Abstract
This article aims to introduce a generalization of the inverse Rayleigh distribution known as exponentiated inverse Rayleigh distribution (EIRD) which extends a more flexible distribution for modeling life data. Some statistical properties of the EIRD are investigated, such as mode, quantiles, moments, reliability, and hazard function. We describe different methods of parametric estimations of EIRD discussed by using maximum likelihood estimators, percentile based estimators, least squares estimators, and weighted least squares estimators and compare those estimates using extensive numerical simulations. The performances of the proposed methods of estimation are compared by Monte Carlo simulations for both small and large samples. To illustrate these methods in a practical application, a data analysis of real-world coating weights of iron sheets is obtained from the ALAF industry, Tanzania, during January-March, 2018. ALAF industry uses aluminum-zinc galvanization technology in the coating process. This application identifies the EIRD as a better model than other well-known distributions in modeling lifetime data.
1. Introduction
In this research life time distribution known as exponentiated inverse Rayleigh distribution (EIRD) was developed and it can be used in reliability estimation and statistical quality control techniques. The Rayleigh distribution is originated from two parameter Weibull distribution and it is appropriate model for life-testing studies. It can be shown by transformation of random variable that if the random variable (r. v) T has Rayleigh distribution, then the r. v. has an inverse Rayleigh distribution (IRD). Reliability sampling plans of IRD are given in Rosaiah and Kantam [1].
Suppose X is a random variable following inverse Rayleigh distribution with scale parameter . Then its pdf, cdf,and reliability function are respectively given byGeneralization of different distribution was discussed in different statistical writings by different authors, mostly applied in reliability estimation, for example, Mudholkar et al. [2], Gupta et al. [3], Nadarajah and Kotz [4], and Mudholkar and Srivastava [5].
Nadarajah and Kotz [4] suggested a method of generating new distribution by using reliability function. Exponentiated Inverse Rayleigh distribution (EIRD) is a generalized form of inverse Rayleigh distribution as suggested by Nadarajah and Kotz [4] as follows: where R(x) in above equation are the reliability function of inverse Rayleigh distribution.
The cumulative density function (CDF) of the EIRD is given by where is the scale parameter and is the shape parameter. The probability density function (PDF) of EIRD isThe inverse Rayleigh distribution is the particular case of (4) for .
Hence for the exponentiated inverse Rayleigh distribution with the scale parameter and shape parameter will be denoted by EIRD .
Reliability and hazard functions of EIRD are as follows:Graphs of PDF, CDF, Reliability, and hazard function of EIRD are depicted in Figures 1–4. From the diagram of the PDF it can be shown that the distribution is left skewed and the CDF shows the increasing pattern as we expected. Also by using reliability function, it can be seen that the distribution can be used in lifetime studies since reliability graph tends to decrease as the time increases. The hazard function curve shows that the first increases and then decreases in shape. The lifetime models that present first increase and then decrease shaped failure rates are very useful in survival analysis. Moreover, the hazard rate will in general converge to a constant value. For an example the infant mortality rises to some extent over time and then mortality decreases after the infants got immunity in the body. For greater details, readers may refer to Kotz and Nadarajah [6].
2. Statistical Properties
In this section, we furnish some significant statistical and mathematical properties of the EIRD such as moments, moment generating function, ordered statistics, mode, quantiles, median, skewness, and kurtosis.
2.1. Moments
The moment about origin is given by Also, Mean of the EIRD can be obtained by putting r=1 in (10).From (10) it can be shown that EIRD has first moment (mean) only and we can only obtain first moment but we cannot obtain variance of this distribution.
2.2. Moment Generating Function
The moment generating function is given byBut we know thatThen (12) becomesThen the moment generating function of EIR distribution is given by
2.3. Order Statistics
Moments of order statistics have great role in quality control testing and reliability to predict time to fail of a certain item by considering few early failures. Suppose are ordered statistics of a random sample drawn from EIRD with CDF and PDF , then the PDF of is given byThe PDF of kth order statistics for the EIRD are as follows:
2.4. Mode
Mode can be obtained by using the following approach:Solution of (18) is called mode of the distribution.
2.5. Quantile and Random Number Generation
Quantiles are very needful for estimation purposes, basically quantile estimators, and also it is used in simulation. The quantile of the EIRD is obtained by the following formulae:Generally, we have the first three, Q1=Q(1/4), Q2=Q(1/2), and Q3=Q(3/4), that is by substituting value of p=1/4=0.25, p=1/2=0.5, and p=3/4=0.75 in , respectively. Quantile is also used in finding the skewness and kurtosis of the distribution.
