A New Three-Parameter Weibull Inverse Rayleigh Distribution: Theoretical Development and Applications

In this work, a three-parameter Weibull Inverse Rayleigh (WIR) distribution is proposed. The new WIR distribution is an extension of a one-parameter Inverse Rayleigh distribution that incorporated a transformation of the Weibull distribution and Log-logistic as quantile function. The statistical properties such as quantile function, order statistic, monotone likelihood ratio property, hazard, reverse hazard functions, moments, skewness, kurtosis, and linear representation of the new proposed distribution were studied theoretically. The maximum likelihood estimators cannot be derived in an explicit form. So we employed the iterative procedure called Newton Raphson method to obtain the maximum likelihood estimators. The Bayes estimators for the scale and shape parameters for the WIR distribution under squared error, Linex, and Entropy loss functions are provided. The Bayes estimators cannot be obtained explicitly. Hence we adopted a numerical approximation method known as Lindley’s approximation in other to obtain the Bayes estimators. Simulation procedures were adopted to see the effectiveness of different estimators. The applications of the new WIR distribution were demonstrated on three real-life data sets. Further results showed that the new WIR distribution performed credibly well when compared with five of the related existing skewed distributions. It was observed that the Bayesian estimates derived performs better than the classical method.


Introduction
In probability theory and statistics, a generalized lifetime model suitable for fitting survival and engineering data is often of interest in the survival and reliability analysis [1]. Convoluted distributions derived from the T-R{Y} framework by [2]and [3] have the probability density function (PDF) that are weighted hazard function of a baseline distribution, which can provide insight into characteristics of failure times and hazard functions that may not be available with classical models.
Many authors have derived hybrid or convoluted distributions that are members of this family, especially the ones that were derived using the standard quantile function of the log-logistic distribution such as Weibull-Uniform{Log-logistic} by [2]; Exponential-Normal{Log-logistic} [3]; Lomax-Cauchy{Log-logistic} [4]; Weibull-Gamma {Log-logistic} [5]; Weibull-Exponential{Log-logistic}, Gamma-Exponential{Log-logistic} and Normal-Exponential { Log-logistic } [ 6 ]; Weibull-Normal { Log-logistic } [7]; and [8]which exploited the statistical properties of Odd Lomax-Exponential{Log-logistic} distribution. All distributions are members of the T-R{log-logistic} family that have a probability density function with three components vis-a-vis the hazard function, survival function, and the odd-ratio. These three components are very important in the survival analysis. The distributions of interest in this research are the Weibull and Inverse Rayleigh distributions.
Several researchers have exploited the importance of Inverse Rayleigh distribution in social, medical, and engineering fields. The extension and generalization of Inverse Rayleigh distribution were studied by some authors. [9] studied some properties of Exponentiated Transmuted Generalized Rayleigh distribution, [10] proposed Half-Logistics Inverse Rayleigh distribution, [11] studied type II Topp-Leone inverse Rayleigh distribution, [12] Transmuted inverse Rayleigh distribution (TIR), [13] fitted the same data on odd Frechet inverse Rayleigh distribution (OFIR) and one parameter Inverse Rayleigh distribution. The possibility of using inverse Rayleigh distribution in this regard has not been accessed.
In this present study, a new member of the family of Weibull-G distribution defined by [14] through the T-R{Y} transformation was proposed. This new proposal is called the Weibull-Inverse Rayleigh { Log-logistic } distribution having three parameters. Different functional properties of this new distribution are derived and their applications to survival and reliability analysis were presented using three real-life datasets, two of which were on corona-Virus 2019 (COVID- 19), and the third one was on the material strength of carbon fiber.
The remaining parts of this article are arranged thus. In Section 2, the new three-parameters Weibull-inverse Rayleigh (WIR) distribution was proposed and its functional forms were presented. Some of its statistical properties were investigated in Section 3 while the procedures for parameter estimations were presented in Section 4. Applications of the new WIR distribution to three real-life data sets were provided in Section 5 while Section 6 presented the conclusion to the work.

