ABSTRACT
Statistical probability distributions are vital in explaining real world situations. It has been shown overtime that extreme value distributions are found to be useful in various field of studies such as engineering, financial sector, biological sciences etc. Generalizing existing distributions is being studied frequently so as to enhance and improve on the flexibility of an existing distribution. In this work, a three-parameter distribution known as Exponentiated New Weighted Exponential distribution was proposed by introducing an exponentiated (shape) parameter to an existing distribution, it can be used to model real life situations. Structural statistical properties of the proposed distribution such as the hazard, survival, moments and moment generating function were derived. The order statistics and quantile function were also studied. The method of maximum likelihood estimation procedure was used in estimating the parameters of the proposed distribution. The kernel density estimates and the classical Kolmogorov-Smirnov two sample test was also utilized to see how the estimated kernel density function from the proposed distribution mimic the parent distribution and also to check if the data sets used (waiting time of bank customers and accelerated life test) and the sample drawn from the fitted ENWED are indeed from the same distribution. Finally, to prove the flexibility and performance of the proposed distribution, we applied two real life data sets to the proposed distribution, exponentiated-new weighted exponential distribution and to competing distributions such as weighted exponential distribution, new-weighted exponential distribution and exponentiated weighted exponential distribution. The results indicate that the newly proposed distribution outperformed the competing distributions.
TABLE OF CONTENTS
DECLARATION ………………………………………………………………………………………………………………….. II
CERTIFICATION ……………………………………………………………………………………………………………….. IV
DEDICATION ……………………………………………………………………………………………………………………… V
ACKNOWLEDGEMENTS …………………………………………………………………………………………………… VI
ABSTRACT ……………………………………………………………………………………………………………………….. VII
CHAPTER ONE …………………………………………………………………………………………………………………… 1
INTRODUCTION ………………………………………………………………………………………………………………… 1
1.1 BACKGROUND OF THE STUDY ………………………………………………………………………………………….. 1
1.2 NEW WEIGHTED EXPONENTIAL DISTRIBUTION …………………………………………………………………. 2
1.3 STATEMENT OF PROBLEM ………………………………………………………………………………………………. 5
1.4 SIGNIFICANCE OF THE STUDY ………………………………………………………………………………………….. 6
1.5 AIM AND OBJECTIVES …………………………………………………………………………………………………….. 6
1.6 LIMITATION OF THE RESEARCH ………………………………………………………………………………………. 6
CHAPTER TWO ………………………………………………………………………………………………………………….. 8
LITERATURE REVIEW ………………………………………………………………………………………………………. 8
2.1 INTRODUCTION ……………………………………………………………………………………………………………… 8
2.2 ON EXPONENTIATED DISTRIBUTION ………………………………………………………………………………… 8
CHAPTER THREE ………………………………………………………………………………………………………………13
METHODOLOGY ……………………………………………………………………………………………………………….13
3.1 INTRODUCTION ……………………………………………………………………………………………………………..13
3.2 THE EXPONENTIATED-NEW WEIGHTED EXPONENTIAL DISTRIBUTION (ENWED) ………………..13
3.2.1 PROBABILITY DENSITY FUNCTION OF ENWED ………………………………………………………….. 14
3.2.2 THE CUMULATIVE DISTRIBUTION FUNCTION OF ENWED ……………………………………………. 15
3.3 RELIABILITY ANALYSIS OF EXPONENTIATED-NEW WEIGHTED EXPONENTIAL DISTRIBUTION .16
3.3.1 SURVIVAL FUNCTION ……………………………………………………………………………………………. 16
ix
3.3.2 ASYMPTOTIC BEHAVIORS ………………………………………………………………………………………. 17
3.4 STATISTICAL PROPERTIES OF THE EXPONENTIATED-NEW WEIGHTED EXPONENTIAL DISTRIBUTION ………………………………………………………………………………………………………………………19
3.4.1 THE QUANTILE FUNCTION AND THE MEDIAN …………………………………………………………….. 19
3.4.2 MOMENTS AND MOMENT GENERATING FUNCTION …………………………………………………….. 21
3.4.3 DISTRIBUTION OF ORDER STATISTICS ………………………………………………………………………. 23
3.5 PARAMETER ESTIMATION AND APPLICATION ……………………………………………………………………24
3.5.1 MAXIMUM LIKELIHOOD ESTIMATION ………………………………………………………………………. 24
3.6 DATA ANALYSIS PROCEDURE…………………………………………………………………………………………..27
3.6.1 THE KERNEL-BASED DENSITY ESTIMATION ………………………………………………………………. 28
3.6.2 KOLMOGROV-SMIRNOV TWO SAMPLE TEST ……………………………………………………………… 29
3.7 APPLICATION ………………………………………………………………………………………………………………..30
