EXTENSION OF BURR V DISTRIBUTION: ITS PROPERTIES AND APPLICATION TO REAL-LIFE DATA
CHAPTER ONE:
INTRODUCTION
1.1Â Â Â Â Â Â Â Â 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 generator 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
p
that this is a genuine distribution, in such a way that ò2p g (x )dx =1, where g ( x) is the PDF of the – 2 proposed distribution. Then, we will also be define and discuss some properties of the extended distribution.
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EXTENSION OF BURR V DISTRIBUTION: ITS PROPERTIES AND APPLICATION TO REAL-LIFE DATA
