ABSTRACT
TheNormal distribution is a very important and well known probability distribution for dealing with problems in several areas of life, however there are numerous situations when the assumption of normality is not validated by the data. In this work, we propose a new model calledWeibull-Normal distribution, an extension of the normal distribution by adding two skewness parameters to the normal distribution using the Weibull generator proposed by Bourguignon et al., (2014). This study has derived explicit expressions for some of its basic statistical properties such as moments, moment generating function, the characteristics function, reliability analysis and the distribution of order statistics. The implications of the plots for the survival and hazard functions indicate that the Weibull-Normal distribution would be appropriate in modeling time or age-dependent events, where survival and failure rate decreases with time or age. Also, the plots for the pdf of the distribution showed that it is negatively skewed. The method of maximum likelihood estimation is used to estimate the parameters of our proposed model. The usefulness of the Weibull-normal distribution has been illustrated by some applications to two real data sets. The results showed that new distribution (Weibull-Normal distribution) performs better (provides better fits) than the generalizations of the normal distribution such as Kumaraswamy-Normal, Beta-Normal, Gamma-Normal, Kummer Beta-Normal and the normal distributions when the data set is negatively skewed however, the results from the second data confirmed that this distribution is more flexible and appropriate for modeling negatively skewed data sets.
TABLE OF CONTENTS
Contents
DECLARATION …………………………………………………………………………………………………………….. iii
CERTIFICATION ………………………………………………………………………………………………………….. iv
DEDICATION …………………………………………………………………………………………………………………. v
ACKNOWLEDGEMENT ………………………………………………………………………………………………… vi
ABSTRACT …………………………………………………………………………………………………………………… vii
LIST OF FIGURES ………………………………………………………………………………………………………….. x
LIST OF TABLES …………………………………………………………………………………………………………… xi
CHAPTER ONE: INTRODUCTION………………………………………………………………………………….. 1
1.1 Background of the study …………………………………………………………………………………………… 1
1.2 Probability Distribution and Estimation Theory ……………………………………………………………….. 2
1.3 The Weibull and the Normal Distributions ………………………………………………………………………. 3
1.4 Aim and Objectives of the Study ……………………………………………………………………………………. 4
1.5 Limitation ……………………………………………………………………………………………………………………. 5
1.6 Motivation …………………………………………………………………………………………………………………… 5
1.7 Significance of the Study ………………………………………………………………………………………………. 6
1.8 Statement of the Problem ………………………………………………………………………………………………. 6
1.9 Definition of Terms ………………………………………………………………………………………………………. 7
1.9.1 Probability distribution ……………………………………………………………………………………………………………………… 7
1.9.2 Moments …………………………………………………………………………………………………………………………………………. 8
1.9.3 Moment Generating Function …………………………………………………………………………………………………………….. 9
1.9.4 Characteristics Function ………………………………………………………………………………………………………………….. 10
1.9.5 Reliability Analysis ………………………………………………………………………………………………………………………… 10
1.9.6 Order Statistics ………………………………………………………………………………………………………………………………. 10
1.9.7 Maximum Likelihood Method ………………………………………………………………………………………………………….. 11
1.9.8 Lifetime data ………………………………………………………………………………………………………………………………….. 11
CHAPTER TWO: LITERATURE REVIEW ……………………………………………………………………….. 12
CHAPTER THREE: METHODOLOGY …………………………………………………………………………… 17
3.1 The Definition of the Weibull-Normal Distribution (WND) ……………………………………………… 17
3.2 Moments …………………………………………………………………………………………………………………… 30
3.2.1 Ordinary Moments ………………………………………………………………………………………………………………………….. 30
3.2.2 The Central Moments ……………………………………………………………………………………………………………………… 33
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3.2.3 Incomplete Moments ………………………………………………………………………………………………………………………. 34
3.3 Moment Generating Function ………………………………………………………………………………………. 36
3.4 The Characteristics Function ……………………………………………………………………………………….. 37
3.5 Reliability Analysis of the WN Distribution …………………………………………………………………… 38
3.5.1 The survival function ………………………………………………………………………………………………………………………. 38
3.5.2 The Hazard Function ………………………………………………………………………………………………………………………. 39
3.6 Order Statistics …………………………………………………………………………………………………………… 40
3.7 Estimation of Parameters of the Weibull-Normal Distribution …………………………………………. 42
CHAPTER FOUR: ANALYSIS AND DISCUSSION …………………………………………………………… 49
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS ……………………… 59
5.1 SUMMARY ………………………………………………………………………………………………………………. 59
5.2 CONCLUSION ………………………………………………………………………………………………………….. 60
5.3 RECOMMENDATIONS …………………………………………………………………………………………….. 60
5.4 CONTRIBUTION TO KNOWLEDGE …………………………………………………………………………. 61
5.5 AREAS OF FURTHER RESEARCH …………………………………………………………………………… 61
REFERENCES ……………………………………………………………………………………………………………… 62
APPENDIX ………………………………………………………………………………………………………………….. 66
