Odd Generalized Exponential-inverse-exponential Distribution: It’s Properties and Applications
Chapter One
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:
- deriving the new distribution, named Odd Generalized Exponential-Inverse-Exponential distribution (OGE-IED) and check its validity;
- 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;
- estimating the parameters of the proposed distribution using the method of maximum likelihood estimation (MLE); and
- comparing the fitness of the proposed model with other competing models in the literature using real life dataset.
CHAPTER TWO:
LITERATURE REVIEW
This chapter present a review of the literature on the techniques used in proposing new probability distribution, the classical distribution used (baseline distribution) and its related generalization, the generator and the probability distributions proposed so far, and finally the parameter estimation methods.
An approach to establishing flexible probability distribution is to insert appropriate probability function into a larger model by adding shape parameter (Aryal and Tsokos, 2009). Tahir et al., (2015a) stated and explained two main approaches for adding new shape parameter(s) to a baseline distribution.
In the first approach, Marshall and Olkin (1997) pioneered the generalization of probability distributions by inducing a parameter to the survival function, , where is the cumulative distribution function (CDF) of any baseline distribution. Gupta et al., (1998) induced one parameter to the CDF, of a baseline distribution to define the Exponentiated- G (“Exp-G” for short) class of distributions based on Lehmann-type alternatives (see Lehmann 1953). Following Gupta et al.’s proposed family of distributions, Gupta and Kundu (1999) studied two-parameter generalized-exponential (GE) distribution as an extension of the exponential distribution based on Lehmann type I alternative. The GE distribution is also known as the Exponentiated exponential (EE) distribution. As one of the early generalizations of the exponential distribution, the GE distribution has received increased attention and many researchers studied its various properties and also offered some comparisons with other distributions. In fact, the GE distribution has been proven to be a good alternative to the gamma, Weibull and log-normal distributions, all of them being two-parameter distributions. The GE distribution can be used effectively for analyzing lifetime data with monotonic hazard rate function but unfortunately it cannot be used if the hazard rate function is upside-down, J or reversed-J shapes.
The second approach is achieved by adding two or more shape parameters. This approach of generalization was pioneered by Eugene et al., (2002) and Jones (2004) who defined the beta-generated (beta-G) class from the logit of the beta distribution. Further work on generalized distributions are the Kumaraswamy-G (Kw-G) by Cordeiro and de Castro (2011), McDonald-G (Mc-G) by Alexander et al., (2012), Odd gamma-G type 3 by Torabi and Montazari (2012), Odd Exponentiated Generalized (odd exp-G) by Cordeiro et al., (2013), transformed-transformer (T-X) (Weibull-X and gamma-X) by Alzaatreh et al., (2013), Exponentiated T-X by Alzaghal et al., (2013), gamma-G type 1 by Zografos and Balakrishanan (2009) and Amini et al., (2014), gamma-G type 2 by Ristic and Balakrishanan (2012) and Amini et al., (2014), logistic-G by Torabi and Montazari (2014), Odd Weibull-G by Bourguignon et al., (2014), Exponentiated half-logistic by Cordeiro et al., (2014), T-X{Y}-quantile based approach by Aljarrah et al., (2014) and TR{Y} by Alzaatreh et al., (2014), a new Weibull-G by Tahir et al., (2015b) and Kumaraswamy odd log-logistic-G by Alizadeh et al., (2015), Lomax-G by Cordeiro et al., (2014), logistic-X by Tahir et al., (2016b).
A one-parameter Inverse Exponential distribution introduced by Keller and Kamath (1982) has an inverted bathtub failure rate and was found to be competitive in comparison to Exponential distribution. It is one of the distributions that are used in modelling lifetime data. Recently, several generalization of inverse-exponential distribution were obtained. Abouammoh and Alshangiti (2009) introduced a generalized version of the inverted exponential distribution (GIED). The study investigated some statistical properties and reliability functions of the GIED. Furthermore, it can be considered as the inverse of the well-known generalized exponential distribution (GED) introduced by Gupta and Kundu (1999). Oguntunde et al., (2014) studied a three-parameter distribution called the Kumaraswamy Inverse Exponential distribution (K-IED). K-IED parameters were estimated and an explicit expression for the moment was given. The distribution is believed to be leptokurtic. A one-parameter Inverse Exponential distribution is being identified as a special case of K-IE distribution when the two additional parameters and have taken a unit value of one. It is expected that, the K-IE distribution would have an advantage over the Beta Inverse Exponential distribution in terms of tractability because the former does not involve any special function like the incomplete beta function ratio. Singh and Goel (2015) developed a three-parameter beta inverted exponential distribution which contains generalized inverted exponential and inverted exponential distributions as special sub models. The distribution can be applied effectively in modelling lifetime data, due to its ability to accommodate non-monotonic, uni-modal and inverse bathtub-shaped hazard functions.
