Marshall-Olkin generalized Erlang-truncated exponential distribution: Properties and applications

Abstract: This article introduces the Marshall–Olkin generalized Erlang-truncated exponential (MOGETE) distribution as a generalization of the Erlang-truncated exponential (ETE) distribution. The hazard rate of the new distribution could be increasing, decreasing or constant. Explicit-closed form mathematical expressions of some of the statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimation was used to estimate the model parameters. The usefulness and flexibility of the new distribution was illustrated with two real and uncensored lifetime data-sets. The MOGETE distribution with a smaller goodness of fit statistics always emerged as a better candidate for the data-sets than the ETE, Exp Fréchet and Exp Burr XII distributions.


Introduction
The exponential distribution is about the simplest distribution in terms of expression and analytical tractability and widely used in reliability engineering. There is no doubt that the wide applicability of the exponential distribution even in inappropriate scenarios is motivated by its simplicity. However, ABOUT

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The simple structure of the proposed MOGETE distribution makes it easy to work with analytically and in practical situations, it provides a better fit to data-sets than some of the already existing distributions like the Erlang-truncated exponential distribution, Exponentiated Fréchet distribution and the Exponentiated Burr XII distribution. In addition to the example of possible applications of the new distribution as shown here, MOGETE distribution is suitable for modelling infant mortality rate and failure rate of some devices/equipments due to ageing.
the exponential distribution has a major problem of constant failure/hazard rate property which makes it inappropriate for modelling data-sets from various complex life phenomena that may exhibit increasing, decreasing or bathtub hazard rate characteristics. El-Alosey (2007) extended the standard one parameter exponential distribution to a two parameter Erlang-truncated exponential (ETE) distribution. The pdf f (x) of the ETE distribution is given by with cdf F (x) and hazard rate function (hrf ) h (x) where is the shape parameter while is the scale parameter. It is important to note that the ETE distribution has a constant hazard rate function.
The inability of the existing standard distributions to adequately model a variety of complex real data-sets; particularly, lifetime ones has stirred huge concern amongst distribution users and researchers alike and has summoned enormous research attention over the last two decades. Interestingly, tremendous research breakthroughs have been recorded by many researchers in their quest to the solution of the lack of fits limitation of the standard probability distributions. Among others is Marshall and Olkin (1997) who introduced the family of distributions that is known as the Marshall-Olkin extended/generalized distributions. The Marshall-Olkin's technique of adding an extra parameter to the original distribution has remarkably been known for its ability of producing more flexible and robust distributions that can represent a wide-ranging coverage of data-sets that emanates from a variety of complex phenomena. The Marshall-Olkin family of distributions can be obtained as follows, where Ḡ (x) and g(x) are the complementary cumulative density function (survival/reliability function) and density function corresponding to the baseline distribution (original distribution).
A lot of standard probability distributions have been generalized by various researchers using the Marshall-Olkin procedure. For example, Ristić and Kundu (2015) introduced the Marshall-Olkin generalized exponential distribution generalizing the exponentiated exponential distribution. Ghitany, Al-Hussaini, and Al-Jarallah (2005) introduced the Marshall-Olkin extended Weibull distribution as a generalization of the standard Weibull distribution. Ghitany (2005) introduced the Marshall-Olkin extended Pareto distribution as a generalization of the standard Pareto distribution. Ristić, Jose, and Ancy (2007) introduced the Marshall-Olkin extended gamma distribution as a generalization of the standard gamma distribution. Ghitany, Al-Awadhi, and Alkhalfan (2007) introduced the Marshall-Olkin extended Lomax distribution as a generalization of the standard Lomax distribution. Jose and Krishna (2011) introduced the Marshall-Olkin extended continuous uniform distribution as a generalization of the standard continuous uniform distribution. Al-Saiari, Baharith, and Mousa (2014) introduced the Marshall-Olkin extended Burr type XII distribution as a generalization of the standard Burr type XII distribution. Alizadeh et al. (2015) introduced the Marshall-Olkin extended Kumaraswamy distribution as a generalization of the standard Kumaraswamy distribution. Gui (2013) introduced the Marshall-Olkin extended log-logistic distribution as a generalization of the standard log-logistic distribution. Pogány, Saboor, and Provost (2015) introduced the Marshall-Olkin extended exponential Weibull distribution generalizing the exponential Weibull distribution. Jose (2011) gave a comprehensive review of the Marshall-Olkin family of distributions and their applications to reliability, time series and stress-strength analysis. For more extensive reviews of the Marshall-Olkin generalized family of distributions see, Nadarajah (2008) and Barreto-Souza, Lemonte, and Cordeiro (2013). Sandhya and Prasanth (2014) introduced the Marshall-Olkin extended discrete uniform distribution as a generalizion of the standard discrete uniform distribution; etc.
Motivated by the idea of additional parameter for extra flexibility to the distribution, we introduce the three-parameter Marshall-Olkin generalized Erlang-truncated exponential (MOGETE) distribution as a generalization of the standard two parameter ETE distribution. The importance of the new distribution is the ability of describing real data-sets with unimodal density as well as decreasing or increasing hazard rate function better than some already existing distributions as we show later. Hence, the MOGETE distribution has a superior fitting ability than the ETE distribution.
The remaining part of this article is organized as follows: Section 2 introduces the MOGETE distribution; Section 3 presents some reliability characteristics of the distribution such as the reliability function, hazard rate function and the mean residual life time; Section 4 presents some statistical properties of the new distribution such as the kth crude moment, mean, variance, coefficient of variation, skewness, kurtosis, moment generating function, pth quantile function, Rényi entropy measure of the new distribution and the distribution of order statistics of the distribution; Section 5 proposes the estimation of the distribution parameters through the method of maximum likelihood estimation; Section 6 presents the application of the new distribution to two real data sets; Section 7 presents the discussion of results and lastly and Section 8 is the conclusion of the study.

