ABSTRACT
In 2001, Surles & Padgett introduced Generalized Rayleigh Distribution (GRD). This
skewed distribution can be used quiet effectively in modeling life time data. In this work,
Bayesian estimates of the shape parameter of a GRD were determined under the
assumption of both informative (gamma) and non-informative (Extended Jeffery’s and
Uniform) priors. The Bayes estimates were obtained under both symmetric and asymmetric
loss functions. The performances of these estimates were compared to the Maximum
Likelihood Estimates (MLEs) using Monte Carlo simulation.
TABLE OF CONTENTS
Declaration …………………………………………………………………………………………………………….. iv
Certification ……………………………………………………………………………………………………………. v
Dedication ……………………………………………………………………………………………………………… vi
Acknowledgments ………………………………………………………………………………………………….vii
Abstract ………………………………………………………………………………………………………………. viii
Table of Contents ……………………………………………………………………………………………………. ix
List of Figures ………………………………………………………………………………………………………… xi
List of Tables ………………………………………………………………………………………………………. xiii
CHAPTER ONE: INTRODUCTION ……………………………………………………………………… 1
1.1 Background to the Study …………………………………………………………………………………….. 1
1.1.1 Theory of estimation ……………………………………………………………………………………………………………… 1
1.1.2 Generalized Rayleigh distribution (GRD) ………………………………………………………………………………… 2
1.2 Statement of the Problem ……………………………………………………………………………………. 4
1.3 Aim and Objectives of the Study ………………………………………………………………………….. 5
1.4 Significance of the Study …………………………………………………………………………………….. 5
1.5 Motivation …………………………………………………………………………………………………………. 6
1.6 Limitation …………………………………………………………………………………………………………. 6
1.7 Definition of Terms …………………………………………………………………………………………….. 6
1.9.1 Estimator ……………………………………………………………………………………………………………………………… 6
1.9.2 Prior distribution …………………………………………………………………………………………………………………… 7
1.9.3 Posterior distribution …………………………………………………………………………………………………………….. 7
1.9.4 Loss function ……………………………………………………………………………………………………………………….. 7
CHAPTER TWO: LITERATURE REVIEW ………………………………………………………….. 8
CHAPTER THREE: METHODOLOGY …………………………………………………………………. 15
3.1 Maximum Likelihood Method ………………………………………………………………………………… 15
3.2 Bayes Estimation of the Shape Parameter of GRD ………………………………………………. 16
3.2.1 Posterior risk and Bayes estimator …………………………………………………………………………………………. 16
3.2.2 Symmetric loss function ………………………………………………………………………………………………………. 18
3.2.3 Asymmetric loss function …………………………………………………………………………………………………….. 19
3.3 Bayesian Estimates under the Extended Jeffrey’s Prior ……………………………………….. 21
3.3.1 Transformation of the random variable M and its distribution …………………………………………………… 25
3.3.2 Convolution ……………………………………………………………………………………………………………………….. 27
3.3.3 Variance and relative efficiency of the estimates under extended Jeffrey’s prior using the various loss
functions ……………………………………………………………………………………………………………………………. 29
x
3.3.4 Posterior Risk …………………………………………………………………………………………………………………….. 32
3.4 Bayesian Estimates under the Uniform Prior ………………………………………………………. 33
3.5 Bayes Estimates under the Gamma Prior …………………………………………………………… 38
3.5.1 The distribution of the random variable H ………………………………………………………………………………. 42
3.5.2 Determination of variance, relative efficiency and posterior risk of the shape parameter under the
squares error, entropy and precautionary loss functions ……………………………………………………………. 44
CHAPTER FOUR: ANALYSIS AND DISCUSSION …………………………………………….. 47
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS …… 58
5.1 Summary ………………………………………………………………………………………………………… 58
5.2 Conclusion ………………………………………………………………………………………………………. 58
5.3 Recommendations ……………………………………………………………………………………………. 59
5.4 Contribution to Knowledge ……………………………………………………………………………….. 59
5.5 Areas of Further Research ………………………………………………………………………………… 60
References ……………………………………………………………………………………………………………. 60
Appendix I …………………………………………………………………………………………………………… 64
CHAPTER ONE
INTRODUCTION
1.1 Background to the Study
Statistical Inference is the branch of statistics concerned with using probability concept to
deal with uncertainty in decision-making. It refers to the process of selecting a sample and
using a sample statistic to draw inference about a given population parameter. The field of
statistical inference is divided into the theory of estimation and hypothesis testing.
