Bayesian Estimation of the Shape Parameter of Generalized Rayleigh Distribution Under Symmetric and Asymmetric Loss Functions

Bayesian Estimation of the Shape Parameter of Generalized Rayleigh Distribution Under Symmetric and Asymmetric Loss Functions

Chapter One

Aim and Objectives of the Study

The aim of this work is to estimate the shape parameter of GRD using Bayesian approach.

We wish to achieve the stated aim through the following objectives

By estimating the shape parameter (α) when the scale parameter (λ) is known using both informative and non-informative priors under symmetric lossfunction
By estimating the shape parameter (α) when the scale parameter (λ) is known using both informative and non-informative priors under asymmetric lossfunction
To compare the performances of the proposed estimators with that of Maximum Likelihood Estimators in terms of Mean SquareError

CHAPTER TWO:

LITERATURE REVIEW

Kundu and Raqab (2005) considered different estimation procedures (maximum likelihood estimation, modified moment estimator, least square estimator, weighted least square estimator, percentile-based estimator and modified L-moment estimator) to estimate the unknown parameter(s) of a GRD. The study compared the performance of the different estimators using Monte-Carlo simulations mainly with respect to their biases and mean square errors for different sample sizes and different parameter values. (Kundu and Raqab, 2005) showed that, when the sample size is small (say, n=10), the performance of most of the methods are quite bad. In particular, the estimation of the shape parameter (α) becomes very difficult for small sample sizes. The biases of all the methods are quite severe for small and moderate sample sizes (n ≤ 20). The study also showed that, the least square estimate performs quite well for n≤20 and for n ≥ 30, the weighted least square estimates outperforms the least square estimate marginally. If mean square error is considered, the weighted least square performs better than the rest in most of the cases considered.

Different methods of estimating R = P (Y < X ) when Y and X both follow GRD with different shape parameters but the same scale parameter were compared by (Raqab and Kundu, 2005). When the scale parameter is unknown, it is observed that the MLEs of the three unknown parameters can be obtained by solving one non-linear equation. An iterative procedure for computing the MLEs of the unknown parameters and the MLE of R was developed. The asymptotic distribution of R was obtained and this was used to compute the asymptotic confidence intervals. It was observed that, even when the sample size is quite small, the asymptotic confidence intervals work quite well. Two bootstrap confidence intervals were also proposed and their performances were also quite satisfactory. On the other hand, when the scale parameter is known they compare MLE and UMVUE with different Bayes estimators. It was observed that the Bayes estimators with non-informative priors behave quite similarly with the MLEs

In similar studies, Mahdi (2006), estimated the parameters of a Rayleigh distribution using five different estimation techniques namely: maximum likelihood, method of moment, probability weighted moment method, least square method and least absolute deviation method. He proposed the modified maximum likelihood estimation method for the parameters and compares it with the above methods. In his comparison, all the methods performed reasonably well except the method of moments. On the other hand, the modified maximum likelihood method provides better estimates for the parameters when the sample sizes are not small (n≥10), while in the case of small samples, the probability weighted moment method outperforms the maximum likelihood method for the estimation of the threshold parameter and performs almost as good as the maximum likelihood method for the estimation of the scale parameter. Hence, Mahdi (2006) recommended the use of modified maximum likelihood method for the parameter estimation of the Rayleigh distribution if the sample size is large.

A conventional method for estimating the two parameters of generalized Rayleigh distribution for different sample sizes (small, medium and large) was suggested by (AL- Naqeeb and Hamed, 2009). The estimators were compared by using mean square error (MSE) as a performance measure based on the simulated data used in the study.

Abdel-Hady (2013) extended the Marshall and Olkin’s bivariate exponential model to the Generalized Bivariate Rayleigh (GBR) Distribution. The CDF, pdf and the conditional distribution of the GBR distribution were computed. The maximum likelihood estimation procedure for the estimation of the GBR parameters when all parameters are unknown and the observed Fisher Information Matrix were also derived.

Al – Kanani and Abbas, (2014) studied the Bayesian and non-bayesian methods in estimating the parameters of the GRD. The non-bayesian methods considered are the maximum likelihood estimator and the moment estimator while the single stage shrinkage estimator was determined and derived for the Bayesian method. Reliability function and hazard function were also obtained.

Pak, et.al., (2013) study the problem of estimating the scale parameter of Rayleigh distribution under doubly type-II censoring scheme when the lifetime observations are fuzzy and are assumed to be related to underlying crisp realization of a random sample. A new method to determine the maximum likelihood estimate of the scale parameter was proposed and its asymptotic variance was derived using the missing information principle.