The MOGETE distribution
The cdf of the MOGETE distribution is given by with the corresponding pdf as where and are the shape parameters and is the scale parameter.
Proof By setting we have that ∀ x ≥ 0, the above equality is true ∀ 1 = 2 , 1 = 2 , and 1 = 2 . ✷ Theorem 2.2 If a random variable say X is distributed according to the MOGETE distribution then the shape of its pdf as x → ∞ is decreasing when ≤ 2 and unimodal when > 2.
Proof The first derivative of the pdf f ′ in Equation (7) is given by Setting f � = 0 gives the critical point x 0 at which the pdf is maximized. x 0 is the root of the equation which is given by this implies that as x → ∞ and > 2 there exists some x < x 0 such that f (x) > 0 and some x > x 0 such that f (x) < 0, hence; f(x) has a single mode at x 0 . Now, it makes sense to conclude that the pdf have decreasing shape as the only alternative shape when ≤ 2; since, f � (x 0 ) ≠ 0 and both conditions ( ≤ 2 and > 2) cannot be jointly satisfied in each case (monotonic decreasing and unimodality). ✷ The asymptotic behaviour of the pdf of the MOGETE distribution is f (0) = (1 − e − )∕ and f (∞) = 0.

Reliability analysis with the MOGETE distribution
In this section, we present some reliability characteristics of the MOGETE distribution that is necessary for reliability analysis, they are: the reliability (survival) function F (x) (R(x)), the hazard rate function h(x) and the mean residual life time M(t).

Reliability function
The reliability function R(x) is an important tool in reliability analysis for characterizing life phenomenon. R(x) is mathematically expressed as R(x) = 1 − F(x). Under certain predefined conditions the reliability function generally gives the estimated probability that a system will operate without failure until a specified time x. The reliability function of the MOGETE distribution is given by For various parameter values R(x) is generally a decreasing function of x and the asymptotic behaviour of the reliability function of the MOGETE distribution is R(0) = 1; and R(∞) = 0.

Hazard rate function
The hazard rate function (hrf) gives the probability of failure for a system that has survived up-to time x. It is mathematically expressed as h(x) = f (x)∕R(x). The hazard rate function of the MOGETE distribution is given by Theorem 3.1 The shape of the hrf of the MOGETE distribution is constant (a special case of the ETE distribution when = 1), decreasing when < 1 and increasing when > 1.
Proof The first derivative of the hrf h ′ in Equation (9) is given by It is easy to see that h � = 0 has no unique root; h � < 0, ∀ < 1 (i.e. the hrf is decreasing) and h � > 0, ∀ > 1 (i.e. the hrf is increasing) and when = 1, h � = 0 (i.e. the hrf is constant). ✷ The asymptotic behaviour of the hrf of the MOGETE distribution is

The mean residual life time
Theorem 3.2 The remaining lifetime of a system that has survived up-to time t is random, as a result the failure time cannot be determined. The expected value of the random failure times is referred to as the mean residual lifetime denoted by M(t). M(t) only exists for F (t) > 0 and it is mathematical expressed as The mean residual lifetime of the MOGETE distribution is given by

Some statistical properties of the MOGETE distribution
Application of any distribution can only be possible if its basic distributional properties are available. In this section, we present explicit derivations of some important distributional properties of the MOGETE distribution.

