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

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1 STATISTICS RESEARCH ARTICLE Marshall-Olkin generalized Erlang-truncated exponential distribution: Properties and applications Idika E. Okorie 1 *, Anthony C. Akpanta 2 and Johnson Ohakwe 3 Received: 8 November 216 Accepted: 15 January 217 First Published: 21 January 217 *Corresponding author: Idika E. Okorie, School of Mathematics, University of Manchester, Manchester M13 9PL, UK idika.okorie@manchester.ac.uk Reviewing editor: Hiroshi Shiraishi, Keio University, Japan Additional information is available at the end of the article 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. Subjects: Science; Mathematics & Statistics; Physical Sciences Keywords: Erlang-truncated exponential distribution; Marshall-Olkin; reliability; failure rate; AIC 1. 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 THE AUTHORS Idika E. Okorie obtained his BSc in Statistics (29 from the Abia State University, Nigeria and MSc in Statistics (Financial Statistics (214 from the University of Manchester, UK. He is presently studying for a PhD in Statistics at the University of Manchester, UK. His research interest includes Statistical modelling; and Distribution theory with applications. Anthony C. Akpanta obtained his PhD in Statistics (28 from the Abia State University, Nigeria. He is currently working as an associate professor of Statistics at the Department of Statistics, Abia State University, Nigeria. His area of research includes Time series analysis and forecasting; Sampling methods and distribution theory. Johnson Ohakwe obtained his PhD in Statistics (29, MSc Statistics (25 and BSc Statistics (1998 from the Abia State University, Nigeria. He is currently a Senior lecturer at the Department of Mathematics, Computing and Physical sciences, Federal University Otuoke, Nigeria. His research interest include Time series and Forecasting and Distribution theory. PUBLIC INTEREST STATEMENT 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. 217 The Author(s. This open access article is distributed under a Creative Commons Attribution (CC-BY 4. license. Page 1 of 19

2 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 (27 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 f (x β(1 e λ e β(1 e λ x ; with cdf F(x x <, β, λ>, (1 F(x 1 e β(1 e λ x ; x <, β, λ>, (2 and hazard rate function (hrf h(x h(x β(1 e λ ; β, λ>, (3 where β is the shape parameter while λ 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, F(x Ḡ(x ; < x < ; << 1 (1 Ḡ(x (4 It follows that F(x 1 F(x and f (x g(x ( 1 (1 Ḡ(x ; < x < ; << 2 (5 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 (215 introduced the Marshall Olkin generalized exponential distribution generalizing the exponentiated exponential distribution. Ghitany, Al-Hussaini, and Al-Jarallah (25 introduced the Marshall Olkin extended Weibull distribution as a generalization of the standard Weibull distribution. Ghitany (25 introduced the Marshall Olkin extended Pareto distribution as a generalization of the standard Pareto distribution. Ristić, Jose, and Ancy (27 introduced the Marshall Olkin extended gamma distribution as a generalization of the standard gamma distribution. Ghitany, Al-Awadhi, and Alkhalfan (27 introduced the Marshall Olkin extended Lomax distribution as a generalization of the standard Lomax distribution. Jose and Krishna (211 introduced the Marshall Olkin extended continuous uniform distribution as a generalization of the standard continuous uniform distribution. Al-Saiari, Baharith, and Mousa (214 introduced the Marshall Olkin extended Burr type XII distribution as a generalization of the standard Burr type XII distribution. Alizadeh et al. (215 introduced the Marshall Olkin extended Page 2 of 19

3 Kumaraswamy distribution as a generalization of the standard Kumaraswamy distribution. Gui (213 introduced the Marshall Olkin extended log-logistic distribution as a generalization of the standard log-logistic distribution. Pogány, Saboor, and Provost (215 introduced the Marshall Olkin extended exponential Weibull distribution generalizing the exponential Weibull distribution. Jose (211 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 (28 and Barreto-Souza, Lemonte, and Cordeiro (213. Sandhya and Prasanth (214 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. 2. The MOGETE distribution The cdf of the MOGETE distribution is given by F(x 1 F(x 1 e β(1 e λ x ; x <, >, β, λ> 1 (1 e β(1 e λ x with the corresponding pdf as { f (x β(1 e λ e β(1 e λ x 1 (1 e x} 2; β(1 e λ x <, >, β, λ> (6 (7 where and β are the shape parameters and λ is the scale parameter. Theorem 2.1 The MOGETE distribution with pdf in Equation (7 is identifiable i.e. f (x; 1, β 1, λ 1 f (x; 2, β 2, λ 2 as we show now. Proof By setting f (x; 1, β 1, λ 1 f (x; 2, β 2, λ 2 we have that log( log(β 1 β 2 +λ 1 λ 2 3β 1 (1 e λ 1 β2 (1 e λ 2 ]x x, the above equality is true 1 2, β 1 β 2, and λ 1 λ 2. Page 3 of 19

