## A STUDY OF PROPERTIES AND APPLICATIONS OF WEIBULL-BURR XII DISTRIBUTION

**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 modelling 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 distributionâ€™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 perform 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).

**CHAPTER ONE**

**INTRODUCTION**

**1.0.1 Background of the study**

Probability distributions are recently receiving a lot 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 prob-ability 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 two-parameter exponential-geometric distribution introduced by Adamidis and Loukas in 1998 which has a decreasing failure rate. Following the same idea of the exponen-tial 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), exponentiated T-X proposed by Alzaghal et al.(2013). Most recently, a New Weibull-G Family of Distributions by Tahir, (2016), The Weibullâ€“G family of probability distributions by Bourguignon et al. (2014). This research is motivated by the work done by Bourguignon et al. (2014) – The Weibullâ€“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 conventional 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 modelling.

**1.0.3 Purpose of the Study**

Existing literature focus on generalizations or modifications of the Weibull distribution that can provide more flexibility in modelling 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 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 the Weibull-Burr XII distribution perform when compared with other Weibullâ€“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 ParanaĂba et al. (2011) where it was concluded that application of the Beta-BXII 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 ParanaĂba 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 RĂ©nyi 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 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.