x
CHAPTER ONE
1.1 Background of the study
The celebrated Normal (Gaussian) distribution has been known for centuries. Its popularity has been driven byits analytical simplicity and the associated Central Limit Theorem. The multivariate extensionis straightforward because the marginal and conditionals are both normal,a property rarely found in most of the other multivariate distributions. Yet there have been doubts, reservations, and criticisms about the unqualified use of normality. There are numerous situations when the assumption of normality is not validated by the data. In fact Geary (1947) remarked, “Normality is a myth; there never was and never will be a normal distribution.” As an alternative, many near normal distributions have been proposed. Some families of such near normal distributions, which include the normal distribution and to some extent share its desirable properties, have played a crucial role in data analysis. For description of some such families of distributions, see Mudholkar and Hutson (2000). See also Azzalini (1985), Turner (1960) and Prentice (1975).Many of the near norma1 distributions mentioned above deal with effects of asymmetry. These families of asymmetrical distributions are analytically tractable, accommodate practical values of skewness and kurtosis, and strictly include the normal distribution. These distributions can be quite useful for data modeling and statistical analysis.
Distribution functions, their properties and interrelationships play a significant role in modeling naturally occurring phenomena. For this reason, a large number of distribution functions which are found applicable to many events in real life have been proposed and defined in literature. Various methods exist in defining statistical distributions. Many of these arose from the need to model naturally occurring events. For example, the Normal distribution addresses real-valued
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variables that tend to cluster at a single mean value, while the Poisson distribution models discrete rare events. Yet few other distributions are functions of one or more distributions. Numerous standard distributions have been extensively used over the past decades for modeling data in several fields such as Engineering, Economics, Finance, Biological, Environmental and Medical Sciences etc. However, generalizing these standard distributions has produced several compound distributions that are more flexible compared to the baseline distributions. For this reason, several methods for generating new families of distributions have been studied.
1.2 Probability Distribution and Estimation Theory
In real life there is no certainty about what will happen in the future, but decisions still have to be made. Therefore, decision processes must be able to deal with the problems of uncertainty. Events that cannot be predicted precisely are often called random events. Many, if not most, of the inputs to, and processes that occur in, our systems are to some extent random. Hence, so too are the outputs or predicted impacts, and even people’s reactions to those outputs or impacts. To ignore this randomness or uncertainty is to ignore reality. One of the commonly used tools for dealing with uncertainty in planning and management is probability. Probability is a branch of mathematical statistics that is used for quantitative modeling of random variables. The probability of an event represents the proportion of times under certain conditions that the outcome can be expected to occur. A probability density functionis a mathematical description that approximately agrees with the frequencies or probabilities of possible events of a random variable.
Maximum Likelihood is a popular estimation technique for many distributions because it picks the values of the distribution’s parameters that make the data more likely” than any other valueof
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the parameter. This is accomplished by maximizing the likelihood function of the parameters
given the data. Some appealing features of Maximum Likelihood estimators include their
asymptoticunbiasedness, in that the bias tends to zero as the sample size n increases; they are
asymptotically efficient, in that they achieve the Cramer-Rao lower bound as n approaches 1;
and they are asymptotically normal.
1.3 The Weibull and the Normal Distributions
The WeibullDistribution is a very popular continuous probability distribution named after a
Swedish Engineer, Scientist and Mathematician, WaloddiWeibull (1887 – 1979). He proposed
and applied this distribution in 1939 to analyze the breaking strength of materials. Since then, it
has been widely used for analyzing lifetime data in reliability engineering. It is a versatile
distribution that can take on the characteristics of other types of distributions, based on the value
of the shape parameter. The Weibull distribution is a widely used statistical model for studying
fatigue and endurance life in engineering devices and materials.