CHAPTER FOUR …………………………………………………………………………………………………………………31
ANALYSIS AND DISCUSSIONS …………………………………………………………………………………………..31
CHAPTER FIVE ………………………………………………………………………………………………………………….41
SUMMARY, CONCLUSION AND RECOMMENDATION ……………………………………………………..41
5.1 SUMMARY …………………………………………………………………………………………………………………….41
5.2 CONCLUSION …………………………………………………………………………………………………………………41
5.3 RECOMMENDATION ……………………………………………………………………………………………………….42
5.4 CONTRIBUTION TO …………………………………………………………………………………………………………42
REFRENCE …………………………………………………………………………………………………………………………43
x
List of Tables Table 4.1a: Waiting Time in Minutes of Bank Customer for n=100 Table 4.1b: Random Sample of Size 100 drawn from ENWED Table 4.2: Performance of Distribution ENWED and NWED using Data on Waiting Time in Minutes of Bank Customers Table 4.3a: Accelerated Life Test Conductors for n=59 Table 4.3b: Random Sample of Size 59 drawn from ENWED Table 4.4: Performance of Distribution ENWED and NWED using Data on Accelerated Life Test Conductors
xi
List of Figures Figure 1: The Behavior of PDF of ENWED and NWED for Some Selected Parameter Values. Figure 4.1 Estimated PDF for the Data on Waiting Time of Bank Customers. Figure 4.2 Kolmogrov-Smirnov Two-Sample Test: Graph of the Empirical Distribution Function for Data on Waiting Time of Bank Customers. .Figure 4.3: Estimated Survival and Hazard Function for the Data on Waiting Time of Bank Customers. Figure 4.4 Estimated PDF for the Data on Accelerated Life Test of Conductors. Figure 4.5 Kolmogrov-Smirnov Two-Sample Test: Graph of the Empirical Distribution Function for Data on Accelerated Life Test of Conductors.
Figure 4.6: Estimated Survival and Hazard Function for the Data on Accelerated Life Test of Conductors.
1
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
Statistical probability distributions are very important in explaining real world situations. It is
known that extreme value distributions have been found to be very useful in the field of
engineering, biological sciences and financial sector. In statistical modeling, the quality of the test
used heavily relies on the assumed probability distribution being adopted. Generalized
distributions have been studied extensively in statistics, although numerous statisticians have
developed different types of generalizations, there is still need to develop new probability
distribution by generalizing existing distributions. For instance, the Kumaraswamy Generalized
(KW-G) proposed by Cordeiro and de Castro (2011), the Beta Generalized (B-G) introduced by
Eugene et al.,(2002) and the Kumaraswamy Odd log-logistic Generalized (KWOLL-G)
introduced by Alisa del et al (2015).
In reliability engineering, the Weibull and exponential distribution are the mostly used
distributions due to their simplicity and easy mathematical manipulations. Also in reliability
engineering and life testing, it has been asserted that the exponential distribution is a suitable
alternative to both the Gamma and Weibull as shown by Gupta et al., (2001). Numerous
researchers have used the exponential distribution as a better alternative to some related
distributions.
Gupta et al., (1998) introduced a class of exponentiated distribution which is based on the
cumulative distribution function, it is derived by raising the cdf of an arbitrary parent distribution
to an additional positive parameter, say, “ ”.
2
Let X denotes a random variable from an arbitrary parent distribution G, the cdf of the resulting
exponentiated distribution is given by:
F(x) G(x) ; 0
(1.1)
where: G(x) is the cdf of the parent distribution. Therefore, the corresponding pdf is obtained by
differentiating equation (1.1) with respect to x to get;
1
f (x) G(x) g(x)
(1.2)
where:
( )
( )
dG x
g x
dx
1.2 New Weighted Exponential Distribution
According to Badmus et al., (2014), the weighted distributions are introduced to an existing
distribution to adjust the likelihood of the events as observed and recorded. Following Nasiru
(2015), the Azzalini’s method was used and applied to the exponential distribution by Oguntunde
et al., (2016) to produce a new weighted exponential distribution, n-WED , , which is
simpler and mathematically easier to handle than the weighted exponential distribution proposed
by Gupta and Kundu (2009) as a generalization of the exponential distribution.