1.1.1 Theory of estimation
Statistical estimation or simply estimation is concerned with the methods by which
population characteristics are estimated based on information drawn from a sample. The
theory of estimation is further sub-divided into Point and Interval Estimation. A point
estimator is a random variable varying from sample to sample and its value is called point
estimate i.e. a point estimate is a single value estimate for the parameter. There are several
methods of finding a point estimator which can all be broadly classified into the Classical
Methods and Non-classical/ Bayesian Methods.
In classical approach, the unknown parameter is assumed to be fixed quantity. Inferences
in classical approaches are based on a random sample only i.e. if a random sample
,,⋯, is drawn from a population with probability function
; and based on
the sample knowledge the estimate of is obtained. The classical approach is based on the
concept of sampling distribution and it does not use any of the prior information available
as a result of familiarity with previous studies. There are different methods of point
estimation under the classical approach. These include: Maximum Likelihood Estimation
(MLE), Method of Moment Estimation, Percentile Estimation, Least Square Estimation,
Weighted Least Square Estimates, and so on. On the other hand, in Non-classical/ Bayesian
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approach, the parameter is assumed to be a random variable which can be described by a
probability distribution (known as prior distribution) that is, the unknown parameter θ
(being random) follows a prior distribution. Hence, Bayesian approach combines new
information that is available with prior information to form basis for inference.
The fundamental difference between the Bayesian and frequentist approaches to statistical
inference is characterized in the way they interpret probability, represent the unknown
parameters, acknowledge the use of prior information and make the final inferences. The
frequentist approach considers probability as a limiting long-run frequency, while the
Bayesian approach regards probability as a measure of the degree of personal belief about
the value of an unknown parameter θ.
1.1.2 Generalized Rayleigh distribution (GRD)
Twelve different families of cummulative distribution function used in modeling lifetime
data were suggested by Burr (1942). Burr Type X distribution is among the most popular
distributions that receives the most attention among these families of cummulative
distributions.
In 2001, two-parameter Burr Type X distribution was introduced by Surles & Padgett.
Kundu and Raqab (2005), Lio, et.al, (2011) and Abdel-Hady (2013) prefer to call this
distribution GRD which will be adopted in this work. For α>0 and λ>0, the Cumulative
Distribution Function (CDF) of the two-parameter GRD is given by:
F
x; α, λ = 1 −
for x, α , λ >0 (1.1)
Its probability density function (pdf) is given by:
3
f
x; α, λ = 21 −
, > 0
0, “#$ℎ&
‘ (1.2)
where α and λ are shape and scale parameters respectively.
Shape and scale parameters are used to determine the shape and location of a distribution.
Shape parameter allows a distribution to take on a variety of shapes depending on the value
of the shape parameter, while the scale parameter stretches or squeezes the graph of a
probability distribution. In general, the larger the scale parameter, the more spread out the
distribution and the smaller the parameter, the more compressed the distribution appears to
be.
GRD is widely used in modeling events that occur in different fields such as medicine,
social and natural sciences. In Physics for instance, the GRD is used in the study of various
types of radiation such as light and sound measurements. It is used as a model for wind
speed and is often applied to wind driven electrical generation. For details, see Samaila and
Cenac, (2006). It is also used in modeling strength and lifetime data (Surles and Padgett
(2001), Lio, et.al (2011), Kundu and Raqab, (2005)). Hence, the GRD has a survival and
hazard functions as shown in equations (1.3) and (1.4) respectively.
Survival function S
x; α, λ = 1 − F
x; α, λ = 1 − 1 −
(1.3)
Hazard function h
x; α, λ = *
; ,
+
; , =
,-.
