ABSTRACT
In recent times, lots of efforts have been made to define new probability distributions
that cover different aspect of human endeavors with a view to providing alternatives
in modeling real data. A five-parameter distribution, called Weibull-Burr XII (Wei-
Burr XII) distribution is studied and investigated to serve as an alternative model
for skewed data set in life and reliability studies. Some of its statistical properties
are obtained, these include moments, moment generating function, characteristics
function, quantile function and reliability (survival) functions. The distribution’s
parameters are estimated by the method of maximum likelihood. We evaluated the
performance of the new distribution compared with other competing distributions
based on application on real data and it was concluded that Weibull-Burr XII distribution
perfom best using BIC, AIC and CAIC. It was also concluded that the
distribution can be used to model highly skewed data (skewed to the right)
TABLE OF CONTENTS
Flyleaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
CHAPTER ONE
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.0.1 Background of the study . . . . . . . . . . . . . . . . . . . . . 1
1.0.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . 2
1.0.3 Purpose of the Study . . . . . . . . . . . . . . . . . . . . . . 2
1.0.4 Aim and Objectives of the Study . . . . . . . . . . . . . . . 3
1.0.5 Significance of the study . . . . . . . . . . . . . . . . . . . . . 3
1.0.6 Limitations of the study . . . . . . . . . . . . . . . . . . . . 4
1.0.7 Definition of terms . . . . . . . . . . . . . . . . . . . . . . . . 4
CHAPTER TWO
LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
CHAPTER THREE
RESEARCH METHODOLOGY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.0.8 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 9
3.0.9 Burr XII distribution . . . . . . . . . . . . . . . . . . . . . . . 9
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3.0.10 Weibull distribution . . . . . . . . . . . . . . . . . . . . . . . 10
3.0.11 The pdf of the generalized Weibull-G family . . . . . . . . . . 10
3.0.12 The Cumulative Distribution Function of Weibull-G Family . 11
3.0.13 The pdf of the Weibull-Burr XII Ditribution based on the
generalized Weibull-G pdf . . . . . . . . . . . . . . . . . . . . 11
3.0.14 Validity of the pdf of Weibull-Burr XII distribution and the
Cumulative Distribution Function (cdf) . . . . . . . . . . . . . 13
3.0.15 Cumulative Distribution Function (cdf) of Weibull-Burr XII
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.0.16 Survival function of Weibull-Burr XII distribution . . . . . . 16
3.0.17 Hazard Rate Function of Weibull-Burr XII distribution . . . . 17
3.0.18 Quantile function of Weibull-Burr XII Distribution . . . . . . 21
3.0.19 Expansion of the pdf ofWeibull-Burr XII by using power series
expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.0.20 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.0.21 Moment Generating Function of Weibull-Burr XII Distribution 26
3.0.22 Characteristic Function of Weibull-Burr XII Distribution . . . 27
3.0.23 Estimation of Parameters of Weibull-Burr XII Distribution . . 28
CHAPTER FOUR
RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS . . . . . . . . . 34
5.0.24 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.0.25 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.0.26 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ix
LIST OF FIGURES
3.0.1 Graph of the CDF of the Weibull-Burr XII distribution . . . . . . . 17
3.0.2 Graph of the pdf of the Weibull-Burr XII distribution . . . . . . . . 18
3.0.3 Graph of the Survival function of the Weibull-Burr XII distribution 19
3.0.4 Graph of the Hazard function of the Weibull-Burr XII distribution 20
4.0.1 Histogram representing the survival times of 121 patients with breast
cancer obtained from a large hospital in a period from 1929 to 1938 31
4.0.2 Graph of a Histogram showing the Survival Times (in days) for the
Patients in Arm A of the Head-and-Neck-Cancer Trial . . . . . . . 32
x
CHAPTER ONE
INTRODUCTION
1.0.1 Background of the study
Probability distributions are recently receiving alot of attention with regards to introducing
new generators for univariate continuous type of probability distributions
by introducing additional parameter(s) to the base line distribution. This seemed
necessary to reflect current realities that are not captured by the conventional probability
distributions since it has been proven to be useful in exploring tail properties
of the distribution under study (Tahir, et.al; 2016).
This idea of adding one or more parameter(s) to the baseline distribution has been
in practice for a quite long time. Several distributions have been proposed in
the literature to model lifetime data. Some of these distributions include: a twoparameter
exponential-geometric distribution introduced by Adamidis and Loukas
in 1998 which has a decreasing failure rate. Following the same idea of the exponential
geometric distribution, the exponential-Poisson distribution was introduced by
Kus (2007) with also a decreasing failure rate and discussed some of its properties.
Marshall and Olkin (1997) presented a simpler technique for adding a parameter to
a family of distributions with application to the exponential and Weibull families.
Adamidis et al. (2005) suggested the extended exponential-geometric (EEG) distribution
which generalizes the exponential geometric distribution and discussed some
of its statistical properties along with its hazard rate and survival functions.