Pathak and Chaturvedi (2014) explored and compared the performance of uniformly minimum variance unbiased estimators as well as maximum likelihood estimators of the reliability functions R(t) = P (X > t) and P = P (X > Y ) for the two-parameter exponentiated Rayleigh distribution. (Pathak and Chaturvedi, 2014) showed that, the uniformly minimum variance unbiased estimators of R(t) and “P” are superior to their maximum likelihood counterparts.

CHAPTER THREE:

METHODOLOGY

Maximum Likelihood Method

In this section, the definition of Maximum Likelihood Estimator is given and the definition will be use in finding an estimator for the shape parameter of GRD.

Let x₁, x₂, ⋯ , x_n be a random sample from a population X with probability density function f(x; θ), where θ is an unknown parameter. The likelihood function, L(θ), is defined to be the joint density of the random variables x₁, x₂, ⋯ , x_n. That is,

L(q ) = Õ f (xi ;q )

i=1

The sample statistic that maximizes the likelihood function, L(θ) is called the maximum

likelihood estimator of θ and is denoted by

qɵ . The GRD(α, λ) has probability density function equation (1.2). If the scale parameter λ is known and without loss of generality let

λ=1, then the probability density function of GRD(α, 1) is given by

f(x; α, 1) = 2αx(1 − e^–^s²)α–1e^–^s² (3.1)

If x₁, x₂, ⋯ , x_n is a random sample from the GRD(α, 1) distribution as in (3.1), then the joint density function is

CHAPTER FOUR:

ANALYSIS AND DISCUSSION

In this chapter, results of some numerical experiments were obtained in order to compare the performance of the different estimators proposed in the previous chapter with that of maximum likelihood estimator. Extensive Monte Carlo simulations were carried out to compare the performance of the different estimators with respect to biases and mean- squared errors (MSEs) for different sample sizes (n= 15, 20, 30, 50 and 100) and different shape (α) parameter values of 0.25, 0.5, 1.0, 1.5, 2.0 and 2.5. All results were replicated 10000 times and averaged over.

CHAPTER FIVE:

SUMMARY, CONCLUSION AND RECOMMENDATIONS

Summary

The primary goal of this work as initially stated in the objectives, was to estimate the shape parameter of a GRD when the scale parameter is known using Bayesian approach. The shape parameter was assumed to follow an Extended Jeffrey’s prior distribution, Uniform prior distribution and Gamma prior distribution. Symmetric and asymmetric loss functions were considered in estimating the parameter. Squared error loss function which is classified as symmetric loss function because it assigns equal importance to the losses due to both over estimation and under estimation was assumed while in the asymmetric case Entropy and Precautionary loss functions were considered. The Bayes estimators of the aforementioned loss functions were obtained. Relative efficiency and Posterior risk of the estimators under the various loss functions were obtained and finally, Monte Carlo simulation was carried out in order to compare the performance of the estimators using Mean Square Error as a performance measure.

Conclusion

In this work, MLE and Bayesian estimation of the shape parameter of a generalized Rayleigh distribution is considered. Three prior distributions on the shape parameter is assumed and the Bayes estimators under the symmetric loss function (squared error) and asymmetric (Entropy and precautionary) loss functions were provided. Comparison between the performances of the MLEs and the Bayes estimators were carried out and it was observed that the Non-classical/ Bayesian estimators were more stable than the classical estimators in terms of Mean Square Error. The estimates under the extended Jeffrey’s prior were found to be more stable than the estimates under uniform and gamma priors. It was also observed that under the informative prior, that the estimates under the entropy loss function is better than the estimates under the squared error and precautionary loss functions.

Recommendations

Based on our observations in the previous chapters, we recommend the following for the estimation of the shape parameter of GRD when the scale parameter is known

If you have enough information about the shape parameter that is, it follows gamma, then use entropy loss function in order to estimate the shape
If on the other hand you have little or no information about the prior distribution of the shape parameter, assumed extended Jeffrey’s prior under the assumption of squared error or entropy loss

Contribution to Knowledge

In this research work, we were able to contribute to the body of knowledge in so many ways, which include the following:

We were able to give some formulae for estimating the shape parameter of GRD when the scale parameter is known under three different prior distributions and three different loss
We were also able to give the variances of the various estimators proposed in this work.
Finally, we were able to show that the estimators under extended Jeffery’s prior is morestable than estimators under uniform and gamma priors in terms of MSE and

we further show that, when loss functions are considered, the estimators under Entropy loss function is more stable in terms of MSE than the estimators under Square error and precautionary loss functions.

Areas of Further Research

The scale parameter of GRD can be estimated when the shape parameter isknown
The shape parameter can be estimated when the scale parameter is unknownand
The scale parameter of GRD can be estimated when the shape parameter is unknown

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