The pth quantile function of the MOGETE distribution
The pth quantile function of the MOGETE distribution is given by Random variables can be simulated from the MOGETE distribution through the method of the inversion of cdf by simply substituting p in Equation (12) with a U(0, 1) variates. Also, the median of the  Bowley (1901Bowley ( -1920 and Moors kurtosis denoted by M due to Moors (1986) depends on the quantile function.
The Bowley skewness is given by and, the Moors kurtosis is given by

The kth crude moment of the MOGETE distribution
Theorem 4.1 If the kth crude moment of any random variable X exists then other essential characteristics of the distribution could be derived from it, such as the mean, variance, coefficient of variation, skewness and kurtosis statistics. The kth crude moment of any continuous random variable X is generally given by E( Hence, it follows that the kth crude moment of the MOGETE distribution is given by

✷
The mean is the first-order crude moment of the distribution and could be obtained by evaluating Equation (15) at k = 1 as While evaluating Equation (15) at k = 2 gives the second-order crude moment of the MOGETE distribution as The variance V(X) could be obtained by substituting E(X) and E(X 2 ) into the following expression V(X) = E(X 2 ) − {E(X)} 2 . Hence, the variance of the MOGETE distribution is given by Setting E(X k ) = � k the coefficient of variation (CV), skewness ( 1 ) and kurtosis ( 2 ) statistics of the MOGETE distribution could be obtained as follows

The kth central moment of the MOGETE distribution
Theorem 4.2 The kth central moment of a continuous random variable X is given by Hence, the kth central moment of the MOGETE distribution is given by Proof Substituting y = (1 − e − )x into Equation (16) gives Substituting z = y(j + 1) into Equation (17) gives

The moment generating function of the MOGETE distribution
Recently, a lot of advancement both in theory and application has been achieved in statistics and probability through the moment generating function (mgf) of a random variable X. The usefulness of the mgf has been found to surpass the very trivial derivation of distributional order moments. For instance; Villa and Escobar (2006) obtained mixture distributions with mgf, Meintanis (2010) used the mgf for testing skew normality, McLeish (2014) performed simulation of random variables using the mgf and the saddle point approximation, von Waldenfels (1987) gave a proof of an algebraic central limit theorem using the mgf, and Inlow (2010) also, proved the Lindeberg-Lévy's central limit theorem with the mgf. The moment generating function is generally defined by It follows from Equations (15) and (18) that the mgf of the MOGETE distribution is given by

Rényi entropy of the MOGETE distribution
The Rényi entropy denoted by H (x) is used to quantify the uncertainty of variation in a random variable X. The limiting value of H (x) as → 1 is the Shannon entropy. Song (2001) (1 − ) j + j http://dx.doi.org/10.1080/23311835.2017.1285093

Order statistics of the MOGETE random variable
The distribution of the rth order statistics denoted by f X (r) (x) of an n sized random sample X 1 , X 2 , X 3 , … , X n is generally given by The density of the rth order statistics of the MOGETE distribution could be obtained by substituting Equations (6) and (7) into Equation (20) as The density of the rth smallest order statistics of the MOGETE distribution is given by The density of the rth largest order statistics of the MOGETE distribution is given by

Estimation
Here, we propose to estimate the parameters of the MOGETE distribution by the method of Maximum likelihood estimation.

Maximum likelihood estimation method
Suppose the random sample x 1 , x 2 , x 3 , … , x n of size n is drawn from a probability distribution with pdf f(x) then the maximum likelihood estimates (mle) of its parameters could be obtained as follows: The likelihood () equation is given by and the log-likelihood ( ) equation is given by then; taking the partial derivatives of Equation (22) w.r.t to ; and and equating to zero gives: The analytical solution to the system of nonlinear equations in Equations (23), (24) and (25) does not exist thus, we require some nonlinear numerical optimization methods such as the Newton Raphson technique to solve the equations. Let = (̂,̂,̂) � . Under some standard regularity conditions, √ n(̂ − ) is asymptotically multivariate normal distributed 3 ( , −1 n ( )), where n ( ) is the expected information matrix defined by ( 2 ( )∕ � ). The asymptotic behaviour of the expected information matrix can be approximated by the observed information matrix, denoted by n (̂ ). Generally speaking, the diagonal elements of −1 n (̂ ) gives the variance of (̂ ) while the off-diagonal elements is the covariances. The observed information matrix of the MOGETE distribution is expressed as where the corresponding elements are:

Simulation study
One major problem of extended probability distributions is parameter estimation. In this section, we present a Monte Carlo simulation study to evaluate the performance of the mle method in estimating the parameters of the distribution by drawing different samples (n = 50, 100, … , 300) from the MOGETE distribution with selected parameter values. Estimation of the parameters was carried out with the simulated random variables through the mle method to investigate the stability of the parameters and sample size effect on the estimates via bias, standard error (se), and mean square error (mse). Application of the following algorithm in  (Statistical software) provides us with the results in Table 1.