Let g(x) be a pdf and G(x)
be the corresponding survival (or reliability) function such that the
cdf G(x) exists, then the new weighted family of distribution is given by:
f (x) kg(x)G( x), x 0, 0
(1.3)
Where k is a constant such that
3
0
f (x)dx 1
(1.4)
The pdf g(x) and survival function G(x)
of an exponential distribution are given as
( ) ; 0, 0 x g x e x
and
(1.5)
G(x) 1 G(x)
1 (1 ) x e
x e (1.6)
Where is a shape parameter
Hence substituting equations (1.5) and (1.6) into (1.3) gives the new weighted exponential
distribution
( ) x x f x k e e
( x x) k e
x(1 ) k e ; or
(1 ) ( ) x f x k e
By equation (1.4)
(1 )
0
1 x k e dx
Let t (1)x
dt (1)dx
4
1
(1 )
dx dx
0
0 1
(1 ) (1 ) (1 )
t
t e k k
k dt e
1
(1 )
k
k 1
Therefore (1 ) ( ) (1 ) ; 0, 0, 0 x f x e x x (1.7)
Where both and are shape parameters. Note that if (1) or 0 , equation (1.7)
reduces to a simple exponential density function with parameters or , that is
(1 )
( ) ,
0
x
x
e if
f x
e if
(1.8)
And and are all shape parameters. Therefore from equation (1.7), both and are shape
parameters. The corresponding cdf of the new weighted exponential distribution is given as
(1 ) ( ) 1 x F x e (1.9)
Unlike in Oguntunde et al., (2016), both and are shape parameters; implying that a product
of two shape parameters is also a shape parameter. Oguntunde et al., (2016) erred by saying that
is a scale parameter.
The new weighted exponential distribution whose pdf and cdf are defined in equations (1.7) and
(1.9) are much easier to handle mathematically than the ones proposed by Gupta and Kundu
(2009):
5
1
( ) 1 ; 0, 0, 0 x x f x e e x
and
(1.10)
(1 ) 1 1
( ) 1 1
1
x x F x e e
(1.11)
Where; is a shape parameter and is a scale parameter.
The hazard function of the new weighted exponential distribution is given by:
(1 )
(1 )
(1 )
( )
x
x
e
h x
e
(1.12)
(1)
In literature, other weighted distributions have also been defined, for instance, the weighted
Weibull distribution by Mhady (2013), the weighted Inverted Exponential distribution by Hussain
(2013), a weighted three parameter Weibull by Essam and Mohammed (2013).
1.3 Statement of Problem
Oguntunde (2015) generalized the weighted exponential distribution in equation (1.10) using the
exponentiated family of distributions to propose the exponentiated Weighted Exponential
Distribution.
The NWED , cannot be used in fitting some real-life data set because some existing real
datasets do not follow NWED and therefore cannot be appropriately described using it, as a result
of this there is a need to generalize the NWED , by adding an additional shape parameter
which will increase its flexibility, applicability and provides a better fit.
6
1.4 Significance of the Study
The addition of another shape parameter into the NWED , will increase its flexibility in
analyzing real data sets and will also improve the statistical test associated with the distribution.
1.5 Aim and Objectives
The aim of this research work is to propose a new distribution called Exponentiated-New
Weighted Exponential Distribution and derive some of its structural properties which will be
attained through the following specific objectives; to:
1. Derive the exponentiated-new weighted exponential distribution using the class of
exponentiated distribution proposed by Gupta et al., (1998).
2. Determine the reliability function (hazard and survival function) and obtain various
structural properties (moments, moment generating function, quantile and order
statistics) of the proposed distribution.
3. Estimate the parameters of the proposed exponentiated-new weighted distribution
using the maximum likelihood estimation,
4. Compare the performance of the proposed exponentiated-new weighted exponential
distribution with competitive alternatives, such as the new weighted exponential
distribution, exponentiated weighted exponential distribution and weighted
exponential distribution using real life datasets; and
5. Apply the kernel based density estimation method and the Kolmogorov-Smirnov two
sample test method to give an enhancement to outlook of the proposed model.
1.6 Limitation of the Research
This research work is limited to using only Alkaike Information Criterion (AIC) for testing the
goodness of fit for the best model
7
IF YOU CAN'T FIND YOUR TOPIC, CLICK HERE TO HIRE A WRITER»