4 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 f β2 (1 e λ 2 e β(1 e λ x 2(1 β 2 (1 e λ 2 e 2β(1 e λ x <. (1 (1 e β(1 e λ x 2 (1 (1 e β(1 e λ x 3 Setting f gives the critical point x at which the pdf is maximized. x is the root of the equation which is given by x log( 1 β(1 e λ ; >2, this implies that as x and >2 there exists some x < x such that f (x > and some x > x such that f (x <, hence; f(x has a single mode at x. Now, it makes sense to conclude that the pdf have decreasing shape as the only alternative shape when 2; since, f (x 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 ( β(1 e λ and f (. 3. 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 e β(1 e λ x R(x ; x <, >, β, λ> 1 (1 e β(1 e λ x (8 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( 1; and R( 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 β(1 e λ h(x ; x <, >, β, λ>. 1 (1 e β(1 e λ x (9 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 h (1 β2 (1 e λ 2 e β(1 e λ x (1 (1 e β(1 e λ x 2. Page 4 of 19

5 It is easy to see that h has no unique root; h <, <1 (i.e. the hrf is decreasing and h >, >1 (i.e. the hrf is increasing and when 1, h (i.e. the hrf is constant. The asymptotic behaviour of the hrf of the MOGETE distribution is h( β(1 e λ and h( β(1 e λ 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 > and it is mathematical expressed as M(t E(X t X > t 1 R(xdx R(t t The mean residual lifetime of the MOGETE distribution is given by M(t ( 1 2j (1 j e β(1 e λ t](j+1. R(tβ(1 e λ j + 1 (1 Proof M(t 1 R(t e β(1 e λ x t 1 (1 e β(1 e λ x] 1dx Substituting y β(1 e λ x into Equation (11, we have M(t R(tβ(1 e λ R(tβ(1 e λ R(tβ(1 e λ R(tβ(1 e λ β(1 e λ t e y β(1 e λ t e y 1 (1 e y] 1 dy ( 1 2j (1 j ( 1 j ( j j e y(j+1 dy β(1 e λ t λ t](j+1 ( 1 2j (1 j e β(1 e j + 1. ( 1 j (1 j e yj dy (11 4. 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 ] 1 π(p β(1 e λ log 1 p 1 p(1 (12 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(, 1 variates. Also, the median of the Page 5 of 19

6 Figure 1. Plots of the Bowley skewness (left panel and Moors kurtosis (right panel of the MOGETE distribution. MOGETE distribution could be obtained from Equation (12 by setting p 1 2. The Bowley skewness denoted by B due to Bowley ( and Moors kurtosis denoted by M due to Moors (1986 depends on the quantile function. The Bowley skewness is given by B π(3 4+π(1 4 2π(2 4, π(3 4 π(1 4 and, the Moors kurtosis is given by M π(3 8 π(1 8+π(7 8 π(5 8. π(6 8 π(2 8 Figure 1 illustrates the variability of both the Bowley skewness and Moors kurtosis on (shape parameter for the MOGETE distribution. The Bowley skewness and Moors kurtosis shrinks with increasing value of 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(X k xk f (xdx. Hence, it follows that the kth crude moment of the MOGETE distribution is given by E(X k Γ(k + 1 β(1 e λ ] k ( 1 2j (1 j (j + 1 k. Proof E(X k x k β(1 e λ e β(1 e λ x 1 (1 e x] 2dx β(1 e λ β(1 e λ x k e β(1 e λ x 1 (1 e x] 2dx β(1 e λ (13 Page 6 of 19

