BAYESIAN ESTIMATION OF THE SHAPE PARAMETER OF GENERALIZED RAYLEIGH DISTRIBUTION UNDER SYMMETRIC AND ASYMMETRIC LOSS FUNCTIONS
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BAYESIAN ESTIMATION OF THE SHAPE PARAMETER OF GENERALIZED RAYLEIGH DISTRIBUTION UNDER SYMMETRIC AND ASYMMETRIC LOSS FUNCTIONS
ABSTRACT
Surles and Padgett presented the Generalised Rayleigh Distribution (GRD) back in 2001. This skewed distribution is good for modelling life-time data. This study analysed Bayesian estimates of a GRD’s shape parameter using both informative (gamma) and non-informative (Extended Jeffery’s and Uniform) priors.
Bayes estimates were achieved using both symmetric and asymmetric loss functions. These estimates were compared to Maximum Likelihood Estimates (MLEs) by Monte Carlo simulation.Chapter one
INTRODUCTION
1.1 Background for the Study
Statistical Inference is the branch of statistics that uses probability concepts to address uncertainty in decision-making. The procedure involves picking a sample and utilising a statistic to derive conclusions about a population parameter. Statistical inference encompasses both estimating theory and hypothesis testing methods.
1.1.1 Theory of Estimation
Statistical estimation refers to the process of estimating population characteristics using sample data. There are two types of estimate theories: point estimation and interval estimation.
A point estimator is a random variable that varies from sample to sample. Its value represents a single estimate of the parameter. There are two types of methods for finding a point estimator: classical and non-classical/bayesian methods.
In the conventional approach, the unknown parameter is believed to be a fixed value. Classical techniques to inference rely solely on a random sample collected from a population with a probability function. Estimates are then made based on sample knowledge.
The conventional approach relies on sampling distribution and ignores past knowledge from earlier investigations. The traditional technique includes various approaches for point estimate.
Various estimation methods exist, such as Maximum Likelihood Estimation (MLE), Method of Moment Estimation, Percentile Estimation, Least Square Estimation, and Weighted Least Square Estimates.
In the Non-classical/Bayesian approach, the parameter θ is considered to be random and follows a probability distribution (also known as the prior distribution).
The Bayesian technique mixes fresh and prior knowledge to make inferences.
The Bayesian and frequentist approaches to statistical inference differ in their interpretation of probability, represen
tation of unknown parameters, use of previous information, and final inference. The frequentist method views probability as restricting long-run frequency, while the Bayesian approach sees probability as a measure of personal conviction in the value of an unknown parameter (θ).
1.1.2 Generalised Rayleigh Distribution (GRD)
Burr (1942) proposed 12 types of cumulative distribution functions for modelling lifespan data. The Burr Type X distribution is one of the most widely studied cummulative distributions.
Surles and Padgett introduced the Burr Type X distribution with two parameters in 2001.
Kundu and Raqab (2005), Lio et al. (2011), and Abdel-Hady (2013) refer to this distribution as GRD, which we shall use in this study. For α>0 and λ>0, the Cumulative Distribution Function (CDF) of the two-parameter GRD is: F x; α, λ = 1 −
for x, α , λ >0 (1.1)
The probability density function (pdf) is defined as: 3 f x; α, λ = 21 −
, > 0, “#$ℎ&” (1.2), where α and λ represent form and scale parameters, respectively.
Shape and scale characteristics are used to determine a distribution’s shape and placement.
The shape parameter moulds a distribution based on its value, while the scale parameter alters the graph of a probability distribution. Generally, greater scale parameters result in more spread-out distributions, whereas smaller parameters result in more compressed distributions.
GRD is frequently used to model occurrences in various domains, including health, social sciences, and natural sciences. In Physics, GRDs are used to examine radiation, including light and sound measurements.
This model represents wind speed and is commonly used in wind-powered electricity generation. For more information, read Samaila and Cenac (2006). Surles and Padgett (2001), Lio et al. (2011), and Kundu and Raqab (2005) all use it to model strength and longevity data.