/0 1
2.3
-.
/0
,-.
/0 1
2 (1.4)
Survival function is the probability that the survival time X takes a value greater than a
specific value x ie 4
= 5
> , while the hazard function is a measure of how likely
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an individual experiences an event as a function of his/her age. These two functions are
used to describe the distribution of survival time data.
Figure 1. 1: The Graph of Generalized Rayleigh Distribution for different values of shape parameters when the
scale parameter takes the value one
The graph of the distribution is shown in Figure 1.1 for different shape parameter values. It
is clear from the Figure that the pdf of a GRD is a decreasing function if α ≤ ½ and it is
right skewed uni-modal when α > ½. (See also Kundu and Raqab, 2005).
1.2 Statement of the Problem
Bayesian inference requires appropriate choice of priors for parameters. But there is no way
to conclude that one prior is better than another. In a situation where one have enough
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information about the parameter(s) then using informative prior(s) will be the best practice
for choosing a prior, otherwise, a non-informative prior suffices.
1.3 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
i. By estimating the shape parameter (α) when the scale parameter (λ) is known using
both informative and non-informative priors under symmetric loss function
ii. By estimating the shape parameter (α) when the scale parameter (λ) is known using
both informative and non-informative priors under asymmetric loss functions
iii. To compare the performances of the proposed estimators with that of Maximum
Likelihood Estimators in terms of Mean Square Error
1.4 Significance of the Study
In Bayesian approach, the parameter is viewed as random variable behaving according to a
subjective (prior) probability distribution that describes our confidence about the actual
behavior of the parameter, whereas in classical approach, the parameter is assumed to be
fixed but unknown. In Bayesian inference, conclusions are made conditional on the
observed data i.e. there is no need to discuss sampling distribution using this method. While
in the classical approach one needs not be concerned about any prior knowledge other than
the available information observed. Bayesian inference also provides a convenient model
for implementing scientific method. The prior distribution is used to state the prior
knowledge we have about the parameter of interest, while the posterior distribution reflects
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the updated knowledge about the population parameter in line with the new information
collected from data.
1.5 Motivation
Statistically, modeling of real life scenario help us to better understand and explain
unforeseen eventualities when they take place, thereby enabling us to reproduce such a
scenario either on a large and/ or on a simplified scale aimed at describing only critical
parts of the phenomenon. These real life phenomena are captured by means of statistical
distribution models, which are extracted or learned directly from data gathered about them.
Every distribution model has a set of parameters that needs to be estimated. These
parameters specify any constant appearing in the model and provide a mechanism for
efficient and accurate use of data.
1.6 Limitation
The study will focus only on estimating the shape (α) parameter when the scale (λ)
parameter is known under the symmetric and asymmetric loss functions assuming
informative and non-informative priors.
1.7 Definition of Terms
1.9.1 Estimator
Let X be a random variable that follows a probability distribution function
; indexed
by a parameter . Let , ,⋯, be a random sample from the given population. Any
statistic that can be used to estimate the parameter is called an estimator of . The
numerical value of this statistic is called an estimate of and is denoted by 6.
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1.9.2 Prior distribution
A prior distribution is a probability distribution that captures the information about a
parameter(s) before data are taken into account. Prior distributions are sub-divided into two
classes: Informative and non-informative priors.
Let ,,⋯ be a random sample from a distribution with density
/ , where
(assumed random) is the unknown parameter to be estimated. The probability distribution
of is called the prior distribution of and is usually denoted as 8
.
1.9.3 Posterior distribution
Let ,,⋯ be a random sample from a distribution with density
/ , where is
an unknown parameter to be estimated. 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
$â„Ž< =# >=#?&@
A
/ 8
, $ℎ< =# ?B<@=<BC# ∞
D
‘ (1.6)
where 8
is the prior distribution of .
1.9.4 Loss function
Let ,,⋯ be a random sample from a distribution with density
/ , where is
an unknown parameter to be estimated. Let 6 be an estimator of . The function â„’6,
represents the loss incurred when 6
is used in place of .
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