## EXPONENTIATED-EXPONENTIAL WEIBULL DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

ABSTRACT

A four-parameter distribution called Exponentiated-Exponential Weibull (EEW) distribution was proposed using a generator introduced in earlier research. Statistical properties of the distribution were derived along with its reliability functions and asymptotic behaviour. The distribution parameters were estimated using the method of Maximum Likelihood Estimation (MLE). The distribution generalizes some well-known special lifetime models such as Exponential, Exponential Weibull and Generalized Exponential distribution. The probability density of the distribution indicate that it is skewed to the right (Positively skewed). We compared the performance of EEW distribution with some related distributions from the literature using the AIC and CAIC method of comparison. Applicability of the distribution was also observed using two real datasets and it was found that the distribution fitted the dataset. Finally it is recommended that the distribution can be used for modelling real life data especially if it is positively skewed.

CHAPTER ONE

INTRODUCTION

1.1    Background to the Study

Several classical distributions have been widely used over the past decades for modelling lifetime data in many areas such as reliability, engineering, economics, biological studies, environmental actuarial, environmental and medical sciences, demography, and insurance. However, in many applied areas such as lifetime analysis, finance, and insurance, there is a clear need for extended forms of these distributions. This is because there still remain many important problems where the real data does not follow any of the classical or standard probability models. For that reason, numerous methods for generating new families of distributions have been considered (Bourguignon et al., 2014). To handle this, there is a strong need to propose useful models for the better study of the real-life marvel. Introducing new probability models or their classes is an old practice and has ever been considered as valuable as many other practical problems in statistics. According to Tahir and Cordeiro (2016), the idea simply started with defining different mathematical functional forms, and then adding of location, scale or shape parameter(s).

Inducing of a new shape parameter(s) introduces a model into greater family of distributions and can give significantly skewed and heavy-tailed distributions and also provides greater flexibility in the form of new distribution. Gupta et al., (1998) proposed a generator for defining a new univariate continuous family of distributions by adding one shape parameter to the baseline (parent) distribution. Gupta and Kundu (1999) pioneered the study of two-parameter generalization, in which they studied two-parameter Generalized-Exponential (GE) distribution also called Exponentiated-Exponential (EE) distribution, after which several other authors worked on GE distribution due to its attractive features, among which are Gupta and Kundu (2001a, 2001b, 2002 and 2003), Kundu et al., (2005) among others. The authors present two real life data sets, where it is observed that in one data set Exponentiated exponential distribution has a better fit compared to Weibull or gamma distribution and in the other data set Weibull has a better fit than Exponentiated exponential or gamma distribution. The induction of parameter(s) has been proved useful in discovering tail properties and also for improving the goodness-of-fit of the proposed distribution (Saboor et al., 2015).

The Generalized-Exponential (GE) distribution also known as Exponentiated Exponential (EE) is a continuous probability distribution belonging to the Exponentiated Family (EF) of distributions. Al-hussaini and Ahsanullah (2015) stated that the history of adding a parameter by exponentiation can be traced back to Gom-pertz (1825) and Verhulst (1838, 1845, 1847) where they used the CDF to compare known human mortality tables and represent mortality growth. The CDF is given as G(t) = (1 − e−t ) for t > 1 ln , where ; and are all positive. So also, in twentieth century, Ahuja and Nash (1967) considered this model and some further generalization. The Gompertz-Verhulst’s CDF happened to be the first member of the EF of distributions. The GE distribution is a particular case of this family for = 1.

The CDF and the pdf of the GE distribution is given by equation (1.3.1) and (1.3.2) respectively.

 G(x) = (1 − e− x) (1.1.1) g(x) =   e− x(1 − e− x) −1 (1.1.2)

where x; and are greater than zero.

The Weibull distribution is a continuous probability distribution. It was named after Swedish mathematician Waloddi Weibull, who describe it in detail in 1951, although it was first recognized by Frechet (1927) and first applied by Rosin and Rammler (1933) to described a unit size of distribution. Weibull distribution has the scale and shape parameters. This distribution has become very popular in analysing lifetime data and for many applications where a skewed distribution is required. It also has increasing and decreasing failure rates depending on the shape parameter. The Weibull distribution is often preferred for analysing lifetime data because, in presence of censoring, it is much easier to handle than the gamma distribution. Also in many positive datasets it is observed that the Weibull distribution fits very well. As one of the disadvantages of the distribution it maximum likelihood estimators has a very low asymptotic convergence to normality (Bain, 1976).