Suppose UUniform (0,1), then (19) can be used in simulation to generate random number of size n from EIRD as shown below:
2.6. Skewness and Kurtosis Using Quantile Approach
There are different methods which are used to find skewness and kurtosis in a certain distribution. The most common method is the one which uses moments of the distribution, but in EIR distribution we have first moment only. Due to this reason the appropriate approach of finding kurtosis and skewness is by using quantiles. Consider,
When , we get , when , we get , and when , we get .Moors [7] suggested a robust alternative measure of kurtosis as follows:
3. Estimation of Unknown Parameters of EIRD
In this section, different methods which can be used in estimation of parameters α and of the EIRD distribution have been discussed. The various methods of estimation discussed are maximum likelihood, ordinary and weighted least square, and percentile based estimation. In literature many authors studied for various distributions to compare different methods of estimation; for more details see Kundu and Raqab [8], Alkasasbeh and Raqab [9], Mazucheli et al. [10], Do Espirito Santo and Mazucheli [11], and Dey et al. [12–17].
3.1. Maximum Likelihood Estimation Method
Maximum likelihood estimation (MLE) method is mostly used in many writings. This is because estimates MLE satisfy many properties of good estimator. Some of the properties are consistency, asymptotic efficiency, and invariance property. Let be random sample of size n drawn from EIR distribution, then the MLE can be obtained as follows:Equation (26) above can be solved by iterative procedure like Newton Raphson method to obtain the ML estimator of .
Information matrix (I) of EIRD is given bywhere
3.2. Ordinary and Weighted Least Square Estimation Methods
This method was suggested by Swain et al. [18], when they used to estimate parameters of Beta distribution. Suppose are the order statistics of n random samples from EIRD with pdf given above, then the least square estimator (LSE) of the unknown parameters and can be found by differentiating equation (29).With respect to unknown parameters and , solve the following nonlinear equation: The weighted least square estimator (WLSE) can be obtained from the first derivative of the (31) with respect to and and set the result equal to zero.Then solve the following obtained nonlinear equation:
3.3. Percentile Estimation Method
This method was originally proposed by Kao [19, 20]. He supposed that let is an estimate of , then the percentile estimator (PCE) of and can be found by differentiating with respect to and and set the result equal to zero
4. Simulation Study for Comparing Estimation Methods
In this part, a Monte Carlo simulation study is conducted to evaluate the performance of different estimation method for estimating unknown parameters of EIRD. The performance of the different estimators is evaluated in terms of mean square error (MSE). The simulation is conducted by using R- software, 10000 random samples of EIRD was generated with values of n = (20, 40, 50, 100) while choosing = (0.5, 1), (1.5, 1), (2, 1), (2.5, 1), (0.5, 2), (1.5, 2), (2, 2), and (2.5, 2). Average bias and MSE values obtained by the method of MLE, LSE, WLSE, and PCE are shown in Tables 1–4. All methods show that they have consistency property since the values of average bias and MSE decrease as the sample size increases. Based on the value of MSE, the method of MLE shows good performance since it has small value of MSE compared to other methods.
We have also obtained bias and MSE estimates of reliability function and hazard rate functions using MLE. In Table 5 we present the corresponding estimates of reliability and hazard rate functions. From Table 5, it can be seen that the value bias and MSE of hazard and reliability are very small, which indicates that the method is good. Also the values of bias and MSE tend to decrease as sample size increases, which implies that the method is consistent. The entire simulation is done using R software and R code is given in the Appendix.
5. Industrial Application
In this section we investigate two data sets of coating weights (gm/m∧2) from ALAF industry, Tanzania. ALAF industry is the part of Safal group which is the major producer of the most accepted and trusted steel roofing brand. Safal group is operated in the 11 countries which are found in the Eastern and Southern part of the African continent. Safal group brought a major advanced coating technology in Africa with four coating mills located at Kenya, Uganda, Tanzania, and South Africa. ALAF industry is the one of the coating mills of Safal group, and then it deals with improving the quality steel roofing. There are several processes which are done in order to improve the quality of steel roofing; one of those processes is coating process. ALAF industry uses aluminum-zinc galvanization technology in the coating process. We analyze two data sets to illustrate the model validity on EIRD. For the first data set of 72 observations on coating weight by chemical method on top center side (TCS) and for the second data set of 72 observations on coating weight by chemical method on bottom center side (BCS), the data sets are presented in Table 6.