CHAPTER THREE:
METHODOLOGY.
The Definition of the Generator
For any continuous distribution with cumulative distribution function (CDF), and
probability density function (pdf), , Tahir et al., (2015a) proposed a new generator for the class of univariate distributions called the Odd Generalized Exponential-G (OGE-G) family of distributions that provides greater flexibility in modelling of real data sets. The cumulative distribution function (CDF) of the OGE-G family of distributions according to Tahir et al., (2015a) is defined as follows:
Given a cumulative distribution function (CDF) of Generalized Exponential distribution (3.1.1)
The CDF of the OGE-G is obtained by replacing the variable in equation (3.1.1) with , as given in equation (3.1.2) below:
CHAPTER FOUR:
ANALYSIS AND DISCUSSION
Introduction
In this chapter, we illustrated the performance of the Odd Generalized Exponential-Inverse-Exponential distribution by fitting real dataset to the model and its baseline model (Inverse-exponential distribution) together with some of the baseline generalizations.
CHAPTER FIVE:
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
In this dissertation, we introduced a new three parameter probability model named the Odd Generalized Exponential-Inverse-Exponential distribution (OGE-IED), which is a hybridization of Generalized Exponential distribution with Inverse-Exponential distribution by adding skewness to the classical Inverse-Exponential distribution. An obvious reason for generalizing a classical distribution is the fact that the generalization provides more flexibility to the new distribution which will enable it to capture more information contained in the real life data that cannot be assessed by the classical distribution. Some of its statistical properties like its asymptotic behaviour, moments, moment generating function, quantile function, reliability analysis and order statistics are studied. The parameter estimates has been determined by the method of maximum likelihood. The usefulness of the OGE-IED has been illustrated by an application to real data set. The new distribution provides better fits compared to the baseline distribution and four of its generalizations. However, the results showed that the new distribution is more flexible and appropriate for modelling right skewed data sets.
Conclusion
A new three parameter probability model named the Odd Generalized Exponential-Inverse-Exponential distribution (OGE-IED) was developed. Some statistical properties of the proposed distribution have been studied appropriately and derived explicit expressions for its moments, moment generating function, quantile function, median, survival function, hazard function, ordered statistics and limiting behaviour. Some plots of the distribution revealed that the distribution is also positively skewed and has only one mode. The model parameters were estimated using the method of maximum likelihood. The implications of the plots for the survival and hazard functions indicate that the OGE-IED would be appropriate in modelling time-dependent events, where survival and failure rate decreases with time.
Recommendations
Based on the findings of this research, (i) the proposed distribution should be used for modelling lifetime events if the data set in question is right skewed (ii) this distribution can be used confidently in modelling time-dependent events, systems, components or random variables.
Contribution to Knowledge
A three parameter distribution which is useful for modelling right skewed data sets better than some existing compound inverse-exponential based probability distributions has been proposed.
Areas for Further Research
Based on the review in this area of research, interested researchers can look at estimation of confidence intervals of the proposed distribution parameters. Also, researchers can estimate the parameters of the new distribution using Bayesian approaches for the purpose of theoretical comparison and validation of methodology.
REFERENCES
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- Abouammoh, A. and Alshangiti, A. M. (2016). The Modified Generalized Inverted Exponential Distribution. Universal Journal of Applied Mathematics, 4(3), 43-50.
- Alizadeh, M. Emadi, M. Doostparast, M. Cordeiro, G. M., Ortega, E. M. M. and Pescim, R. R. (2015): A new family of distributions: the Kumaraswamy Odd Log-logistic: Properties and Applications. Hacet. J. Math Stat.44(6), 1491-1512.
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