7 Substituting y β(1 e λ x into Equation (13 gives β(1 e λ ] y k e y 1 (1 e y] 2 k dy β(1 e λ ] y k e y k β(1 e λ ] k ( 1 j ( j j ( 1 2j (j + 1(1 j y k e y(j+1 dy ( 1 j (1 j e yj dy (14 Substituting z y(j + 1 into Equation (14 gives β(1 e λ ] k Γ(k + 1 β(1 e λ ] k ( 1 2j (1 j (j + 1 k ( 1 2j (1 j (j + 1 k. z k e z dz (15 The mean is the first-order crude moment of the distribution and could be obtained by evaluating Equation (15 at k 1 as E(X β(1 e λ ( 1 2j (1 j j + 1. While evaluating Equation (15 at k 2 gives the second-order crude moment of the MOGETE distribution as E(X 2 2 β(1 e λ ] 2 ( 1 2j (1 j (j 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 V(X 2 β(1 e λ ] 2 ( 1 2j (1 j (j + 1 ( 1 2j (1 j 2 β(1 e λ j + 1 Setting E(X k μ k the coefficient of variation (CV, skewness (γ 1 and kurtosis (γ 2 statistics of the MOGETE distribution could be obtained as follows CV μ 2 1 μ 2 1 γ 1 μ 3 3μ 2 μ + 1 2μ 3 1 (μ 2 μ γ 2 μ 4 4μ 3 μ 1 + 6μ 2 μ 2 1 3μ 4 1 (μ 2 μ ] 2 Page 7 of 19

8 4.3. The kth central moment of the MOGETE distribution Theorem 4.2 The kth central moment of a continuous random variable X is given by E((X μ k (x μk f (xdx. Hence, the kth central moment of the MOGETE distribution is given by E(X μ k ] Proof β(1 e λ ] k i ( 1 i+2j Γ(k + 1 μβ(1 e λ ] i (1 j (j + 2Γ(k i + 1 Γ(i + 1Γ(k i + 1(j + 1 k i+1. E(X μ k ] (x μ k β(1 e λ e β(1 e λ x 1 (1 e x] 2dx β(1 e λ β(1 e λ (x μ k e β(1 e λ x 1 (1 e β(1 e λ x] 2dx (16 Substituting y β(1 e λ x into Equation (16 gives E(X μ k ] β(1 e λ ] e y y μβ(1 e λ ] k k 1 (1 e y] 2 dy ( k β(1 e λ ] e y ( 1 i μβ(1 e λ ] i k y k i i i ( j ( 1 j ( 1 j (1 j e yj dy j β(1 e λ ] k y k i e y(j+1 dy Substituting z y(j + 1 into Equation (17 gives ( 1 i+2j μβ(1 e λ ] ( i k (1 j (j + 2 i i (17 E(X μ k ] β(1 e λ ] k z k i e z dz β(1 e λ ] k i ( 1 i+2j( k i ( 1 i+2j i μβ(1 e λ ] i (1 j (j + 2 (j + 1 k i+1 Γ(k + 1 μβ(1 e λ ] i (1 j (j + 2Γ(k i + 1 Γ(i + 1Γ(k i + 1(j + 1 k i 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 Page 8 of 19