If a random variable X has the Weibull distribution with scale parameter α>0 and shape
parameter β>0, then its cdf and pdf are, respectively, given by
( ) 1
x
F x e
(1.1)
and
1 ( )
x
f x x e
(1.2)
For x 0, 0, 0 where α and β are the scale and shape parameters respectively.
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The Normal distribution is also called Gaussian distribution, after a German Mathematician Carl
Freidrich Gauss (1777 -1855), who introduced it in connection with the theory of error. It is an
extremely important probability distribution in many fields. The pdf and the cdf of the normal
distribution with location parameter and scale parameter 0 are respectively
given by
2
2
( )
2
1 1
; , exp
2
x x
g x
(1.3)
where
2
2
1
exp
2 2
x x
and
2
2
( )
2
1
( ; , ) exp
2
x
x v
F x dv
(1.4)
1.4 Aim and Objectives of the Study
The aim of this research is to propose a Weibull-Normal distribution, propose some of its
properties and evaluate its performance using a real life data. The stated aim is expected to be
achieved through the following objectives. By
i. defining and expressing the pdf of the proposed distribution as a mixture of exponentiated-
G density functions.
ii. deriving some statistical properties of the proposed distribution such as the moments
(mean and variance), moment generating function, characteristics function, reliability
functions and the distribution of order statistics.
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iii. estimating the parameters of the proposed distribution using the method of Maximum Likelihood Estimation (MLE) technique.
iv. evaluating the performance of the proposed model compared to other generalizations of the normal distribution.
1.5 Limitation
The study is focused only on extending the normal distribution and deriving mathematical expressions for some selected properties of the proposed distribution such as mean, variance, moment generating function, characteristics function, density functions for the minimum and maximum order statistics and estimating the model parameters by using only the method of MLE.
1.6 Motivation
Bourguignon et al. (2014) highlighted some special distributions that can be obtained by their Weibull generator. These distributions include the Uniform, Exponential, Weibull, Normal, Frechet, Half-logistic, Power function, Pareto, Burr XII, Log-Logistic, Lomax, Gumbel and Kumaraswamy distribution. Also, Johnson et al. (1994) stated that, the use of four-parameter distributions should be sufficient for most practical purposes. According to them, at least three parameters are needed but they doubted any noticeable improvement arising from including a fifth or sixth parameter.
Statistically, modeling of real life scenario help us to better understand and explain random events when they occur, 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
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learned directly from data gathered about them. Every distribution model has a set of parameters that needs to be estimated. These parameters specify the model and provide a mechanism for efficient and accurate use of data. Besides these facts, the normal distribution is a well-known and commonly used standard continuous distribution with many areas of application. However, there has been no study on the hybrid (Weibull-Normal) distribution since the introduction of the Weibull generator by Bourguignon et al. (2014). Therefore we proposed the Weibull-Normal distribution, study some of its properties and its application to a real life data.
1.7 Significance of the Study
The study of the proposed distribution, its properties and the parameter estimates has increased the flexibility of the normal distribution and itwill make it model more easily and appropriatelydata sets that do not properly fit the normaldistribution. This study compared the proposed model to some existing generalizations of the normal distribution using three information criteria (AIC, CAIC and BIC) to identify the most fitted model under two real life data sets.
1.8 Statement of the Problem
The quality of the procedures used in statistical analysis depends heavily on the assumed probability model or distribution that the random variable follows. Many lifetime data used for statistical analysis follow a particular probability distribution and therefore knowledge of the appropriate distribution that any phenomenon follows greatly improves the sensitivity, reliability and efficiency of the statistical analysis associated with it.
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Furthermore, it is true that several probability distributions exist for modeling lifetime data;
however, some of these lifetime data do not follow any of the existing and well known standard
probability distributions (models) or at least are inappropriately described by them. This,
therefore, creates room for developing new distributions which could better describe some of
these phenomena and therefore provide greater flexibility and wider acceptability in the
modeling of lifetime data.