Algorithm
(i) Simulate u i ∼ Uniform(0, 1), for i = 1, 2, 3, … , n(50, 100, … , 300); (17) evaluated at U i for some parameter values (see Table 1) and X ~ MOGETE distribution; (iii) Using x and the nlm function under the stats package in , calculate the mle estimates of the parameters of the MOGETE distribution; (iv) Repeat steps (i-iii) in 5,000 (N) iterations; (v) For each n and parameter, compute the mean (parameter estimate), standard deviation (standard error), bias and mse of the sequence of 5,000 parameter estimates.
) decreases with increasing n, where = ( , , ) � . These results suggest that the mle method does well in estimating the parameters of the MOGETE distribution.

Applications
This section illustrates the applicability and flexibility of the MOGETE distribution with two real datasets. The goodness of fit of the new lifetime distribution would be assessed by a comparison of its performance in modelling the real data-sets with that of its competing sub-model (ETE distribution) and the following three-parameter distributions: • Exponentiated Burr XII • Exponentiated Fréchet based on the Akaike information criterion (AIC) statistic, Akaike (1981), the AIC with a correction statistic (AICc), Sugiura (1978), where ̂ , k, and n corresponds to the estimate of the model maximized/minimized log-likelihood function, number of model parameters and sample size, respectively. The Chen and Balakrishnan (1995) W ⋆ and A ⋆ goodness of fit measures were also considered. See Oluyede, Foya, Warahena-Liyanage, and Huang (2016) for detail on the computational steps of the W ⋆ and A ⋆ statistics. The distribution with the smallest goodness-of-fit measure is the best. Table 2 gives the waiting times in minutes of 100 bank customers in a queue before service. The data-set was first published in Ghitany, Atieh, and Nadarajah (2008). Merovci and Elbatal (2013) and Bhati, Malik, and Vaman (2015) have also fitted the data to different distributions. The results we obtained from the data fitting are tabulated in Table 3.
, c, k > 0, and The variance-covariance matrix of the MOGETE distribution under the fitted 100 bank customers waiting times data is given by The second example is on the annual maximum daily precipitation in millimetre that was recorded in Basan, Korea, from 1904 to 2011. The data are presented in Table 4. The data-set has been analysed by Jeong, Murshed, Am Seo, and Park (2014) and was recently reported in Mansoor et al. (2016). Results from the data fitting to the distributions are presented in Table 5.   The variance-covariance matrix of the MOGETE distribution under the fitted Rainfall data is given by

Discussion of results
The density plots in Figure 2 (left panel) depict some monotonic decreasing function of x for ≤ 2 and for α > 2 the distribution is unimodal, while the cdf plot (right panel) shows some monotonic increasing curves for all < 1 . The plots in Figure 2 indicate that the reliability function (left panel) is a monotonically decreasing function of x for all while the hazard rate function (right panel) could increasing (if < 1), decreasing (if < 1), or constant (if = 1), these characteristics make it more reasonable for analysing complex lifetime data-sets. The results in Tables 3 and 5 show that the MOGETE distribution with smaller minimized log-likelihood value and smaller information statistics provides better fit to the data-sets than the ETE and the other competing distributions. Also, the P-P plots in Figures 4 and 5 does not raise any alarm against the suggestion of the AIC, AICc, W* and A ⋆ statistics.

Conclusions
This article introduces a new lifetime distribution-the (MOGETE) distribution. The new distribution generalizes the ETE distribution and has the ETE distribution as a sub-model. We have given explicit mathematical expressions for some of its basic statistical properties such as the probability density function, cumulative density function, kth raw moment,kth central moment, mean, variance, coefficient of variation, skewness, kurtosis, moment generating function, pth quantile function, the rth order statistics and the Rényi's entropy measure. Also, some of its reliability characteristics like the     reliability function, hazard rate function and the mean residual life time was given. Estimation of the model parameters was approached through the method of maximum likelihood estimation. The applicability, flexibility and robustness of the new lifetime distribution was demonstrated with the 100 bank customers waiting times data and 105 Rainfall data, and the results obtained show that the MOGETE distribution provides a more reasonable fit than the ETE, Exp Fréchet and Exp Burr XII distributions. We hope that the MOGETE distribution would receive a high rate of application, particularly, because of its hazard rate characteristics.