5.1. Model Validity and Selection Criteria
To verify that this EIRD model is suitable for the data sets which have been used, we have used negative log-likelihood value, Akaike information criteria (AIC), Bayesian information criteria (BIC), K-S distance, and p value,
wherewhere denotes the log-likelihood at MLEs, q is the number of parameters, and n is a sample size.
The values of goodness-of-fit measures, the values of negative likelihood obtained by MLE method, and their standard errors (in brackets) of the models fitted are given in the Tables 7 and 8.
The strength of fit of EIRD is compared with the other four distributions, namely: IRD, RD, IWD, and GIED, and the results in Tables 7 and 8 show that EIRD has the highest p value compared with the four distributions. Also histograms and theoretical densities and Q-Q plot verify how well used data set fits to EIRD compared with other four distributions in Figures 5 and 6. The data set of coating weights for March, 2018 from ALAF industry is used to obtain the results.
The parameters of EIRD are estimated by using four different estimation methods, known as, maximum likelihood, least square, weighted least square, and percentile estimation method. The efficiency of the estimation methods differs from one data set to another as shown in Tables 9 and 10. For example least square estimation method is the best among the others in TCs while maximum likelihood is the best estimation method for BCs.
6. Conclusions
EIRD which is derived from this study performs well; from the diagram of the PDF it can be shown that the distribution is positively skewed and the CDF shows the increasing pattern as other distributions. Also by using reliability function it can be seen that the distribution can be used in lifetime studies since reliability graph tends to decrease as the time increases. The hazard function shows the upside down bath-tub curve shape. The unique characteristic of the distribution has only one moment and kurtosis and skewness are found in terms of quantile.
Four methods of estimation was used in parameter estimation; the methods are maximum likelihood, least square, weighted least square, and percentile estimation. From the simulation study it is observed that the method of maximum likelihood is the best compared to other methods since it has minimum value of MSE. Also findings revile that all methods are consistent since the values of bias and MSE decrease as sample size increases. The data set of coating weights for March, 2018 from ALAF industry is used to study the performance of the proposed distribution. It is shown that EIRD is better performed more than the existing distributions namely: IRD, RD, IWD, and GIED.
Appendix
R codes for estimating parameters of exponentiated inverse Rayleigh distribution # Probability density function deird<-function(x,sigma,alpha) p<-(2alphasigma∧2/x∧3)exp(-(sigma/x)∧2)(1- exp(-(sigma/x)∧2))∧(alpha-1) return(p) # cumulative distribution function peird<-function(x,sigma,alpha) d<-1-(1-exp(-(sigma/x)∧2))∧alpha return(d) # quantile function qeird<-function(p,sigma,alpha) q<-sigma/(-log(1-(1-p)∧(1/alpha)))∧(1/2) return(q)
# random number generation reird<-function(n,sigma,alpha) x<-qeird(p=runif(n),sigma,alpha) return(x) # Pdf Curves x <- seq(0, 3, by=.001) plot(x, deird(x, 1,0.5), type=＇＇l＇＇, ylim=c(0,3.5), ylab=＇＇Density＇＇, main=＇＇ ＇＇,lwd=3) lines(x, deird(x, 1,1.5), col=2,lwd=3) lines(x, deird(x, 1.0, 2.5), col=3,lwd=3) lines(x, deird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(＇＇exp＇＇, ＇＇exp＇＇, ＇＇exp＇＇), pch=c(20,20,20),col=c(1,2,3)) legend(par(＇usr＇), par(＇usr＇), xjust=1, c(c(as.expression(substitute(EIRD(sigma==1, alpha==0.5))), as.expression(substitute(EIRD(sigma= =1,alpha==1.5))), as.expression(substitute(EIRD(sigma= =1,alpha==2.5))), as.expression(substitute(EIRD(sigma= =1,alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4))