1.9 Definition of Terms
1.9.1 Probability distribution
The probability density function of a random variable X depends on whether X is a discrete or a
continuous random variable. A random variable is a variable whose value changes from one
subject to the other. Probability is used to describe the likelihood or the chances that these
random variables will equal specific values or be within a given range of specific values. A
probability density function is a mathematical expression that approximately agrees with the
frequencies of possible events of a random variable. A random variable X is said to be discrete if
it has a finite or countablyinfinite range. Discrete random variables are used to model
experiments which have to do with count data such as the number of defectives, the number of
boys in a family of n children, the sex of a baby etc. Similarly, a random variable is said to be
continuous if its range contains an interval of real numbers. They represent measured data such
as weight, height, temperature, time, blood pressure etc.
For a continuous random variable X, the cumulative distribution function (cdf) is defined as;
( ) ( ) ( )
x
F x P X x f u du
(1.5)
whilethe probability density function (pdf) is defined as;
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( )
( )
dF x
f x
dx
(1.6)
with the following properties:
1. f (x) 0
2. f (x)dx 1
3. ( ) ( )
b
a
P a X b f x dx
1.9.2Moments
Moments are used to study some of the most important features and characteristics of a random
variable such as mean, variance, skewness and kurtosis.Let X be a continuous random variable,
the nth moment of X about the origin can be defined as;
‘
( ) n n
n
E X x f x dx
(1.7)
Also The nth central moment of X, say , or moment about the mean can be obtained as
‘ ‘ ‘
1 1
0
( 1)
n
n
i i
n n i
i
n
E X
i
(1.8)
Hence, mean, , is the 1st moment about the origin while the variance ( ) is the 2ndcentral
moment or moment about the mean.
Skewness is a measure of the degree of asymmetry or lack of symmetry of a distribution. The
coefficient of skewness is the standardized third central moment of X or moment about the mean
and can be obtained using the expression;
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3
‘
1 3
3 ( )
x
Sk E
(1.9)
Kurtosis is a measure of the degree of peakness or flatness of a density near the center.Whereas
the coefficient of kurtosis is the standardized fourth central moment of X or moment about the
mean and is given by;
4
‘
1 4
4 ( )
x
Ku E
(1.10)
Where σ can be obtained using equation (1.8) for n=2 and and using (1.8) for n=3 and n=4
respectively.
1.9.3Moment Generating Function
This is a clever way of organizing all the moments into one mathematical quantity, and that
object is called the moment generating function. The moment generating function of a random
variable X can be obtained as;
( ) ( ) tx tx
x M t E e e f x dx
(1.11)
In other words, the moment generating functiongenerates the moments ofof a random variableX
by differentiation i.e., for any real number say k, the kth derivative of evaluated at is
the kthmoment of X.
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1.9.4Characteristics Function
The characteristics function has many useful and important properties which give it a central role
in statistical theory. Its approach is particularly useful for generating moments, characterization
of distributions and in analysis of linear combination of independent random variables.
A representation for the characteristics function is given by
( ) cos( ) sin( ) cos( ) sin( ) itx
x t E e E tx i tx E tx E i tx (1.12)
1.9.5Reliability Analysis
Survival Function
Survival function is the probability function that a system or an individual will survive beyond a
given time. Mathematically, the survival function is given by:
S x 1Fx (1.13)
where F x is cdf of a baseline distribution
Hazard Function
Hazard function is also called the failure or risk function and is the probability that a component
will fail or die for an interval of time. The hazard function is defined as;
1
f x f x
h x
F x S x
(1.14)
whereF(x) and f(x) are the cdf and pdf of a baseline distribution.
1.9.6 Order Statistics
Order statistics are used in a wide range of problems including robust statistical estimation and
detection of outliers, characterization of probability distributions and goodness of fit tests,
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entropy estimation, analysis of censored samples, reliability analysis, quality control and strength
of materials.Suppose 1 2 , ,……, n X X X is a random sample from a distribution with pdf, f(x), and
let 1: 2: : , ,……, n n i n X X X denote the corresponding order statistic obtained from this sample. The
pdf, i:n f x of the ith order statistic can be defined as;
1
:
!
