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

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

CHAPTER ONE

INTRODUCTION

1.0.1Â Â Â  Background of the study

Probability distributions are recently receiving alot of attention with regards to in-troducing 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) distri-bution 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) intro-duced 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-ponentiated 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 dis-tributions by Bourguignon et al. (2014). This research is motivated by the work done by Bourguignon et al. (2014) – The Weibullâ€“G family of probability distri-butions 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 distribu-tions 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.