9 instance; Villa and Escobar (26 obtained mixture distributions with mgf, Meintanis (21 used the mgf for testing skew normality, McLeish (214 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 (21 also, proved the Lindeberg-Lévy s central limit theorem with the mgf. The moment generating function is generally defined by ] M X (t E(e tx (tx k t k E k! k! E(xk. k k It follows from Equations (15 and (18 that the mgf of the MOGETE distribution is given by (18 M X (t k ( 1 2j t k Γ(k + 1 k! β(1 e λ ] k (1 j (j + 1 k Rényi entropy of the MOGETE distribution The Rényi entropy denoted by H δ (x is used to quantify the uncertainty of variation in a random variable X. The limiting value of H δ (x as δ 1 is the Shannon entropy. Song (21 compared tails and shapes behaviour of some standard probability distributions using the Rényi entropy. The Rényi entropy measure is generally given by H δ (x lim n (I ρ (f n log(n 1 1 ρ log f ρ (xdx 1 1 ρ log(i ρ. Theorem 4.3 If X follows the MOGETE distribution, then its Rényi entropy measure is given by ( ( ] H ρ (x 1 β 1 e λ ρ 1 ρ log β(1 e λ Proof ( 1 2j Γ(j + 2ρ Γ(j + 1Γ(2ρ (1 j. ρ + j I ρ β ( 1 e λ ( 2 ] ρ e β(1 e λ x 1 (1 e x β(1 e λ dx β ( 1 e λ] ρ ( 2ρdx e ρβ(1 e λ x 1 (1 e x β(1 e λ (19 Substituting y β(1 e λ x into Equation (19 gives I ρ β ( 1 e λ ] ρ β(1 e λ β ( 1 e λ ] ρ β(1 e λ β ( 1 e λ ] ρ β(1 e λ e ρy( 1 (1 e y 2ρ dy e ρy ( 1 j ( j + 2ρ 1 j ( 1 2j ( j + 2ρ 1 j ( 1 j (1 j e yj dy (1 j e y(ρ+j dy (1 j ( ] β 1 e λ ρ ( j + 2ρ 1 ( 1 2j β(1 e λ j ρ + j ( ( ] H δ (x 1 β 1 e λ ρ 1 ρ log ( 1 2j Γ(j + 2ρ β(1 e λ Γ(j + 1Γ(2ρ (1 j ρ + j Page 9 of 19

10 4.6. 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 f X(r (x n! (r 1!(n r! (F x (xr 1 (1 F x (x n r f x (x. (2 The density of the rth order statistics of the MOGETE distribution could be obtained by substituting Equations (6 and (7 into Equation (2 as n!β ( 1 e λ ] 2 e β(1 e λ x 1 (1 e β(1 e λ x f X(k (x (k 1!(n k! ] k 1 1 e β(1 e λ x 1 (1 e β(1 e λ x e β(1 e λ x 1 (1 e β(1 e λ x ] n k The density of the rth smallest order statistics of the MOGETE distribution is given by f X(1 (x n!β ( 1 e λ e β(1 e λ x ] 2 1 (1 e β(1 e λ x e β(1 e λ x 1 (1 e β(1 e λ x ] n 1 The density of the rth largest order statistics of the MOGETE distribution is given by f X(n (x n!β ( 1 e λ e β(1 e λ x ] 2 1 (1 e β(1 e λ x 1 e β(1 e λ x 1 (1 e β(1 e λ x 5. 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: ] n 1 The likelihood ( equation is given by n i1 β(1 e λ e β(1 e λ x i { 1 (1 e β(1 e λ x i } 2 β(1 e λ ] n e β(1 e λ and the log-likelihood (l equation is given by n x i i1 n } 2 {1 (1 e β(1 e λ x i i1 l n log ( β(1 e λ n n } β(1 e λ x i 2 log {1 (1 e β(1 e λ x i i1 i1 (21 (22 then; taking the partial derivatives of Equation (22 w.r.t to ; β and λ and equating to zero gives: Page 1 of 19

11 l n n 2 e β(1 e λ x i 1 (1 e β(1 e λ x i i1 l β n β (1 e λ l λ ne λ n n x i 2 i1 βe λ λ 1 e i1 i1 n n x i 2 i1 (1 (1 e λ x i e β(1 e λ x i 1 (1 e β(1 e λ x i (1 βx i e λ β(1 e λ x i 1 (1 e β(1 e λ x i (23 (24 (25 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 Ω (, β, λ. Under some standard regularity conditions, n( Ω Ω is asymptotically multivariate normal distributed N 3 (, J 1 n (Ω, where J n (Ω is the expected information matrix defined by E( 2 l(ω Ω Ω. The asymptotic behaviour of the expected information matrix can be approximated by the observed information matrix, denoted by I n ( Ω. Generally speaking, the diagonal elements of I 1 ( n Ω gives the variance of ( Ω while the off-diagonal elements is the covariances. The observed information matrix of the MOGETE distribution is expressed as I n ( Ω 2 l(ω 2 2 l(ω β 2 l(ω λ 2 l(ω β 2 l(ω β 2 2 l(ω λ β 2 l(ω λ 2 l(ω β λ 2 l(ω λ 2. where the corresponding elements are: 2 l n n e 2β(1 e λ x i, 2 1 (1 e β(1 e λ x i ] 2 2 l β 2 n β i1 n (1 (1 e λ 2 x 2 x i i e β(1 e λ i1 1 (1 e β(1 e λ x i n (1 2 (1 e λ 2 x 2 x i i + 2 e 2β(1 e λ, i1 (1 (1 e β(1 e λ x i 2 2 l λ λ ne (1 e λ +e 2λ ] n + βe λ x 2 (1 e λ 2 i i1 n (1 βxi e λ β(1 e λ x i( 1 βxi e λ 2 i1 1 (1 e β(1 e λ x i (1 ] 2 β 2 x 2 λ i e 2(λ+(1 e x i, (1 (1 e β(1 e λ x i 2 2 l n β 2 (1 e λ x i e β(1 e λ x i i1 1 (1 e β(1 e λ x i n (1 (1 e λ x + 2 i e 2β(1 e λ x i, i1 1 (1 e β(1 e λ x i] 2 2 l n λ 2 βx i e λ β(1 e λ x i 1 (1 e β(1 e λ x i + 2 n i1 i1 (1 βx i e λ 2β(1 e λ x i 1 (1 e β(1 e λ x i] 2, Page 11 of 19