1.2    Statement of the Problem

Most of the lifetime data does not follow any of the classical distribution due to its uniqueness. Therefore, the need to provide univariate distributions that lifetime data follows is of great importance to address this problem. Adding new shape parameter(s) to the baseline distribution increases the flexibility and provide goodness-of-fit to new univariate continuous distributions. Induction of parameter(s) has been proved useful in discovering tail properties and also for improving the goodness-of-fit of the proposed distribution (Saboor et al., 2015). Hence, developing new distributions that would enhance in addressing this problem is of paramount importance. Our interest here is to propose a new probability distribution function which can serve as an extension of the Exponential Weibull (EW) called Exponentiated-Exponential Weibull (EEW) distribution using Gupta et al., (1998) generator so that it will address this problem appropriately to some extent.

1.3    Aim and Objectives

This research is aimed at developing a new probability density function called Exponentiated-Exponential Weibull (EEW) distribution using the generator pro-posed by Gupta et al., (1998).

The specific objectives through which the stated aim will be achieved are to:

1. derive and establish a new Exponentiated-Exponential Weibull distribution;
2. Study some of its statistical properties such as Survival function, Hazard function, Asymptotic behaviour, Quantile function, Median, Moment, Moment Generating Function and Order Statistics;
3. Estimate the parameters of the distribution using the method of Maximum Likelihood Estimation (MLE) and;
4. Compare the fitness of the proposed probability distribution with other related distribution using a real-life data.

1.4    Scope of the Study

The study focused on extending the Exponential Weibull distribution and deriving mathematical expressions for some selected properties of the proposed distribution such as Survival function, Hazard function, Asymptotic behaviour, Quantile function and Median, Moment, Moment Generating Function and Order Statistics and estimating the model parameters by using the method of Maximum Likelihood.

1.5    Significance of the Study

The significance of any research is to deliver a new way of applying the acquired knowledge to solve problems not necessarily a real-life problem. So, the proposed distribution can be used in modelling lifetime data, since induction of new shape parameter(s) has been proved useful in exposing tail properties and also improve the goodness-of-fit of the proposed generator family (Saboor et al., 2015). The proposed distribution will be compared with some existing distributions to identify the distribution that will provide better fit using a real dataset.

1.6    Motivation

Several standard theoretical distributions have been found to be useful in the field of reliability, insurance, engineering, medicine, economics, finance and so on. Yet, generalizing these standard distributions have produced several compound distributions that are more flexible compared to the baseline distributions. Hence, we intend to develop a new probability density function called Exponentiated-Exponential Weibull (EEW) distribution using Exponential Weibull distribution as our baseline and apply it to a real-life dataset.

1.7    Definition of Terms

1.7.1    Probability

Probability is a branch of mathematics that deals with calculating the likelihood of a given event’s occurrence, which is expressed as a number between 1 and 0. An event is considered as “certain” if it has a probability of 1 while an event with the probability of 0 is considered as an impossible or uncertain event.

1.7.2    Probability Distribution

A probability distribution is a statistical function that describes the frequencies or possible values and likelihoods that a random variable can take.

1.7.3    Continuous Random Variable

A non-discrete random variable X is said to be continuous random variable if its distribution function can be represented as;

 x F (X) = P (X ≤ x) = S−∞ f(u)du (1.7.1)

for every real number x and evaluated from −∞ to a real number x. while the probability density function (pdf) is defined as;

dF (x)

f(x) =  (1.7.2)

dx

and it has the following properties:

1. f(x) ≥ 0; ∀x

1. −∞ f(x)dx = 1
2. P (a < X < b) = ab f(x)dx

1.7.4    Moments

Moment is one of the statistical properties of any distribution that is used to study some of the characteristics of a random variable such as mean, variance, skewness (Sk) and kurtosis (Ku). Mathematically, it can be defined as: let X be a continuous random variable, then the rth moment of X about the origin is given in equation (1.8.3)

 r E Xr ∞ Xrf x dx (1.7.3) ′ = ( ) = S−∞ ( )

where f(x) is the proposed pdf.

1.7.5    Quantile Function

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

1.7.6     Skewness

It is a measure of the degree of asymmetric of a probability distribution. The skewness is either to the right (positively skewed) or to the left (negatively skewed) on the normal distribution plot.

1.7.7    Kurtosis

Is a statistical technique that measures the peak of distribution. The kurtosis of a normal distribution is zero and if the kurtosis vary from zero, it means that the distribution is peaked more than normal or flattered.

