## EXTENSION OF BURR V DISTRIBUTION: ITS PROPERTIES AND APPLICATION TO REAL-LIFE DATA

**Abstract**

More than a decade ago, a new general family of continuous distribution known as beta-G Distribution was proposed. This new generator has been used to propose some distributions in recent years with simplified mathematical treatment. An extension of Burr V (BV) distribution was therefore proposed, using the logit of the beta random variable and some of its properties studied. The Probability Density Function (PDF) of the Beta Burr V (BBV) was defined and verified. Hazard Rate Function, Survival Function and Asymptotic Behavior of the distribution were obtained, the distribution was also found to be bimodal. The method of maximum likelihood was proposed to estimate the parameters of the distribution. The performance of the new distribution was evaluated using a real-life data. The extended distribution was found to be superior and quite flexible when subjected to scrutiny with burr V, beta Dagum and Beta Power Exponential distributions by fitting it to two real empirical datasets, thus; it was found that it can serve as a good alternative distribution to model positive real data in many areas.

**CHAPTER ONE **

**INTRODUCTION**

**1.1 Background to the Study**

Extended or generalized distributions have been extensively studied in recent years. Amoroso (1925) was the pioneer researcher to start generalizing continuous distributions, discussing the generalized gamma distribution to fit observed distribution of income rate. Since then, numerous authors have developed various classes of generalized distributions. Well-known distributions have been generated or extended in many ways. Some of the well-established generators are Marshal-Olkin generated family (MO-G) by Marshall and Olkin (1997), the beta-G by Eugene et al. (2002), Jones (2004), gamma-G (type 1) by Zografos and Balakrishnan (2009), Kumaraswamy-G (Kw-G for short) by Cordeiro and De Castro (2011), gamma-G (type 2) by Ristic and Balakrishnan (2012), gamma-G (type 3) by Torabi and Hedesh (2012), McDonald-G (Mc-G) by Alexander et al. (2012), log-gamma-G by Amini et al. (2014), exponentiated generalized-G by Cordeiro et al. (2013), Weibull-G by Bourguignion, et al. (2014) among others. Recent developments have been geared to define new families by introducing shape parameters to control skewness, kurtosis and tail weights thus providing great flexibility in modeling skewed data in practice (Jones, 2004; Cordeiro et al., 2013).

Evidently, one of the most used generalized distribution generators is the beta-G. The earliest of the beta extended distributions is the class of distributions generated from the logit of a beta random variable with cumulative distribution function that involve employing two parameters whose role is to introduce skewness and to vary tail weights (Eugene et al., 2002). Jones (2004) discusses general beta family influenced by its order statistics and shows that it has beautiful distributional properties and potential for interesting statistical applications. The generalization method used is the logit of beta distribution.

In this Research, we will define and study a four-parameter beta-Burr type V distribution. To attain this, we will let F(x) be the CDF of the Burr type V distribution as given in Equation (1.7) such that the emerging distribution is being referred to as the beta-Burr type V distribution (BBV) using appropriate transformations and employing the logit of beta. We will then prove

that this is a genuine distribution, in such a way that ò2__p__*g* (*x* )*dx* =1, where *g* ( *x*) is the PDF of the

– _{2}

proposed distribution. Then, we will also be defined and discuss some properties of the extended distribution.

**1.2 Aim and Objectives of the Study**

The aim of this work is to extend Burr V (BV) distribution and study some of its properties and evaluate its performance using a real-life data. This will be achieved through the following objectives:

i. To define and express the pdf of the Beta Burr V (BBV) distribution as a mixture of beta- G density function using the logit of beta and derive some of its statistical properties.

ii. To estimate the parameters of the BBV distribution by Maximum Likelihood procedure.

iii. To evaluate the performance of the BBV compared to other traditional distributions.

**1.3 Scope of the study**

The study will focus only on building or proposing a new BBV distribution and then exploring the verification of its PDF. The expression of the CDF will be defined; the asymptotic behavior of the proposed distribution, hazard rate function, and order Statistics will be given too. Its parameter will then be estimated using the method of Maximum Likelihood Estimation (MLE) procedure.

**1.4 Statement of the problem**

One of the interests of statisticians and researchers is to simplify and get better flexibility out of probability models or distributions. Many lifetime data used for statistical analysis follow a particular probability distribution and hence the technical know-how of the appropriate distribution that any circumstances follow greatly better the sensitivity, strength and efficacy of the statistical tests related to it. As it was found in the work by Jones (2004) generalization of continuous distribution yield a better and more flexible distribution, though the baseline distributions can sometimes outperform the resulting hybridized distribution- this is mostly depending on the parameters in place.

Different distributions are used for modeling these real-life data. Some of these real-life data do not follow any of the existing well-known probability distributions or are ineptly described by the models. It is as a result of these that building or developing a new distribution that could better explain most of these situations or circumstances and thus provide greater elasticity in the modeling of real-life data.

**1.5 Significance of the Study**

At the end of the proposition, the distribution, its properties and the parameter estimates will increase the flexibility of the baseline distribution- Burr type V distribution and easily model various data sets that cannot be properly fitted by the existing distributions. The hazard and survival functions of the proposed distribution will allow for its applications in reliability analysis. Afterwards, the study will compare the proposed model to some existing generalizations of the Beta distribution and other generalized distribution to identify the most fitted model using a real-life data set.

**1.6 Motivation**

Ever since the introduction of the Burr distribution, it has somewhat been neglected as an option in the analysis of lifetime data. Even though there are researches that indicate that this distribution possesses sufficient flexibility to make it a possible model for various types of data; for example, beta-Burr X was used to examine the strengths of 1.5 cm glass fibers (Merovci et al. 2016) and Paranaiba et al. (2011) introduced beta-Burr XII to discuss the cancer recurrence.

However, the distributions such as Burr type V and beta individually fail to address the variety of shapes the distribution should exhibit and to what direction it should take. That is, to combine both characteristic of shape and scale parameters, and lacking in-terms of flexibility to model real-life data set in statistics.

Hence, it has become imperative to have an extension of the Burr V distribution using the logit of beta. This we hope will be better than each of them individually in terms of the estimate of their characteristics in their parameters and application to real-life data.