Thesis Abstract

<p> In 2001, Surles &amp; Padgett introduced Generalized Rayleigh Distribution (GRD). This<br>skewed distribution can be used quiet effectively in modeling life time data. In this work,<br>Bayesian estimates of the shape parameter of a GRD were determined under the<br>assumption of both informative (gamma) and non-informative (Extended Jeffery’s and<br>Uniform) priors. The Bayes estimates were obtained under both symmetric and asymmetric<br>loss functions. The performances of these estimates were compared to the Maximum<br>Likelihood Estimates (MLEs) using Monte Carlo simulation. <br></p>

Thesis Overview

<p> </p><p>INTRODUCTION<br>1.1 Background to the Study<br>Statistical Inference is the branch of statistics concerned with using probability concept to<br>deal with uncertainty in decision-making. It refers to the process of selecting a sample and<br>using a sample statistic to draw inference about a given population parameter. The field of<br>statistical inference is divided into the theory of estimation and hypothesis testing.<br>1.1.1 Theory of estimation<br>Statistical estimation or simply estimation is concerned with the methods by which<br>population characteristics are estimated based on information drawn from a sample. The<br>theory of estimation is further sub-divided into Point and Interval Estimation. A point<br>estimator is a random variable varying from sample to sample and its value is called point<br>estimate i.e. a point estimate is a single value estimate for the parameter. There are several<br>methods of finding a point estimator which can all be broadly classified into the Classical<br>Methods and Non-classical/ Bayesian Methods.<br>In classical approach, the unknown parameter is assumed to be fixed quantity. Inferences<br>in classical approaches are based on a random sample only i.e. if a random sample<br>,,⋯, is drawn from a population with probability function<br>; and based on<br>the sample knowledge the estimate of is obtained. The classical approach is based on the<br>concept of sampling distribution and it does not use any of the prior information available<br>as a result of familiarity with previous studies. There are different methods of point<br>estimation under the classical approach. These include: Maximum Likelihood Estimation<br>(MLE), Method of Moment Estimation, Percentile Estimation, Least Square Estimation,<br>Weighted Least Square Estimates, and so on. On the other hand, in Non-classical/ Bayesian<br>2<br>approach, the parameter is assumed to be a random variable which can be described by a<br>probability distribution (known as prior distribution) that is, the unknown parameter θ<br>(being random) follows a prior distribution. Hence, Bayesian approach combines new<br>information that is available with prior information to form basis for inference.<br>The fundamental difference between the Bayesian and frequentist approaches to statistical<br>inference is characterized in the way they interpret probability, represent the unknown<br>parameters, acknowledge the use of prior information and make the final inferences. The<br>frequentist approach considers probability as a limiting long-run frequency, while the<br>Bayesian approach regards probability as a measure of the degree of personal belief about<br>the value of an unknown parameter θ.<br>1.1.2 Generalized Rayleigh distribution (GRD)<br>Twelve different families of cummulative distribution function used in modeling lifetime<br>data were suggested by Burr (1942). Burr Type X distribution is among the most popular<br>distributions that receives the most attention among these families of cummulative<br>distributions.<br>In 2001, two-parameter Burr Type X distribution was introduced by Surles &amp; Padgett.<br>Kundu and Raqab (2005), Lio, et.al, (2011) and Abdel-Hady (2013) prefer to call this<br>distribution GRD which will be adopted in this work. For α&gt;0 and λ&gt;0, the Cumulative<br>Distribution Function (CDF) of the two-parameter GRD is given by:<br>F<br>x; α, λ = 1 −</p><p>for x, α , λ &gt;0 (1.1)<br>Its probability density function (pdf) is given by:<br>3<br>f<br>x; α, λ = 21 −</p><p>, &gt; 0<br>0, “#$â„Ž&amp;<br>‘ (1.2)<br>where α and λ are shape and scale parameters respectively.<br>Shape and scale parameters are used to determine the shape and location of a distribution.<br>Shape parameter allows a distribution to take on a variety of shapes depending on the value<br>of the shape parameter, while the scale parameter stretches or squeezes the graph of a<br>probability distribution. In general, the larger the scale parameter, the more spread out the<br>distribution and the smaller the parameter, the more compressed the distribution appears to<br>be.<br>GRD is widely used in modeling events that occur in different fields such as medicine,<br>social and natural sciences. In Physics for instance, the GRD is used in the study of various<br>types of radiation such as light and sound measurements. It is used as a model for wind<br>speed and is often applied to wind driven electrical generation. For details, see Samaila and<br>Cenac, (2006). It is also used in modeling strength and lifetime data (Surles and Padgett<br>(2001), Lio, et.al (2011), Kundu and Raqab, (2005)). Hence, the GRD has a survival and<br>hazard functions as shown in equations (1.3) and (1.4) respectively.<br>Survival function S<br>x; α, λ = 1 − F<br>x; α, λ = 1 − 1 −</p><p>(1.3)<br>Hazard function h<br>x; α, λ = *<br>; ,<br>+<br>; , =<br>,-.