On Erlang-Truncated Exponential Distribution: Theory and Application

In this article, we introduce a new distribution called the McDonald Erlang-truncated exponential distribution. Various structural properties including explicit expressions for the moments, moment generating function, mean deviation of the new distribution are derived. The estimation of the model parameters is performed by maximum likelihood method. The usefulness of the new distribution is illustrated by two real data sets. The new model is much better than other important competitive models in modeling relief times and survival times data sets.


Introduction and motivation
The Erlang-truncated exponential distribution (ETEx) is widely used in the field of queuing system and stochastic processes. El-Alosey (2007) introduced Erlang-truncated exponential (ETEx) distribution by mixing the Erlang distribution with left truncated one parameter exponential distribution. The ETEx distribution, like the exponential distribution, has a constant failure rate which makes it practically impossible for the model to provide a reasonable parametric fit to data sets with decreasing failure rate, increasing failure rate and non-monotonic failure rate such as the bathtub and unimodal failure rates which are common in reliability studies and other related fields of studies. The cumulative distribution function (CDF) of this distribution is given by where > 0 is the scale parameter and > 0 is the shape parameter. When → ∞ , we have the standard exponential model. The corresponding probability density function (PDF) and hazard rate function (HRF) are given by and ℎ ETEx ( , ) ( ) = (1 − − ) respectively. As illustrated in (3), the HRF of the ETEx distribution is constant and this which makes it inadequate for modelling many complex lifetime data sets that have nonconstant failure rates. So, the main aim of this paper is to extend the ETEx distribution by adding three extra shape parameters to define a new flexible model referred to as the Mc-Donald Erlang-truncated exponential (McETEx) distribution. The role of the three additional parameters is to introduce skewness and to vary tail weights and provide greater flexibility in the shape of the generalized distribution and consequently in modeling observed data. It may be mentioned that although several skewed distribution functions exist on the positive real axis not many skewed distributions are available on the whole real line which are easy to use

Pakistan Journal of Statistics and Operation Research
for data analysis purpose. The main idea is to introduce three shape parameters, so that the McETEx distribution can be used to model skewed data, a feature which is very common in practice.
where > 0 , > 0 and > 0 are additional shape parameters. Note that ( ; ) is the PDF of baseline distribution, ( ; ) = ( ; )/ and is the paramrt vector. The class of distributions (1.4) includes as special sub-models the beta generalized (B-G) family of distributions for = 1 (see Eugene et al. (2002)) and Kumaraswamy generalized (K-G) family of distributions (see Cordeiro and Castro (2011)) for = 1. The corresponding CDF OF (4) is given by where denotes the incomplete beta function ratio (Gradshteyn and Ryzhik, (2000)). Equation (5) can also be rewritten as follows where is the well-known "hypergeometric function" which are well established in the literature (see Gradshteyn and Ryzhik (2000)). The HRF and reverse hazard functions (RHF) of the Mc-G distribution are given by and Mc-G Maximum likelihood estimation is performed in Section 5. In Section 6, we provide application to real data set to illustrate the importance of the new distribution.

The new distribution
Using (1) and (5) the CDF of (McETEx) can be written as the corresponding PDF is given by The new distribution will attract wider applications in reliability as well as in other areas of research and it can be used in a variety of problems in modeling survival data (see Section 7). Figure 1 represents some plots of the probability density function of the McETEx distribution for some different parameter values. Table 1 gives some sub models from the McETEx model.

Useful Expansions
In this Section, we present some representations of PDF of McETEx distribution. The mathematical relation given below will be useful in this subsection. The series representation given below will be useful in this subsection. If is a positive real non-integer and | | < 1 then Substituting from (11) into (9), we get again, using the binomial series expansion in (12) we get where

Statistical Properties Moments
The following theorems give the r ℎ moment ( ) and moment generating function ( ) of the distribution McETEx. Theorem (4.1): If has the McETEx, then the r ℎ moment of is given by the following is the upper incomplete gamma function, 1 1 [⋅,⋅,⋅] is a confluent hypergeometric function and ( , ) + ( , ) = ( ), The MRL has many applications in biomedical sciences, life insurance, maintenance and product quality control, economics and social studies, demography and product technology (see Lai and Xie, 2006). Guess  (2, ).

Bonferroni and Lorenz Curves
The Lorenz curve for a positive random variable is defined as , (2, ) (2, ), where = −1 ( ). If represents annual income, ( ) is the proportion of total income that accrues to individuals having the 100p% lowest incomes. If all individuals earn the same income, then ( ) = for all . The area between the line ( ) = and the Lorenz curve may be regarded as a measure of inequality of income. Also Bonferroni curve is defined by where the Bonferroni curve has many applications not only in economics to study income and poverty, but also in other fields like reliability, medicine and insurance. Also, we analogously discuss the reversed residual life and some of its properties. The reversed residual life can be defined as the conditional random variable − | ≤ which denotes the time elapsed from the failure of a component given that its life is less than or equal to . This random variable may also be called the inactivity time (or time since failure); for more details you may (see Kundu  Using ( ) and 2 ( ) one can obtain the variance and the coefficient of variation of the reversed residual life of the McETEx distribution.

Estimtion
In this section, we determine the maximum likelihood estimates (MLEs) of the parameters of the McETEx distribution from complete samples only. Let 1 , 2 , . . . , be a random sample of size from McETEx where = ( , , , , ) is the parameter vector. The log likelihood function for the vector of parameters can be written easily derived and the components of the score vector as well.

Application
In this Section, we provide two applications to a real data sets to assess the flexibility of the McETEx model. Figure  2 gives the total time test (TTT) plots (see Aarset (1987)), box plots and Quantile-Quantile (Q-Q) plots for the two real data sets. Based on Figure 2 we note that the HRF of the two data are "increasing", the first data contains an extreme value, the second data contains four extreme values. The kernel density estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable. KDE is a fundamental data smoothing problem where inferences about the population are made based on a finite data sample. Let ( 1 , 1 , … , ) be a univariate independent and identically distributed sample drawn from some distribution with an unknown density function. Then, kernel density estimator is where is the non-negative kernel function, and ℎ > 0 is a smoothing parameter called the bandwidth. Figure 3 gives the KDE plots (see Rosenblat (1956) and Parzen (1962)). In order to compare the McETEx model with other fitted distributions, we compare the fits of the McETEx distribution with the Exponential (Ex( )), Odd Lindley Exponential (OLiEx), Marshall-Olkin Exponential (MOEx ( , )), Moment Exponential (MomEx ( ) ), The Logarithmic Burr-Hatke Exponential (Log BrHEx( )), Generalized Marshall-Olkin Exponential (GMOEx ( , , )), Beta Exponential (BEx ( , , ) ), Burr X Ex ( , ), Marshall-Olkin Kumaraswamy Exponential (MOKEx ( , , , ) ), Kumaraswamy Exponential (KEx ( , , )) and Kumaraswamy Marshall-Olkin Exponential (KMOEx ( , , , )) models. The data represents the lifetime data relating to relief times (in minutes) of patients receiving an analgesic (see Gross and Clark (1975)  First data set Second data set First data set Second data set  Tables 2 and 4 gives the MLEs and SEsvalues for the two data sets. Table 3