( ) ( ) ( ) 1 ( )
( 1)!( )!
i n i
i n
n
f x f x F x F x
i n i
(1.15)
1.9.7Maximum Likelihood Method
Let 1 2 , ,……, n x x x be a random sample from a population X with probability density function
f (x; ), where θ is an unknown parameter. The likelihood function, L , is defined to be the
joint density of the random variables 1 2 , ,……, n x x x . That is,
1
( ) ( , )
n
i
i
L f x
(1.16)
The sample statistic that maximizes the likelihood functions, L ,is called the maximum
likelihood estimator of θ and is denoted by.
1.9.8 Lifetime data
Real life data sets are data sets obtained based on real happenings or normal life occurrences.
Real life data are not simulated in any way but based on what has been observed and recorded in
our day to day activities. Meanwhile, lifetime data are data collected on living subjects that has
to do with life. Loosely speaking we can say that lifetime data are mostly found in the clinic or
hospital e.g data on cancer, tuberculosis, HIV/AIDS e.t.c. Hence, lifetime data is also part of real
life data sets.
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CHAPTER TWO: LITERATURE REVIEW
In recent times, researchers have worked tirelessly to produce compound probability distributions which have been evaluated and considered to have performed much better than the well known classical or standard probability distributions. Most of these studies are meant purposely to introduce a higher level of skewness in the existing probability distributions by generalizing on a well-known distribution under certain facts and assumptions. A review of some of these generalizations is as listed: the ßeta generalized family (Beta-G) by Eugene et al. (2002) which generalized the beta distribution under some assumptions, the Kumaraswamy-G by Cordeiro and de Castro (2011) using the Kumaraswamy distribution as a baseline distribution, Transmuted family of distributions by Shaw and Buckley (2007), Gamma-G (type 1) by Zografos and Balakrishnan (2009) using the Gamma distribution, McDonald-G by Alexander et al. (2012), Gamma-G (type 2) by Risticet al. (2012), Gamma-G (type 3) by Torabi and Montazari (2012), Log-gamma-G by Aminiet al. (2012), Exponentiated T-X by Alzaghalet al. (2013), Exponentiated-G (EG) by Cordeiroet al. (2013), Logistic-G by Torabi and Montazari (2014), Gamma-X by Alzaatrehet al., (2013), Logistic-X by Tahiret al. (2015), Weibull-X by Alzaatrehet al. (2013), Weibull-G by Bourguignon et al. (2014) and Beta Marshall-Olkin family of distributions by Alizadehet al. (2015) and several other studies of the same kind. Further work on some of these studies above have led to the development of some compound probability distributions such as skew normal distribution by Azzalini (1985), the generalized Weibull distribution by Mudholkar and Kollia (1994), the exponentiatedWeibull distribution by Mudholkaret al. (1995), the beta normal distribution by Eugene et al. (2002), the beta-Weibull distribution by Famoyeet al. (2005), the generalized normal distribution by Nadarajahet al. (2005) and the kumaraswamy normal distribution by Cordeiro and De’Castro (2011).
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In an attempt to introduce skewness (reduce the level of symmetry in the distribution) into the well known normal distribution, many researchers have proposed many generalizations of this distribution by employing several methodologies. Details of some of the recent extensions of the normal distribution in addition to the aforementioned ones above in the literature include the following: Cordeiroet al. (2012) presented a five parameter life time distribution, referred to as the McDonald normal (MCN) distribution, which includes as special cases most of the commonly used distribution in the lifetime literature such as the normal, skew-normal, exponentiated normal, beta normal and kumaraswamy normal distributions, among others. They obtained their ordinary moments, moment generating function and mean deviation. They also derived the ordinary moments of the order statistics. They used the method of maximum likelihood to fit the new distribution and illustrated its potentiality with three applications to real data. Their results proved that the new model is much more flexible than the exponentiated normal, beta normal, skew normal and kumaraswamy normal models proposed recently. Alzaatrehet al. (2014) proposed a new family of distribution called the gamma – X family where X represents any continuous distribution. They discussed some properties of the gamma – x family and studied a member of the family; the gamma – normal distribution. They provided the limiting behaviours, moments, mean deviations, dispersions, and Shannon entropy for the gamma-normal distribution. They also obtained bounds for the non-central moments. The method of maximum likelihood estimation was used for estimating the parameters of the gamma-normal distribution. Two real data sets were used to illustrate the application of the gamma-normal distribution. The result showed that the gamma-normal distribution provide a good fit to each data set compared to other existing generalized forms of the normal distribution.