Theoretical Development
In the T-R{Y} framework by [2], the random variable is a transformer that is used to transform the random variable into a new family of generalized distributions of using the quantile function of . If follows the log-logistic distribution, the standard quantile function of as a function of the Cumulative Distribution Function (CDF) of gives the odd-ratio of . Thus, the CDF of the generalized family of distributions as defined by [14] is given by (1) and the corresponding probability density function (PDF) of , ( ) is derived by differentiating (1) with respect to to have Alternatively, the PDF in (2) can be written as where ℎ ( ) is the hazard function of , ( ) is the survival function of and ( ) is the odd-ratio of . Let follows the Weibull distribution with CDF given by its corresponding PDF is given by By using the CDF of [14], given in (1), we derived the Weibull-G family by putting (4) in (1) to have where ( ) is the non-decreasing CDF of any known continuous probability distribution. The corresponding PDF to the Weibull-G family is derived by differentiating (5) with respect to to have

The New Three-Parameters Weibull-Inverse Rayleigh (WIR) Distribution
Let be a random variable that follows the Inverse Rayleigh distribution with CDF given by with the corresponding PDF given by By using the Weibull-G framework given in (5), we derived the CDF of Weibull Inverse Rayleigh (WIR) distribution by putting (7) in (5) to have Solving (8) further to have 9) and the corresponding PDF is derived by differentiating (9) with respect to to have where is a shape parameter, and are scale parameters. Thus, the CDF in (9) and the PDF in (10) are the CDF and PDF of the proposed three-parameters WIR distribution. The PDF of WIR distribution is demonstrated with different shape and scale parameters are shown in Figure 1.

Properties of the New WIR Distribution
The structural properties of the new WIR distribution are investigated in detail in this section. For ease, a random variable with PDF ( ) in (10) is said to follow the three-Parameters Weibull Inverse Rayleigh distribution and is denoted by W-IR( , , ).

The Survival Function
The survival function of the WIR( , , ) is given by The function in (11) is a function that gives the probability that an item of interest will survive beyond any given specified time ∈ , also known as the reliability function.

The Hazard Function
The hazard function of the WIR( , , ) is given by The function in (12) of random variables that follows WIR distribution is called the hazard rate of W-IR distribution, also known as the failure rate or force of mortality.

Figure 2. Hazard Function Plot of WIR Distribution
Different shapes of hazard function of WIR are displayed in Figure 2 with different values of scale and shape parameters.

The Cumulative Hazard Function
The cumulative hazard function, ( ) of a continuous random variable that follows the WIR distribution is derived from this definition Put equation (11) into (13) to have Equation (14) defines the probability of failure at time given survival until time ∀ ∈ .

The Reverse Hazard Function
Given a random variable of that has a WIR distribution, the Reverse Hazard function of x is given by

The Median, Skewness, and Kurtosis
Lemma 1: Let be a random variable that follows a Weibull distribution with parameters and , then the quantile function of WIR( , , ) is given by where ( ) is the quantile function of Weibull distribution with parameters and .

Proof.
Using the concept of the T-R{Y} framework. If follows T-Inverse Rayleigh { Y } distribution, then it follows that This implies that where ( ) is the CDF of a standard log-logistic distribution with all parameters equal to one, and it is given by Put equation (18) in (17) to have Equation (19) completes the proof. The quantile function of WIR( , , ) distribution in equation (19) can be computed from the Weibull quantile function by using R code ( = ( , , )), where is uniformly distributed on the closed interval 0 and 1.

Theorem 1:
Let be a random variable that follows a WIR( , , ), its quantile function is given by

Proof.
From Lemma 1, it follows that ( ) is the quantile function of Weibull distribution with parameters and given by Substitute equation (20) into (19) to arrive at the quantile function of WIR( , , ) and it is given by

Median
The median and other measures of partition for the new WIR distribution can be derived from the quantile function in equation (21).
The median of WIR( , , ) is given by

The Skewness and Kurtosis
The coefficient of skewness and kurtosis are mostly derived from the moment of a distribution. But we shall adopt measures of skewness and kurtosis based on quantile functions. The measure of skewness S and kurtosis K defined by [15] and [16] respectively are based on quantile functions and they are defined as Proof. Recall the quantile function of WIR distribution given in (21) as Factorise (25) by factoring out , since all the terms contain to have The same procedure is also used to prove that the kurtosis does not contain parameter . Put (21) in (23) to have The results show that kurtosis and skewness are not a function of  Table 1 shows that the value of does not affect the shape of the WIR distribution. The skewness is only affected by the parameters and .

Proof.
From the CDF in (8), we derive the PDF of WIR distribution as and using Taylor series expansion of from [17], we have the following series if (28) is substituted into (27) to have Then, further expansion of (29) gives Hence the linear representation of the new WIR distribution is given as and where ℎ ( +1)+ ( ) is the PDF of Inverse Rayleigh distribution with scale parameter [ ( + 1) + ] 1/2

The Order Statistic
The th order statistic of a random variable is given by If ~WIR( , , ), then the : ( ) of is derived thus; Inserting (33) and (34) into (32) gives By series expansion, we have Thus, Further series expansion of (37) gives and Therefore, the density function of order statistic is given as

The Monotone Likelihood Ratio Property of WIR Distribution
In statistics, the monotone likelihood ratio property (MLRP) is a property of the ratio of two probability density functions.

