## ODD GENERALIZED EXPONENTIAL-INVERSE-EXPONENTIAL DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

**ABSTRACT**

There remain many problems in real life where observed data do not follow any of the well-known probability models. To address this, there is then, a strong need to propose probability models to better capture the behavior of some real-life phenomenon. In this dissertation, we propose a new lifetime distribution, called the Odd Generalized Exponential-Inverse-Exponential Distribution (OGE-IED) based on the odd generalized exponential generator proposed in earlier study. The statistical properties of the new distribution are studied along with its reliability functions and limiting behavior. Also, the distribution parameters were estimated using the method of maximum likelihood. We illustrate the flexibility of the proposed model by application to real data and concluded that the OGE-IED provides the best fit among the competing distributions. Finally, we recommend that this distribution can be used in modeling time-dependent events, systems, components or random variables.

**CHAPTER ONE **

**INTRODUCTION**

**1.1 Background of the Study**

Statistical data modeling is an important aspect of statistics that attracts researcher’s wide attention. Modeling lifetime data in areas comprising reliability analysis, engineering, economics, biological studies, environmental and medical sciences, requires an appropriate statistical model for proper actualization of the data. However, there still remains problems where the real data does not follow any of the classical or standard probability models. To address this, there is strong need then to propose new models that can better capture real-life phenomenon inherent in a given dataset. Introducing new probability models or their classes is an old practice and has ever been considered as very valuable as many other practical problems in statistics. The idea simply started with defining different mathematical functional forms, and then induction of location, scale or inequality parameters (Tahir and Cordeiro, 2016a). Addition of new shape parameter(s) expands a model into a larger family of distributions and can provides significantly skewed and heavy-tailed new distributions. It also provides greater flexibility in the form of new distributions. This induction of parameter(s) has been proved useful in exploring tail properties and also for improving the goodness-of-fit of the proposed generator family (Saboor et al., 2015).

The one parameter Inverse Exponential distribution otherwise known as the Inverted Exponential distribution was introduced by Keller and Kamath (1982). It has an inverted bathtub failure rate and it is a competitive model for the Exponential distribution. It is an important probability distribution for modeling lifetime data.

**1.2 Statement of the Problem**

There exist several probability distributions for modeling lifetime data.However, some of these dataset do not follow any of the existing and well known standard probability distributions or at least are inappropriately described by them. This brings increased interest in proposing new univariate continuous distributions by adding one or more new shape parameter(s) to the baseline model. In this dissertation, a new probability distribution called Odd Generalized Exponential-Inverse-Exponential distribution (OGE-IED) taking inverse-exponential as the baseline distribution and using Tahir et al., (2015) generator is being proposed, aimed to provide greater flexibility and create more weight to the tails of the new distribution.

**1.3 Aim and objectives of the study**

This work is aimed at developing a new probability density function called Odd Generalized Exponential-Inverse -Exponential distribution(OGE-IED) using Inverse-Exponential distribution as the baseline distribution.

The stated aim is to be achieved through the following objectives:

i. Deriving the new distribution, named Odd Generalized Exponential-Inverse-Exponential distribution (OGE-IED) and check its validity;

ii. Obtaining some statistical properties of the proposed distribution comprising moment, moment generating function, reliability function, hazard function, quantile function and distribution of its order statistics;

iii. Estimating the parameters of the proposed distribution using the method of maximum likelihood estimation (MLE); and

iv. Comparing the fitness of the proposed model with other competing models in the literature using real life dataset.

**1.4 Significance of the Study**

The new proposed distribution can be of significance in modeling lifetime data, since induction of new shape parameter(s) has been proved useful in exploring tail properties and also in improving the goodness-of-fit of the proposed generator family (Saboor et al., 2015). It can be successfully applied to problems arising in several areas of research such as physical and biological sciences, reliability theory, hydrology, medicine, meteorology, engineering and survival analysis.

**1.5 Limitation**

The study focused only on developing new probability distribution and deriving its mathematical expressions for some selected properties of the proposed distribution such as moment generating function, quantile function, median, asymptotic behavior, density functions for the minimum and maximum order statistics and estimating the model parameters by using the method of MLE.

**1.6 Motivation**

Tahir et al., (2015) highlighted some special distributions that can be obtained by their Odd Generalized Exponential generator. These distributions include the Normal, Frechet, and Weibull distributions. Also, Johnson et al. (1994) stated that the use of four-parameter distributions should be sufficient for most practical purposes. According to them, “at least three parameters are needed but they doubted any noticeable improvement arising from including a fifth or sixth parameter”. Besides these facts, the inverse-exponential distribution is a well-known and commonly used standard continuous distribution with many areas of application. However, there has been no study on the hybrid of Odd Generalized Exponential Inverse-exponential distribution since the introduction of the Odd Generalized Exponential Generator by Tahir et al. (2015). Hence the need to propose an Odd Generalized Exponential- Inverse-exponential distribution (OGE-IED), study some of its properties and explore its applicability to a real life dataset.

**1.7 Definition of terms**

**1.7.1 Quantile Function**

Quantile function is use for calculating the median, skewness and kurtosis and for simulation of random numbers.

**1.7.2 Skewness**

Skewness is a measure of the degree of asymmetry or lack of symmetry of a distribution. It is said to be either right skewed (positively skewed) or left skewed (negatively skewed) when the measure is on either sides of zero respectively.

**1.7.3 Kurtosis**

A statistical technique that measures how peaked a distribution is. The kurtosis of a normal distribution is zero. If kurtosis is different from zero, then the distribution is either flatter or more peaked than normal.

**1.7.4 Survival Function**

The reliability function is otherwise known as the survival or survivor function. It is the probability of surviving beyond a specified time and it is obtained mathematically as the complement of the cumulative density function (CDF).

**1.7.5 Hazard Function**

Hazard function is also called the failure or risk function and is the probability that a system or component will fail or die in an interval of time.

**1.7.6 Order Statistics**

Suppose is a random sample from a distribution with pdf , and let

denote the corresponding order statistic obtained from this sample, then the pdf,

of the order statistic can be obtained by

It is used in a wide range of problems including robust statistical estimation and detection of outliers, characterization of probability distributions and goodness of fit tests, entropy estimation, analysis of censored samples, reliability analysis, quality control and strength of materials.

**1.7.7 Maximum Likelihood Method**

Let be a random sample from a population with probability density function

where is an unknown parameter. The likelihood function, is defined to be

the joint density of the random variables . That is,

(1.6.2)

The sample statistic that maximizes the likelihood function is known as maximum likelihood estimator of and is denoted as and the technique is called maximum likelihood method.