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Nadarajahet al. (2014) also proposed a new family of skewed distribution referred to as modified beta distributions. They studied some properties of the new family including estimation procedures. They also studied a member of this family called the modified beta normal distribution. A real data application and simulation studies were used to show superior performance of this new model versus other known models. Lima et al. (2015) proposed a new three parameter distribution called the gamma normal distribution which extends the normal distribution. They defined and studied some of its structural properties. The study provided explicit expressions for the ordinary and incomplete moments, Quantile and generating functions, mean deviations, Renyi entropy, Shannon entropy, order statistics and their moments. They estimated the model parameters using the method of maximum likelihood estimation. This study also determined the observed information matrix of the proposed distribution. The potentiality of the proposed model was evaluated based on three criteria and the results showed that the new distribution provides a better fit than the beta normal distribution, kumaraswamy-normal and the standard normal distribution.
Pescimet al. (2015) introduced a new extension of the normal distribution called the Kummer beta normal distribution. They defined and expressed the pdf of the new distribution as a linear combination of exponentiated normal pdfs which allowed them to derive some of its mathematical properties like its ordinary and incomplete moments, mean deviations and order statistics. The estimation of parameters of the proposed model was done by the method of maximum likelihood and Bayesian analysis. The usefulness of the model was illustrated by an application to a real data set. Their results revealed that the fit of the Kummer beta normal distribution outperforms all the distributions considered in their study based on the data set used. So they concluded that the proposed distribution can yield better fits than the normal, beta
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normal, gamma normal, McDonald normal and the modified beta normal distribution and therefore may be an interesting alternative to these distributions for modeling skewed data sets. Bourguignon et al. (2014) introduced a new wider family of distributions called the Weibull generalized (Weibull-G) family of distributions. According to these authors, this generator can extend several widely known distributions such as the Uniform, Exponential, Weibull, Frechet, Half-logistic, Power Function, Pareto, Burr XII, Logistic, Lomax, Normal, Gumbel and Kumaraswamy distributions. They derived general mathematical properties of the new wider Weibull family of distributions. They also showed that the Weibull-G density function can be expressed as a mixture of exponentiated-G density functions. This mixture representation is important to derive several structural properties of this family in full generality. They provided some of this structural properties such as the ordinary and incomplete moments, Quantile function and order statistics. The study also proposed that for any Weibull-G family distributions, the estimation of model parameters should be done by method of maximum likelihood estimation. These authors also recommended that for each baseline distribution G, their methods and results should be adopted.
Merovciet al. (2015) used the Weibull generalized family of distributions proposed by Bourguignon et al. (2014) to study the Weibull Rayleigh (WR) distribution. They derived expressions for the moments and the moments generating functions. Estimation of parameters of their model was done by the method of maximum likelihood estimation. Their model performance was compared to some baseline distributions such as beta Weibull,ExponentiatedWeibull and the standard Weibull distribution. The results of this comparison using the likelihood ratio statistics showed that the Weibull Rayleigh distribution can be used quite effectively to provide better fit than the Weibull distribution.
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Oguntundeet al. (2015) also used the Weibull-G family of distribution introduced by Bourguignon et al., (2014) to study the Weibull Exponential distribution (WED). They also derived explicit expression for some of its basic mathematical properties like moments, moment generating function, reliability analysis, limiting behavior and order statistics. The method of maximum likelihood estimation was proposed for estimating its parameters and real life applications were provided to illustrate the flexibility of the new distribution over the standard exponential distribution. Afifyet al. (2016) proposed a four parameter distribution named the WeibullFrechet distribution using the Weibull-G family by Bourguignon et al. (2014). They studied some of its mathematical and statistical properties. The applications of the new distribution with some baseline distributions showed that it is more fitted compared to kumaraswamyFrechet (KFr), exponentiatedFrechet (EFr), beta Frechet (BFr), gamma extended Frechet (GEFr), transmitted marshallOlkinFrechet (TMOFr) and Frechet (Fr) distributions. With this understanding, this research seeks to increase the flexibility of the normal distribution using the weibull-G family of distribution proposed by Bourguignon etal. (2014). Hence, this study proposed “Weibull-Normal distribution: its properties and applications”.
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