Definition 4. Let ( ) and
( ) be the PDFs of two distributions with respect to , if for every > , , ∋ , then ( 2 ) ( 2 ) that is if the ratio is non-decreasing in the argument .
If the functions are first-differentiable, the property is sometimes stated as Then ( ) and ( ) have the MLRP in . For two distributions that satisfy the definition with respect to some argument , we say they "have the MLRP in .

The Moment
Theorem 4: If ~W-IR( , , ), the th moment of exist only for ≤ and it is given by Proof.
Recall the linear expansion of the PDF of WIR distribution in (30) as The th moment of a random variable is given by ( ) = ∫ ( ) . So, that we have The sum of integral is equal to the integral of a sum so that we have Using the gamma function on (43) gives Equation (44) completes the proof and it is the th moment of WIR distribution with parameters , , and . If = = 0, the th moment becomes The mean of WIR distribution is derived from (44) if = 1 and it is given by According to [18], to have a reduced form of the moment, we set = = 0, the mean in (46) is then reduced to It is obvious that only the first moment and fractional moment exist. Higher moments of W-IR distribution do not exist. Thus, its variance does not exist.

The Asymptotes of WIR Distribution
The value of the function WIR distribution when ( ) approaches∞.
The functions ( ) and ℎ ( ) of WIR distribution will be undefined if and = ∞. Thus, the vertical asymptote of ( ) is given by and the vertical asymptote of ℎ ( ) is given by So, equations (49) and (50) are the vertical asymptotes of the PDF and hazard function of WIR distribution respectively.

The Maximum Likelihood Estimation
For a random sample x= ( 1 , 2 , … , ) of size from (10) Differentiating the log-likelihood function ( , , ) partially with respect to , , and solving for , and from equations (57) -(59) analytically by equating to zero is impossible. The maximum likelihood estimator of , and says ̂, ̂ and ̂ are the solutions to the equations (57) -(59). Thus, the solution is intractable, therefore, the numerical method of Newton Raphson is adopted as follows: The Jacobian matrix must be a non-singular symmetric matrix so its inverse must exist. So, using the Newton Raphson method we have with error term being the absolute differences between the new and the previous value of and in the iterative algorithm. That is where , and are the initial values of , , and respectively.

The Quantile Function Estimation Method
This method is a method that is not common. For three parameters distribution, we have the system of equations given by where 1 , 2 and 3 are the 1st, 2nd and 3rd quartiles respectively, which can be obtained from the given data. Thus, we solve the system of equations for the parameters , and , which gives the estimates of the three parameters, the method was successfully implemented by [19].

The Bayesian Estimation
In probability and statistics, the Bayesian inference is a statistical inference method in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or facts from data becomes available. It is an important technique in mathematical statistics. It employs the use of the posterior predictive distribution to do predictive inference. Several methods of Bayesian estimation select measurements of central tendency from the posterior distribution [20]. For example, if there exists a finite mean for the posterior distribution, then the posterior mean is a method of estimation [21].
Bayesian inference has been applied in different Bioinformatics scenarios, by building inferences from prior knowledge and available data, such as differential gene expression analysis and general cancer risk model (Continuous Individualized Risk Index), where serial measurements are incorporated to update a Bayesian model [22].
In this section, we consider three types of loss functions namely: Squared Error Loss Function which is also known as Quadratic Loss Function, the Linex Loss Function, and the Generalization of Entropy Loss Function. Therefore, the Bayesian estimators of the parameters of the new WIR distribution were considered under these three loss functions.

The Prior Distribution
To obtain the estimates of , , and , we assumed that all the parameters are real-valued random variables with probability density function ( , , ). The posterior distribution Pr( , , | ) is the conditional probability density function of , , and given data x. We assumed that , and are unknown and certainly their conjugate priors do not exist. So, we consider the following prior distribution on , , and ; An informative prior distribution of the form

