11/2∑m=0∞Δmπ/m+11/2,and(28)BFx=LFxFx=∑m=0∞ΔmΓ1/2,m+1α/t2/i+11/2sinπ/21−1−e−α/x2λ/1+1−e−α/x2λ∑m=0∞Δmπ/m+11/2.


## 2.4. Entropies

The entropy of the SHLIR model can be measured by various measures such as Rényi entropy (RE) [28], Havrda and Charvat entropy (HCE) [29], Arimoto entropy (AE) [30], and Tsallis entropy (TE) [31]. These measures of entropy are mentioned in Table 1.Table 1
Various measures of entropy for a distribution with PDFfx at γ.
The measuresFormulaREIRγ=1/1−γLog∫0∞fγxdx,γ≠1,γ>0.HCEHCRγ=1/21−γ−1∫0∞fγxdx−1,γ≠1,γ>0.AEARγ=γ/1−γ∫0∞fγxdx1/γ−1,γ≠1,γ>0.TETRγ=1/γ−11−∫0∞fγxdx,γ≠1,γ>0.∫0∞fγxdx is very complicated to calculate, so it will be solved numerically.


## 3. Maximum Likelihood Estimation

To obtain the ML estimators (MLEs) of the SHLIR model with parametersα and λ, let X1, …, Xn be observed values from this distribution. As a result, the log-likelihood function can be written as(29)ℓ=nlog2π+nlogλ+2nlogα−3∑i=1nlogxi+λ−1∑i=1nlogTi−2∑i=1nlog1+Tiλ+∑i=1nlogcosπ21−Tiλ1+Tiλ.The ML equations of the SHLIR distribution are given by(30)∂ℓ∂α=2nα+2αλ−1∑i=1ne−α/xi2xi2Ti−4αλ∑i=1ne−α/xi2Tiλ−1xi21+Tiλ+2παλ∑i=1ne−α/xi2Tiλ−1xi21+Tiλ2tanπ21−Tiλ1+Tiλ,and(31)∂ℓ∂λ=nλ+∑i=0∞logTi−2∑i=0∞TiλlnTi1+Tiλ+2∑i=1nTiλln1−e−α/xi21+Tiλ2tanπ21−Tiλ1+Tiλ,where Ti=1−e−α/xi2. Equating ∂l/∂α and ∂l/∂λ with zeros and solving simultaneously, we obtain the ML estimators of α and λ.


## 4. Numerical Results

A numerical result is evaluated and compared to evaluate and compare the behaviour of the estimates in terms of their mean square errors (MSEs). From the SHLIR model, we generate 5000 random samplesX1, …, Xn of sizes n = 10, 20, 30, 50, 100, and 200. Four distinct sets of parameters are taken into account, and their ML estimates (MLEs) are computed. The MSEs of the estimated unknown parameters are then computed. In Table 2, the simulated outcomes are listed, and the following observations are found.Table 2
MLEs and MSEs of SHLIR distribution.
nα=1.5,λ=0.8α=1.5,λ=0.5α=1.5,λ=1α=0.5,λ=0.5MLEsMSEsMLEsMSEsMLEsMSEsMLEsMSEs101.7160.3051.8250.6171.6990.2660.6090.0720.9940.2900.6180.0911.3170.7080.6180.091201.6300.1331.6850.2141.6190.1050.5640.0260.9070.0820.5570.0241.1390.1270.5570.024301.5700.0681.5930.1071.5580.0580.5330.0150.8590.0370.5390.0171.0890.0770.5390.017501.5480.0401.5600.0591.5400.0350.5226.335∗0.8410.0230.5206.079∗1.0460.0360.5206.076∗1001.5260.0211.5250.0281.5160.0170.5102.775∗0.8208.132∗0.5102.736∗1.0200.0160.5102.736∗2001.5108.196∗1.5190.0141.5146.478∗0.5081.225∗0.8104.216∗0.5071.178∗1.0125.964∗0.5071.178∗∗ indicates that the value has multiplied by 10−3.For all estimates, the MSEs decrease as sample sizes increase.


## 5. Applications

Two data analyses are provided in this section to assess the goodness of fit of the SHLIR model in comparison to some known distributions such as type II Topp–Leone IR (TIITLIR) in [32], TIR, and IR distributions.Maximized likelihood (A1), Akaike information criterion (A2), consistent Akaike information criterion (A3), Bayesian information criterion (A4), and Hannan–Quinn information criterion (HQIC) were used to compare the models. The model with the lowest values of A1, A2, A3, A4, and A5 is thought to be the best fit for the proposed data.Data I: Bjerkedal [33] observed and reported the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli.Data II: they represent the waiting times (in minutes) before service of 100 bank customers, observed and reported by Ghitany et al. [34].Figures3 and 4 show the fitted cumulative function (ECDF) of the SHLIR distribution, as well as the ECDFs of the compared models (HLIR, TIITLIR, TIR, and IR) for the first and second datasets.Figure 3
Fitted CDFs of models for data I.Figure 4
Fitted CDFs of models for data II.According to Figures3 and 4, the SHLIR distribution is the best fit when compared to the other models mentioned above for the two datasets.Tables3 and 4 show the ML estimates and standard errors (SEs) for the SHLIR model when compared to some known distributions such as HLIR, TIITLIR, TIR, and IR. Tables 5 and 6 also show the corresponding measures of fit statistic using A1, A2, A3, A4, and A5.Table 3
The MLE and SE of the model parameters for data I.
ModelMLEs and SEsSHLIRα,λ0.439(0.05)0.487(0.061)HLIRα,λ0.436(0.05)0.579(0.07)TIITLIRα,λ0.325(0.036)0.404(0.058)TIRα,λ0.352(0.426)−0.942(0.351)IRα0.68(0.04)Table 4
The MLE and SE of the model parameters for data I.
ModelMLEs and SEsSHLIRα,λ2.419(0.226)0.499(0.052)HLIRα,λ2.404(0.226)0.589(0.06)TIITLIRα,λ1.824(0.162)0.43(0.051)TIRα,λ9.978(1.136)−0.812(0.085)IRα3.619(0.181)Table 5
The values of A1, A2, A3, A4, and A5 for data I.
Goodness-of-fit criteriaModelA1A2A4A5A3SHLIR223.087227.087226.802228.9227.261HLIR260.586264.586264.301266.399264.76TIITLIR280.492284.492284.207286.305284.666TIR280.538284.538284.253286.351284.712IR327.518329.518329.375330.424329.575Table 6
The values of A1, A2, A3, A4, and A5 for data II.
Goodness-of-fit criteriaModelA1A2A4A5A3SHLIR628.185632.185632.185634.293632.308HLIR680.806684.806684.806686.915684.93TIITLIR700.214704.214704.214706.323704.338TIR720.665724.665724.665726.774724.706IR759.629761.629761.629762.683761.67Also, Tables5 and 6 confirm that the SHLIR distribution is the best fit among the other models for the two datasets, as the SHLIR distribution has the lowest values of A1, A2, A3, A4, and A5.


## 6. Conclusion

This article investigates a new two-model distribution known as the sine half-logistic inverse Rayleigh (SHLIR). Some fundamental statistical properties of the SHLIR model are calculated and discussed, including the quantile function, moments, moment generating function, incomplete moment, and Lorenz and Bonferroni curves. Entropy measures such as Rényi entropy, Havrda and Charvat entropy, Arimoto entropy, and Tsallis entropy are investigated. The model parameter estimation is discussed using the ML method. Applications to two real datasets show that the SHLIR model outperforms other well-known competitive models such as the HLIR, TIITLIR, TIR, and IR models in terms of fit.



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