# CDF curves x <- seq(0, 3, by=.001) plot(x, peird(x, 1,0.5), type=＇＇l＇＇, ylim=c(0,1.1), ylab=＇＇Cumulaive density＇＇, main=＇＇ ＇＇,lwd=2) lines(x, peird(x, 1,1.5), col=2,lwd=3) lines(x, peird(x, 1.0, 2.5), col=3,lwd=3) lines(x, peird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(＇＇exp＇＇, ＇＇exp＇＇, ＇＇exp＇＇), pch=c(20,20,20),col=c(1,2,3)) legend(1.5,0.4, c(c(as.expression(substitute(EIRD (sigma==1,alpha==0.5))), as.expression(substitute(EIRD(sigma ==1,alpha==1.5))), as.expression(substitute(EIRD(sigma ==1,alpha==2.5))), as.expression(substitute(EIRD(sigma ==1, alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4))
# Reliability curves x <- seq(0, 3, by=.001) plot(x, rteird(x, 1,0.5), type=＇＇l＇＇, ylim=c(0,1.0), ylab=＇＇R(t)＇＇,xlab=＇＇t＇＇, main=＇＇ ＇＇,lwd=3) lines(x, rteird(x, 1,1.5), col=2,lwd=3) lines(x, rteird(x, 1.0, 2.5), col=3,lwd=3) lines(x, rteird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(＇＇exp＇＇, ＇＇exp＇＇, ＇＇exp＇＇), pch=c(20,20,20),col=c(1,2,3)) legend(par(＇usr＇), par(＇usr＇), xjust=1, c(c(as.expression(substitute(EIRD(sigma ==1, alpha==0.5))), as.expression(substitute(EIRD(sigma ==1, alpha==1.5))), as.expression(substitute(EIRD(sigma ==1, alpha==2.5))), as.expression(substitute(EIRD(sigma ==1, alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4)) # Hazard function curves x <- seq(0, 3, by=.001) plot(x, heird(x, 1,0.5), type=＇＇l＇＇, ylim=c(0,6), ylab=＇＇h(t)＇＇,xlab=＇＇t＇＇, main=＇＇ ＇＇,lwd=3) lines(x, heird(x, 1,1.5), col=2,lwd=3) lines(x, heird(x, 1.0, 2.5), col=3,lwd=3) lines(x, heird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(＇＇exp＇＇, ＇＇exp＇＇, ＇＇exp＇＇), pch=c(20,20,20),col=c(1,2,3)) legend(par(＇usr＇), par(＇usr＇), xjust=1, c(c(as.expression(substitute(EIRD(sigma ==1, alpha==0.5))), as.expression(substitute(EIRD(sigma ==1, alpha==1.5))), as.expression(substitute(EIRD(sigma ==1, alpha==2.5))), as.expression(substitute(EIRD(sigma ==1, alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4))
### To find estimation EIRD_MLE<-function(B,n,psigma,palpha) require(maxLik) old = options(digits=20) set.seed(2018) #Setting seed o<-0 ite<-0 mps <- matrix(nrow=B,ncol=2)
rIR<-function(n,sigma,alpha) U<-runif(n,0,1) t<-sigma/sqrt(-log(1-((1-U)∧(1/alpha)))) return(t)
LL_eird1<-function(theta,x) n<-length(x) sigma<-theta alpha<-theta -nlog-nlog(theta)-2nlog(theta) +3sum(log(x))+sum((theta/x)∧2)- (theta-1)sum(log(1-exp(-(theta/x)∧2))) mles<-optim(c(55,8.52), LL_eird1,x=Tds, hessian=TRUE,control= list(maxit=10000)) ################## Monte Carlo Simulation ####################### while(o<B) amps<-NULL x<-rIR(n, psigma, palpha) mles<-try(optim(c(psigma, palpha), LL_eird1,x=x, hessian=TRUE,control= list(maxit=10000))par) if(is.double(mles) & mles2]>0) o<-o+1 mps[o,] <- mles # cat(o,＇＇ ＇＇,ite+1, ＇＇∖n＇＇) ite<-ite+1 old = options(digits=4) mediaemv<-c(mean(mps), sd(mps), mean(mps), sd(mps)) round(mediaemv,4) sbias<-mean(mps)-psigma abias<-mean(mps)-palpha smse<-sbias∧2+(sd(mps))∧ amse<-abias∧2+(sd(mps))∧ print(c(n,psigma, palpha,sbias,abias,smse,amse))
Abbreviations
| EIRD: | Exponentiated Inverse Rayleigh Distribution |
| r.v.: | Random variable |
| PDF: | Probability Density Function |
| CDF: | Cumulative Density Function |
| MLE: | Maximum Likelihood Estimator |
| LSE: | Least Square Estimator |
| PCE: | Percentile Estimator |
| WLSE: | Weighted Least Square Estimator |
| MSE: | Mean Square Error. |
Data Availability
The used data sets are given in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Methodology and computations are done by GS Rao and writing and data collection done by M. Sauda. Both authors read and approved the final manuscript.