The Lindley's Approximation
The joint posterior distribution of the unknown parameters , , and cannot be solved analytically because it has no closed-form solution. Therefore, in this study we employed the Lindley's approximation method to investigate the quantities of the parameters , , and under the Squared error loss function, Linex loss function, and the General Entropy loss function. According to [23], if n is sufficiently large, any ratio of the integral of the form where ̂,̂ and ̂ are the maximum likelihood estimators of , and while = 1 1 + 2 2 + 3 3 = 1,2,3.
If is the parameter to be estimated by an estimator ̂ then, the square error loss function can be defined as This method has been adopted by several authors such as [24] and [25]. [26] defined the Linex loss function as where q and g are known as the shape and scale parameters of the loss function. We assume q = 1 in this study. According to [27], the Bayes estimator of the linex function is the value ̂ that minimizes (97) stated as provided that [ ] exits. This loss function has been widely used by [28] - [32]. The general Entropy loss function which is also known as modified Linux loss function is defined as

Simulation Study
In this section, we generate 2000 random samples The results presented in Table 2     The results of the simulation exercises are presented in Tables 2 to 4. It can be observed from the various results in the tables that the estimated values for all three parameters are close to their true values. Expectedly, the mean squares errors of the parameters decrease as sample sizes increase for all the estimates provided by the four estimators (ML, BS, LL, and GE). We observed that the results obtained under both the Bayesian and Classical methods become better as the sample size increases. Also, we deduced that as the sample sizes increase the estimated values for the three parameters approached the true value of the parameters. In terms of MSE, the Bayesian estimators with relatively smaller MSEs under the three-loss functions provided better results than the MLE at all the sample sizes considered. However, within the Bayesian context, all the Bayesian estimators under the three-loss functions considered compete favorably with one another, especially at sample sizes above 25.

Applications
In this section, we demonstrate the applications of the new WIR distribution on three real-life datasets. The first two datasets are the cases of the COVID-19 dataset (January 21 to March 27, 2020) as reported by WHO (2020) and Worldometers (2020). The third data were on the Strength measured in GPA for single carbon fibers and impregnated 1000-carbon fiber tows at gauge lengths of 20 mm as reported in [33].
The Akaike Information Criterion (AIC), Bayesian information criterion (BIC), and Hannan-Quinn information criterion (HQIC) are used to compare the performances of all the models on the three data sets employed.  Table 11 shows the summary statistics of the WIR, HLIR, TIR, OFIR, and IR distributions. These five distributions are fitted to the data 3 using maximum likelihood estimation. The MLES of the parameters (with their standard errors) are discussed in Table 9 and their corresponding log-likelihood values. AIC, BIC, HQIC and p-values are displayed in Table 10. Hence, the new WIR model provides the best fit among the other models for data3 as shown in Table 12, since it has the lowest values of AIC, BIC, and K-S Values.

Conclusions
In this research paper, we propose a new three-parameters WIR distribution. Some mathematical properties of the new WIR distribution such as moment, order statistic, asymptotic, skewness, kurtosis, and its monotonic likelihood property were investigated. The method of the maximum likelihood estimation and Bayesian method were adopted to estimate the parameters of the distribution. Figures 1 shows the density plot for different scale and shape parameter values. Figure 2 displayed the hazard function for different values of scale and shape parameters. Figure 3, shows the MSEs for the four types of estimations (ML, BS, LL, and GE) adopted in this study. It can be seen that the MSEs for the Bayesian estimators (BS, LL, and GE) are smaller than that of the classical method (ML). Also, as the sample size increases the MSEs decrease for all four estimations. Figure 4 (a-c), displayed the histogram and empirical density, probability plot and time plot, quantiles distribution, empirical CDF, and theoretical CDF for the WIR distribution on the COVID-19 dataset. Figure 5 (a-c) fitted PDF, time plot, and Q-Q plots of the new WIR distribution for the Strength (20mm) data set (data 3).
The new WIR distribution was fitted to three real-life data sets to demonstrate its application and efficiency. All the results obtained showed that the proposed distribution provided the best fit compared to other competing models of the same baseline distributions based on several models' selection criteria adopted in this work.
Further results from simulation studies showed that the estimated values of all three parameters of the new WIR distribution are close to their true values. Consequently, it was observed that the standard errors of the parameters are relatively smaller than those provided by other competing existing distributions compared and that these standard errors reduce as the sample sizes increase for all the parameter values specified in the simulation study.
Fundamentally, the results from this work showed that the Bayesian estimates of the new WIR distribution are better than those provided by the classical frequentist method of MLE as can be observed from Table 2 to 4 and graphically displayed by Fig 3. The MSEs under the Bayesian methods are less than that of the classical approach.
Although, the performances of both the MLE and Bayesian estimators of the new WIR distribution are not worst off as can be observed in various results, the superiority of the Bayesian estimators over the MLE necessitates our recommendation here that the Bayesian estimation techniques be employed in future applications of the new WIR distribution for better efficiency irrespective of the sample sizes.