The GRD includes survival and hazard functions, as seen in equations (1.3) and (1.4).
The survival function is S = x; α, λ = 1 – F = x; α, λ = 1 – 1
(1.3)
Hazard function h x; α, λ = * (1.4)
The survival function determines the likelihood of an individual experiencing an event based on their age, while the hazard function determines the likelihood of an event occurring at a specified value (e.g., x = 4). These two functions describe the distribution of survival time data.
Figure 1.1: The graph shows the Generalised Rayleigh Distribution for various shape parameter values when the scale parameter is set to one.
Figure 1.1 shows a graph of the distribution for various shape parameter values. The Figure shows that the pdf of a GRD is declining for α < ½ and right skewed uni-modal for α > ½. (See also Kundu & Raqab, 2005).
1.2 Statement of the Problem
Bayesian inference necessitates the selection of appropriate priors for parameters. However, it is impossible to say that one prior is superior to another. If there is sufficient knowledge on the parameter(s), it is recommended to use informative priors. Otherwise, a non-informative prior is sufficient.
1.3 Goals and Objectives of the Study
The goal of this study is to estimate the shape parameter of GRD using a Bayesian technique.
We intend to attain the stated target through the following objectives:
To estimate the shape parameter (α) when the scale parameter (λ) is known, both informative and non-informative priors are used under a symmetric loss function.
ii. Using informative and non-informative priors under asymmetric loss functions, estimate the shape parameter (α) given the scale parameter (λ).
iii. Evaluate the proposed estimators against Maximum Likelihood Estimators in terms of Mean Square Error.
1.4 Significance of the Study
The Bayesian approach views the parameter as a random variable with a prior probability distribution, indicating our confidence in its actual behaviour. In contrast, the classical approach assumes the value to be fixed and unknown.
Bayesian inference draws inferences based on observable data, eliminating the need to consider sample distributions. The classical approach focuses solely on observed facts and does not require prior knowledge.
Bayesian inference is a useful model for applying scientific methods. The prior distribution represents our prior understanding of a parameter, while the posterior distribution reflects updated information based on fresh data.
1.5 Motivation.
Modelling real-life scenarios helps us comprehend and explain unforeseen events, allowing us to repeat them on a larger or simplified scale, focusing on essential aspects of the occurrence. Statistical distribution models are used to capture real-life occurrences based on collected data.
Every distribution model contains a set of parameters that must be calculated. The model’s parameters specify constants and enable efficient and accurate data utilisation.
1.6 Limitation.
The study focuses on estimating the shape (α) parameter while the scale (λ) parameter is known using symmetric and asymmetric loss functions with informative and non-informative priors.
1.7 Definition of Terms.
1.9.1 Estimator
Consider X as a random variable with a probability distribution function, indexed by a parameter. Assume,⋯, is a random sample from the specified population. An estimator refers to any statistic that can estimate a parameter. The numerical value of this statistic is referred to as an estimate, denoted by 6.
1.9.2 Prior distribution.
A prior distribution captures information about a parameter(s) prior to data collection. Prior distributions are classified into informative and non-informative priors.
Assume,,⋯ is a random sample from a distribution with density, and (assumed random) is the unknown parameter for estimation. The probability distribution is referred to as the prior distribution and is typically indicated by 8 (1.9.3).
Assume,,⋯ is a random sample from a distribution with density /, with an unknown parameter to estimate. The conditional density 8 / , ⋯, is called the posterior distribution of and is given by () () () 1 2 () / / , , n f x x x x g x q q q = Õ ⋯ Õ (1.5)
where 9 is the marginal distribution of and is given by 9 = Σ; / 8 $ℎ< =# >=#?&@ A / 8 , $ℎ< =# ?B<@=<BC# ∞ D ‘ (1.6) where 8 is the prior distribution of .
1.9.4 Loss function.
Assume,,⋯ is a random sample from a distribution with density /, with an unknown parameter to estimate. Let 6 represent an estimate of. The function ℒ6 represents the loss suffered by using 6 instead of.
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