## TRANSMUTATION OF WEIBULL-PARETO DISTRIBUTION

**ABSTRACT**

A generalization of the New Weibull-Pareto so called Transmuted Weibull-Pareto Distribution is proposed and studied using the quadratic rank transmutation map (QRTM). The structural properties which include explicit expression for the quantiles, moment and order statistics of the proposed distribution were derived. Its parameters were estimated using the method of Maximum likelihood and least square estimation. The Transmuted Weibull-Pareto Distribution was applied to real life data set to examine its flexibility over the New Weibull-Pareto distribution. The result obtained shows that the Transmuted Weibull-Pareto Distribution outperforms the New Weibull-Pareto distribution. The Performance criteria used are the AIC and BIC values.

**CHAPTER ONE**

**INTRODUCTION**

**1.1 Background to the study**

In the field of Statistics and reliability engineering, the quality of the procedures used in statistical analysis depends heavily on the assumed probability distribution. Due to this fact, significant efforts have been made by many researchers in the development of standard probability distributions which are different from the known classical probability distribution. The standard probability distributions are obtained by generalizing the classical probability distribution such as exponential, Weibull, Pareto and Beta distributions. Application of probability distributions in engineering, medicine, finance, ICT among others, have further shown that many data sets do not follow the existing classical distributions. As a result of this, there is the need for development of standard probability distributions by generalization of some well-known classical distributions.

These distributions are derived by adding one or more parameters to the baseline model of continuous distributions. These families provide more flexibility in modelling and in analyzing real life data in many applied areas. For instance, the generalized transmuted-G family proposed by Nofalet al(2015), Transmuted Weibull Distribution: A Generalization of the Weibull Probability Distribution proposed by Aryal and Tsokos (2011), the transmuted geometric-G family introduced by Afifyet al(2016), the transmuted exponentiated generalized-G class of distributions defined by Yousofet al(2015), transmuted exponential distribution proposed by Enahoroet al(2015)and the Kumaraswamy transmuted-G family introduced by Afifyet al (2016).

Over the years, several attempts have been made to generalize the Weibull distribution by adding new parameters into the distribution which has led to the development of new distributions. For instance theExponentiated Weibull distribution (Pal et al 2003), Transmuted Weibull distribution (Gokarnal and Chris, 2011), Lomax-Weibul distribution (Almheidat et al 2015) ,Beta Weibull Distribution (Cordeiro et al 2012), New Weibull-Pareto distribution ( Suleiman and Albert 2015).These distributions have been found to be more flexible than the Weibull distribution when applied to real life data sets.

**1.2 Weibull Distribution**

. The Weibull distribution is a well-known distribution named after Waloddi Weibull, He

developed the distribution in 1939 and applied it to analyze the breaking strength of materials. A random Variable X is said to have a Weibull distribution with pdf

f ( x ) ( x ) 1 e( x )

where

The distribution is mostly used in reliability engineering for studying the fatigue and endurance life in devices and materials. There is the fact that the Weibull distribution cannot capture the behavior of life time data sets that exhibit bathtub or upside down bathtub failure rate that are usually encountered in reliability engineering. This also led to several generalization of Weibull distribution which have been proposed and studied to address this limitation of Weibull.

**1.3 Pareto Distribution**

The Pareto distribution was developed in the 19th century by Italian economist Vilfredo Pareto to model the distribution of income over a population. Let x be a random variable from a Pareto distribution with its probability density function given by

f ( x; , k ) kk xk 1

where 0 is a scale parameter and k 0 is the shape parameter

`Pareto distribution is often used to model life time of a manufactures item.

**1.4 New Weibull-Pareto Distribution**

According to Suleiman and Albert (2015), a life random variable is said to have a New Weibull-Pareto distribution denoted by NWPD if its distribution function has the form:

G ( x ) 1 e ( x )

(1)

And its probability density function is

x 1 ( )

g ( x ) x

( ) e (2)

where

is the scale parameter, and are shape parameters.

The New Weibull-Pareto distribution (NWPD) is suitable for modelling components that wears faster with time or component that wears slower with time. Another usefulness of NWPD is that it can be used in the characterization of the survival time of a given system because of its analytical structure (Suleiman and Albert 2015).

**1.5 Statement of the Problem**

Many real life data sets are heavily skewed and cannot be fitted by the existing New Weibull-Pareto Distribution. As a result of this, there is the need to generalize the New Weibull-Pareto distribution by introducing a new parameter that will increase the flexibility of the existing New Weibull-Pareto distribution that will provide a better fit than the New Weibull-Pareto Distribution.

**1.6 Significance of the Study**

The introduction of a transmuted parameter to the New Weibull-Pareto Distribution will provide more flexibility in modeling and analyzing real life datasets in many applied areas and greatly improve the sensitivity and efficiency of the statistical tests associated with the distribution. Some real life data sets do not follow the existing New Weibull-Pareto distribution, as a result of this there is the need to generalize the New Weibull-Pareto distribution by adding a new parameter to it so as to increase its flexibility.

**1.7 Aim and Objectives of the Research**

The aim of this research is to propose a new distribution called Transmuted Weibull-Pareto Distribution using the Quadratic Rank Transmutation Map and to derive its properties. The objectives are to:

ii. Derive the Transmuted Weibull-Pareto Distribution using Quadratic Rank Transmutation Map(QRTM)

ii. Determine the reliability behavior (survival and hazard function) and obtain various structural properties.

iii. Estimate the parameters of the transmuted distribution through the Maximum Likelihood Estimation and Least Square Estimation.

iv. Compare the performance of the Transmuted Weibull-Pareto Distribution to New Weibull-Pareto Distribution (Suleiman et al 2015),Weibull-Pareto Distribution

(Tahir et al 2015) and Pareto Distribution.

**1.8 Definition of Terms and Relevant Abbreviations**

**AIC:** Akaike Information Criterion

**BIC**: Bayesian Information Criterion

**CDF:** Cumulative Distribution Function

**Classical Probability Distribution:** These are the common probability distributions such as Exponential, Weibull, Pareto and Beta distributions.

**Common Events:** These are events that occur with a high frequency

**PDF:** Probability Density Function

**Real life Data:** This is a data set obtained through an experiment. It is a non-simulated data set

**Rare Events:** These are events that occur with a low frequency

**Standard Probability Distribution:** These are probability distributions that are obtained by adding one or more parameters into the common probability distributions.

**Transmuted Probability Distribution:** These are probability distributions obtained using the Quadratic Rank Transmutation Map.

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