1.7.8    Moment Generating Function

The Moment Generating Function provides an alternative way to analytical result rather than working directly with the pdf. The MGF of a random variable X can be obtained using equation (1.8.7)

 Mx t ) = E ( etx ∞ etxf ( x dx (1.7.4) ( ) = S−∞ )

1.7.9    Reliability Analysis

Reliability analysis can be defined as the study and modelling of observed product lives. Life data can be lifetime of products in the marketplace such as the time product operated effectively or the time the product operated before it collapsed.

1.7.10    Survival Function

The survival function, also known as the reliability function in engineering, is the distinctive of an independent variable that maps a set of events, frequently related with mortality of some system onto time. It is the probability that the system will survive beyond a specified period of time. Mathematically it is given by;

S(x) = P (X > x) = S  f(u)du = 1 − F (x)     (1.7.5)

x

where F (x) is the CDF of any distribution.

1.7.11    Hazard Function

The hazard function is defined as the probability per unit time that a case which has survived to the beginning of the respective interval will fail in that interval. It is computed as the number of failures per unit time in the respective interval, divided by the average number of surviving cases at the mid-point of the interval. Mathematically its given by

Hx f (x) 1f (x) (1.7.6)

() =S (x) = − F (x)

where f(x) is pdf of any distribution.

1.7.12    Order Statistic

Order statistics is widely used in many areas of statistical theory and practice, for instance, detection of outlier in statistical quality control processes.

Suppose X1; X2; ; Xn is a random sample from a distribution with pdf f(x) and let X1 n; X2 n; ; Xi n denote the corresponding order statistic obtained from this sample. The pdf fi n(x) of the ith order statistic can be express as

 n! fi n(x) = (i − 1)!(n − i)!f(x)F (x)i−1[1 − F (x)]n−i (1.7.7)

1.7.13    Maximum Likelihood Estimation

The method of maximum likelihood estimation is used for estimating parameter of statistical model given observations, by finding parameter value that makes the known likelihood distribution a maximum. It is considered as robust of the parameter estimation techniques. Suppose X1; X2; ; Xn is a random sample from a population with pdf f(x; ), where is an unknown parameter to be estimated. The likelihood function, L( ), is defined to be the joint density of the random variables X1; X2; ; Xn. That is,

 n f(xi; ) L( ) = i 1 (1.7.8) M=

The sample statistic that maximizes the likelihood function L( ) is known as maximum likelihood estimator of and is denoted as.

References

Adeyemi, S. and Adebanji, A. O. (2004). The exponentiated generalized Pareto distribution. Ife J Sci, 6, 127-133.

Ahuja, J. C. and Nash S. W. (1967). The generalized Gompertz-Verhulst family of distributions, Sankhya, 29, 141-161.

Al-Hussaini, E. K. and Ahsanullah, M. (2015). Family of Exponentiated Exponential distribution. Atlantis Studies in Probability and Statistics, vol 5. Atlantis Press, Paris. DOI: 10.2991/978-94-6239-079-9-4.

Ayal, J. R. and Tsokos C. P. (2011). Transmuted Weibull; distribution:A generalization of the Weibull probability distribution, 4, 89-102.

Bain L. J. (1976). Statistical analysis of reliability and life in fatique. Journal of the American Statistic Association 49, 575-597.

Bourguignon, M., Sliver, R. B. and Cordeiro, G. M. (2014). The Weibull-G family of probability distribution. Journal of Data Science, 12, 53-68

Carrasco, J. M.F., Ortega, E. M. M., and Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. Comput Stat Data Anal, 53, 450-462.

Cordeiro, G. M., Gomes, A. E., Da Silva, C. Q. and Ortega, E. M. M. (2013). The beta exponential Weibull distribution. Journal of statistics comput simul, 83, 114-138.

Cordeiro G. M., Ortega, E. M. M. and Da-chunha D. C. C. (2013). Exponentiated Generalized Class of distributions. Journal of Data Science, 11, 01-027.

Cordeiro G. M., Ortega, E. M. and Lemonte A. J. (2013). The Exponential-Weibull distribution. Journal of Statistical Computation and Simulation, 2592-2606.

Cordeiro G. M., Ortega, E. M. M. and Thiago, G. R. (2015). A new generalized weibull family of distributions: mathematical properties and application. Journal of statistical distributions and applications, DOI: 10.1186/s40488-015-0036-6