Some of the well-known class of generators include the following: Kumaraswamy-G
(Kw-G) proposed by Cordeiro and de Castro (2011), McDonald-G (Mc- G) introduced
by Alexander et al. (2012), gamma-G type 1 presented by Zografos and
Balakrishanan (2009), exponentiated generalized (exp-G) which was derived by
Cordeiro et al. (2013), others are weibull-power function by Tahir et. al. (2010), ex-
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ponentiated T-X proposed by Alzaghal et al.(2013). Most recently, a NewWeibull-G
Family of Distributions by Tahir, (2016), The Weibull–G family of probability distributions
by Bourguignon et al. (2014). This research is motivated by the work
done by Bourguignon et al. (2014) – The Weibull–G family of probability distributions
who introduced a generator based on the Weibull random variable called
a Weibull-G family. In this research, we propose an extension of the Burr XII pdf
called the Weibull-Burr XII distribution based on the Weibull-G class of distributions
defined by Bourguignon et al (2014). i.e. we propose a new distribution with
five parameters, referred to as the Weibull-Burr XII (Wei-BXII) distribution, which
contains as special sub-models the Weibull and Burr XII distributions.
1.0.2 Statement of the problem
It has been anticipated that a generalized model is more flexible than a conventional
or ordinary model and its applicability is preferred by many data analysts in analyzing
statistical data. It is imperative to mention that through generalizations, the
convetional logistic distribution with only two parameters (location and scale) has
been propagated into type I, type II and type III generalized logistic distributions
which has three parameters each as indicated in Balakrishnan and Leung (1988).
So, there is a genuine desire to search for some generalizations or modifications of
the Burr XII distribution that can provide more flexibility in lifetime modeling.
1.0.3 Purpose of the Study
Existing literature focus on generalizations or modifications of the Weibull distribution
that can provide more flexibility in modeling lifetime data such as; Weibull-
Log logistic distribution by Broderick (2016), Weibull-Lomax distribution by Tahir,
(2015), etc. Less attention is given to generalization of Weibull and Burr XII distributions.
Where the later distribution was discovered by Burr in1942 as a two
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parameter family. An additional scale parameter was introduced by Tadikamalla in
1980. It is a very popular distribution for modelling lifetime data.
The purpose of this research focuses mainly on generalization of a Burr XII distribution
to a five-parameter distribution, called the Weibull-Burr XII (Wei-BurrXII)
distribution for modelling skewed data set (skewed to the right).
1.0.4 Aim and Objectives of the Study
The aim of this research is to study Weibull-Burr XII probability distribution and
investigate its properties and applications. This is expected to be achieved through
the following objectives by:
1. establishing the Weibull-Burr XII distribution;
2. establishing some statistical properties of Weibull-Burr XII distribution such
as; moments, moment generating function, quantile function, characteristics
function, survival function and hazard rate function;
3. estimating the parameters of the proposed model by the method of maximum
likelihood estimation;
4. evaluating how well theWeibull-Burr XII distribution perform when compared
with other Weibull–G family of distributions based on application on real life
data.
1.0.5 Significance of the study
Many models were introduced in the literature by extending some distributions with
Burr XII distribution. e.g. the Beta- Burr XII (BBXII) distribution discussed by
ParanaÃba et al. (2011) where it was concluded that application of the Beta-BXII
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distribution indicated that it had provided a better fit than other statistical models
used in lifetime data analysis, the Kumaraswamy -Burr XII distribution introduced
by ParanaÃba et. al. (2013). Therefore, the significance of this study is mainly to
propose a new model (Wei-Burr XII distribution) that is much more flexible than
the Burr XII distribution.
1.0.6 Limitations of the study
The limitation of this research is that, it did not consider estimating parameters of
the Weibull-Burr XII distribution using other methods like Bayesian method. Some
other properties of probability distribution are also not considered in this research
work. e.g Rényi entropy, incomplete moments, e.t.c.
1.0.7 Definition of terms
Reliability is generally regarded as the likelihood that a product or service is
functional during a certain period of time under a specified operation.
Survival function is the probability that a patient, device, or other object of
interest will survive beyond a specified time. It is also known as the survivor function
or reliability function.
S(x) = Pr(an object will survive beyond time x).
Hazard function (also known as the failure rate, hazard rate, or force of mortality)
is the ratio of the probability density function to the survival function. Failure rate
is the frequency with which an engineered system or component fails, expressed in
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failures per unit of time (Evans,et.al. 2000)
H(x)= Pr(an object will fail at time x+t given that it survive up to time x)
Akaike Information Criterion (AIC) is a measure of the relative quality of
statistical models for a given set of data. Given a collection of models for the data,
AIC estimates the quality of each model, relative to each of the other models. Hence,
AIC provides a means for model selection. Given a set of candidate models for the
data, the preferred model is the one with the minimum AIC value. Mathematically,
AIC =2k-2ll
Where ll is the log-likelihood function for the model and k is the number of estimated
parameters in the model.
Bayesian Information Criterion (BIC) or Schwarz criterion is also a criterion
for model selection among a finite set of models. The model with the lowest BIC is
preferred. Computed by;
BIC = ln(n)k-2ll where n is the sample size and k is the number of estimated
parameters in the model.
Consistent Akaike Information Criterion (CAIC) is mathematically defined
by
CAIC = -2ll+ 2kn/(n-k-1) where ll = log likelihood.
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