<br>/0 1<br>2.3<br>-.<br>/0<br>,-.<br>/0 1<br>2 (1.4)<br>Survival function is the probability that the survival time X takes a value greater than a<br>specific value x ie 4<br>= 5<br>&gt; , while the hazard function is a measure of how likely<br>4<br>an individual experiences an event as a function of his/her age. These two functions are<br>used to describe the distribution of survival time data.<br>Figure 1. 1: The Graph of Generalized Rayleigh Distribution for different values of shape parameters when the<br>scale parameter takes the value one<br>The graph of the distribution is shown in Figure 1.1 for different shape parameter values. It<br>is clear from the Figure that the pdf of a GRD is a decreasing function if α ≤ ½ and it is<br>right skewed uni-modal when α &gt; ½. (See also Kundu and Raqab, 2005).<br>1.2 Statement of the Problem<br>Bayesian inference requires appropriate choice of priors for parameters. But there is no way<br>to conclude that one prior is better than another. In a situation where one have enough<br>5<br>information about the parameter(s) then using informative prior(s) will be the best practice<br>for choosing a prior, otherwise, a non-informative prior suffices.<br>1.3 Aim and Objectives of the Study<br>The aim of this work is to estimate the shape parameter of GRD using Bayesian approach.<br>We wish to achieve the stated aim through the following objectives<br>i. By estimating the shape parameter (α) when the scale parameter (λ) is known using<br>both informative and non-informative priors under symmetric loss function<br>ii. By estimating the shape parameter (α) when the scale parameter (λ) is known using<br>both informative and non-informative priors under asymmetric loss functions<br>iii. To compare the performances of the proposed estimators with that of Maximum<br>Likelihood Estimators in terms of Mean Square Error<br>1.4 Significance of the Study<br>In Bayesian approach, the parameter is viewed as random variable behaving according to a<br>subjective (prior) probability distribution that describes our confidence about the actual<br>behavior of the parameter, whereas in classical approach, the parameter is assumed to be<br>fixed but unknown. In Bayesian inference, conclusions are made conditional on the<br>observed data i.e. there is no need to discuss sampling distribution using this method. While<br>in the classical approach one needs not be concerned about any prior knowledge other than<br>the available information observed. Bayesian inference also provides a convenient model<br>for implementing scientific method. The prior distribution is used to state the prior<br>knowledge we have about the parameter of interest, while the posterior distribution reflects<br>6<br>the updated knowledge about the population parameter in line with the new information<br>collected from data.<br>1.5 Motivation<br>Statistically, modeling of real life scenario help us to better understand and explain<br>unforeseen eventualities when they take place, thereby enabling us to reproduce such a<br>scenario either on a large and/ or on a simplified scale aimed at describing only critical<br>parts of the phenomenon. These real life phenomena are captured by means of statistical<br>distribution models, which are extracted or learned directly from data gathered about them.<br>Every distribution model has a set of parameters that needs to be estimated. These<br>parameters specify any constant appearing in the model and provide a mechanism for<br>efficient and accurate use of data.<br>1.6 Limitation<br>The study will focus only on estimating the shape (α) parameter when the scale (λ)<br>parameter is known under the symmetric and asymmetric loss functions assuming<br>informative and non-informative priors.<br>1.7 Definition of Terms<br>1.9.1 Estimator<br>Let X be a random variable that follows a probability distribution function<br>; indexed<br>by a parameter . Let , ,⋯, be a random sample from the given population. Any<br>statistic that can be used to estimate the parameter is called an estimator of . The<br>numerical value of this statistic is called an estimate of and is denoted by 6.<br>7<br>1.9.2 Prior distribution<br>A prior distribution is a probability distribution that captures the information about a<br>parameter(s) before data are taken into account. Prior distributions are sub-divided into two<br>classes: Informative and non-informative priors.<br>Let ,,⋯ be a random sample from a distribution with density<br>/ , where<br>(assumed random) is the unknown parameter to be estimated. The probability distribution<br>of is called the prior distribution of and is usually denoted as 8<br>.<br>1.9.3 Posterior distribution<br>Let ,,⋯ be a random sample from a distribution with density<br>/ , where is<br>an unknown parameter to be estimated. The conditional density 8<br>/ , ⋯, is<br>called the posterior distribution of and is given by<br>( ) ( ) ( )<br>1 2 ( )<br>/<br>/ , , n<br>f x<br>x x x<br>g x<br>q q<br>q = Õ â‹¯ Õ<br>(1.5)<br>where 9<br>is the marginal distribution of and is given by<br>9<br>=<br>Σ;<br>/ 8<br>$â„Ž&lt; =# &gt;=#?&amp;@<br>A<br>/ 8<br>, $â„Ž&lt; =# ?B&lt;@=D<br>‘ (1.6)<br>where 8<br>is the prior distribution of .<br>1.9.4 Loss function<br>Let ,,⋯ be a random sample from a distribution with density<br>/ , where is<br>an unknown parameter to be estimated. Let 6 be an estimator of . The function â„’6,<br>represents the loss incurred when 6<br>is used in place of .</p> <br><p></p>