A New Extension of the Exponential Distribution

The present paper considers an extension of the exponential distribution based on mixtures of positive distributions. We study the main properties of this new distribution, with special emphasis on its moments, moment generator function and some characteristics related to reliability studies. We also discuss parameter estimation considering the maximum likelihood and moments approach. An application reveals that the model proposed can be very useful in fitting real data. A final discussion concludes the paper.


Introduction
In lifetime data analysis it is usually the case that models with monotone risk functions are preferred as is the case of the gamma distribution.For some models there are no closed form risk functions (such as the Gamma model) and numerical integration might be required for its computation.In recent statistical literature modified extensions of the exponential distributions have been proposed to contour such difficulties.For example, Gupta & Kundu (1999) and Gupta & Kundu (2001) introduced an extension of the exponential distribution typically called the generalized exponential (GE) distribution.Therefore, it is said that the random variable X follows the GE distribution if its density function is given by g 1 (x; α, β) = αβe −αx (1 − e −αx ) β−1 where x > 0, α > 0 and β > 0. We use the notation X ∼ GE(α, β) for a random variable with such distribution.
More recently, Nadarajah & Haghighi (2011) introduced another extension of the exponential model, so that a random variable X follows the Nadarajah and Haghighi's exponential distribution (N HE) if its density function is given by where x > 0, α > 0 and β > 0. We use the notation X ∼ N HE(α, β).
Both distributions have the exponential distribution (E) with scale parameter α, as a special case when β = 1, that is, where x > 0, α > 0 with the notation X ∼ E(α).Other extensions of the exponential model in the survival analysis context are considered in the Marshall & Olkin's (2007) book.
The main object of this paper is to present yet another extension for the exponential distribution that can be used as an alternative to the ones mentioned above.We discuss some properties for this new distribution like its moments and moment generating function which can be used for parameter estimation as starting values for computing maximum likelihood estimators.
The paper is organized as follows.Section 2 delivers the density and distribution functions, moments, moment generating function, asymmetry and kurtosis coefficients and hazard function.Section 3 is devoted to parameter estimation based on maximum likelihood and moments approach.It is recommended that the moment estimators are used to initialize the maximum likelihood approach.In Section 4 an application to a real data set is presented and comparisons between the proposed model and other extensions of the exponential distribution are reported.The main conclusion is that the new model can perform well in applied situations.

Density and Properties
A random variable X is distributed according to the extended exponential distribution (EE) with parameters α and β if its density function is given by where x > 0, α > 0 and β ≥ 0. We use the notation X ∼ EE(α, β).
Figures 1 and 2 depict the behavior of the distribution for some parameter values.

Properties
Let X ∼ EE(α, β), Y ∼ E(α) and Z ∼ Gamma(2, α).Then, the distribution function for the random variable X is given by The expectation and variance are given by The moment generation function can also be obtained in closed form and is given by It also follows that its density can be obtained as a mixture of two positive ones, namely, Using the representation as a mixture of two positive densities, we can provide a general representation for the distribution moments, namely, where Γ(•) is the usual gamma function.
Using the moments above for the EE model, we can compute asymmetry ( √ β 1 ) and kurtosis (β 2 ) coefficients, which are given by Lemma 1.Note that as β → 0, then √ β 1 → 2 and β 2 → 9 which correspond to the asymmetry and kurtosis respectively for the exponential model.General coefficients of asymmetry and kurtosis are such that √ 2 < √ β 1 ≤ 2 and 6 < β 2 ≤ 9, respectively, as shown in Figures 3 and 4. The hazard function for the random variable X ∼ EE(α, β) is given by The Figure 5 illustrates the behavior of the hazard function for some parameter values.

Inferential Considerations
In this section, we consider inference for the EE using moments and the maximum likelihood approach.

Method of Moments
The moment estimators for the parameters α and β are obtained by solving From the first equation we obtain the moment estimators for β( β) as a function of the moment estimator for α( α).
using (9) and the second equation for the system given in (8) we obtain the moment estimator for α.
x 2 (10) Therefore, α from (10) replacing α in (9) we obtain β.These estimators will be used as initial parts to get the maximum likelihood estimation in the next section.

Maximum Likelihood
Let x 1 , x 2 , . . ., x n a random sample from X ∼ EE(α, β), so that we obtain the log-likelihood function Differentiating the log-likelihood function with respect to α and β, the following equations follow: From ( 12) we obtain and the maximum likelihood estimator for α is obtained by resolving numerically the following equation n i=1 The estimator α is the solution to the equation ( 15), and replacing it in (14) we obtain β.This algorithm leads to the maximum likelihood estimators for α and β.
Table 1 presents basic descriptive statistics for data set.We use the notation √ b 1 and b 2 to represent sample asymmetry and kurtosis coefficients.For comparing model fitting, Akaike (1974), namely where k is the number of parameters in the model under consideration.The AIC specifies that the model that best fits the data is the one with the smallest AIC value.
Table 2 shows parameter estimators for distributions GE, NHE and EE using maximum likelihood (MLE) approach and the corresponding Akaike information criterion (AIC).For these data, AIC shows a better fit for the EE model.Figure 6 reveals model fitting for the three models, and Figure 7 compares the distribution functions for the three models with the empirical distribution function.

Concluding Remarks
This paper introduces a new model positive data.It is shown that the model can be represented as the mixture of two distributions.The scale-exponential distribution can be seen as a particular case of the new model.It is shown that the distribution function, hazard function and moment generating function can be obtained in closed form.Moment estimators are derived and maximum likelihood estimators can be computed using Newton-Raphson type algorithms.The moment estimators can be used as starting values for the maximum likelihood estimators.Asymmetry and kurtosis coefficients are derived and their ranges are plotted.It is illustrated the fact that the model proposed has more flexibility in terms of coefficients of asymmetry and kurtosis.A real data application has demonstrated that the model studied is quite useful for dealing with real data and behaves better in terms of fitting than other models proposed in the literature such as the GE and NHE models.

Table 1 :
Descriptive statistics for rupture time.

Table 2 :
Parameter estimates for GE, NHE and EE models for the stress-rupture life data set.