A Comparative Analysis on the Performance of the Convoluted Exponential Distribution and the Exponential Distribution in terms of Flexibility

Corresponding Author: Pelumi Emmanuel Oguntunde Department of Mathematics, Covenant University, Ogun State, Nigeria Email: peluemman@yahoo.com Abstract: In this article, the convoluted exponential distribution which was derived as the sum of two independent exponentially distributed random variables was compared with the exponential distribution in terms of flexibility when applied to four real data sets. The idea is to verify if the convoluted exponential distribution would perform better than the exponential distribution in modeling real life situations. Some other basic statistical properties of the convoluted exponential distribution were also identified.


Introduction
The concept of convolution is a very useful topic in the theory of statistics. As a result, a number of researchers have worked on the sum of Independent and Identically Distributed (IID) random variables. For instance, Sun (2011) defined and studied the convoluted beta Weibull distribution, Shittu et al. (2012) proposed and studied the convoluted beta exponential distribution, Oguntunde et al. (2014) studied the convoluted exponential distribution. Meanwhile, applications to real data sets were not examined in all these researches.
Let Z denote a random variable, it has the Probability Density Function (PDF) of the convoluted beta Weibull distribution (the sum of two independent beta Weibull variates) given by: The corresponding Cumulative Density Function (CDF) is given by:
Details about how Equation 1 and 2 were derived are rigorously explained in Sun (2011).
In the same way, the PDF of the convoluted beta exponential distribution is given by: The corresponding CDF is given by: For details about the construction of Equation 3 and 4, readers are referred to Shittu et al. (2012).
Another interesting part of the concept of convolution is when the sum of independent random variables from different distributions is considered.
The interest of this research is to further explore the convoluted exponential distribution defined in Oguntunde et al. (2014) by assessing its flexibility over the exponential distribution using four real data sets. The rest of this paper is structured as follows; details about the convoluted exponential distribution (including existing and new properties) are provided in section 2, real life applications are discussed in section 3, followed by a concluding remark.

Convoluted Exponential Distribution: Existing and More Properties
In this section, the PDF, CDF and basic properties of the convoluted exponential distribution are highlighted as available in Oguntunde et al. (2014). Also, some other new properties are given.
The PDF of the convoluted exponential distribution is given by: The corresponding CDF is given by: For λ 1 , λ 2 , z > 0 where; λ 1 and λ 2 are scale parameters The mean is given by: The moment generating function is given by:

Renyi Entropy
The Renyi entropy being one of the functions used in quantifying the uncertainty or randomness in a system is mathematically given by: For p ≠ 1 and p>0 For the Convoluted Exponential distribution, the entropy is derived from:

Emperical Study
The tables for the mean and variance of the convoluted exponential distribution are provided in Table 1 and 2 respectively.
It can be observed from Table 1 that the mean of the Convoluted Exponential distribution decreases as the parameter increases and vice versa. Table 2 reveals that the variance of the Convoluted Exponential distribution decreases as the value of the parameters increases and vice versa.
The data is summarized in Table 3 and the performances of the competing distributions are given in Table 4.

Remark
Considering Table 4, 6, 8 and 10, the model with the lowest AIC or highest log-likelihood is considered to be the best fit. This means that the exponential distribution is considered the best fit and thereby highlighted.

Conclusion
A comparison between the convoluted exponential distribution and the exponential distribution has been successfully done in terms of real life applications. It was observed that the exponential distribution outperformed the convoluted exponential distribution considering the four applications provided in this research. The decisions and conclusion in this study is based on the log-likelihood and AIC values posed by the distributions under study. For all the four data sets, the AIC value of the Exponential distribution is the lowest while its log-likelihood values are higher than that of the Convoluted Exponential distribution. Nevertheless, the authors did not underrate the concept of convolution. Convolution still remains a relevant topic in the theory of statistics. Further research would involve comparing convoluted beta Weibull distribution derived by (Sun, 2011) with beta Weibull distribution derived by (Famoye et al., 2005) and comparing convoluted beta exponential distribution derived by (Shittu et al., 2012) with beta exponential distribution derived by (Nadarajah and Kotz, 2006) to assess their flexibilities in modeling real life data sets.