12 and 2 l n n (1 x β λ e λ x i 2 i e λ β(1 e λ x i i1 i1 1 (1 e β(1 e λ x i n (1 βx 2 x i i + 2 e λ β(1 e λ i1 1 (1 e β(1 e λ x i n (1 2 β(1 e λ x 2 x i i + 2 e λ 2β(1 e λ 1 (1 e β(1 e λ x i] 2 i1 Given that n( Ω Ω N 3 (, I 1 ( n Ω, we can perform statistical inference for functions of Ω. For instance, the approximate 1(1 ε% two-sided confidence interval of the model parameters Ω could be calculated as: Ω ± Z ε 2 where I 1 I 1 2 l(ω Ω 2 l(ω Ω (, are the diagonal entries of the observed information matrix, and Z ε is the upper 2 ε 2th percentile of the standard normal distribution 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 5, 1,, 3 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 Algorithm (, (i Simulate u i Uniform(, 1, for i 1, 2, 3,, n(5, 1,, 3; (ii Set x i F 1 (u i, where F 1 ( is the quantile function in Equation (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, (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, parameter estimates. Page 12 of 19

13 Table 1. Simulation results n β λ se se β se bias λ bias bias mse β λ mse β mse λ 5., β 5., λ , β 3.5, λ , β 2.3, λ , β.5, λ , β 2., λ Page 13 of 19

14 Whatever parameter values we chose in Table 1, the mle estimates ( Ω n 1 N Ω i approximates ( i1 to the actual value as n becomes large and the standard error se Ω N ( i1 Ω i Ω 2 (N 1, bias (bias Ω i Ω; i 1,, n and mse (mse 1 N N i1 Ω i Ω] 2 decreases with increasing n, where Ω (, β, λ. These results suggest that the mle method does well in estimating the parameters of the MOGETE distribution. 6. 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 f (x kcx c 1 (1 + x c k 1 1 (1 + x c k ] 1,, c, k >, and Exponentiated Fréchet f (x β ( x β 1e ( x β s,, β, s >, s s based on the Akaike information criterion (AIC statistic, Akaike (1981, AIC 2 l + 2k the AIC with a correction statistic (AICc, Sugiura (1978, AICc AIC + 2k(k + 1 n k 1 where l, 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 (216 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 1 bank customers in a queue before service. The data-set was first published in Ghitany, Atieh, and Nadarajah (28. Merovci and Elbatal (213 and Bhati, Malik, and Vaman (215 have also fitted the data to different distributions. The results we obtained from the data fitting are tabulated in Table 3. Table 2. 1 bank customers waiting times (min before service Page 14 of 19

15 Table 3. Results from modelling the 1 bank customers waiting times data Models Estimatess s.e l AIC AICc W A MOGETE β λ ETE β λ Exp Fréchet e+3 β e ŝ e+2 Exp Burr XII k ĉ The variance-covariance matrix of the MOGETE distribution under the fitted 1 bank customers waiting times data is given by I 1 ( n Ω The second example is on the annual maximum daily precipitation in millimetre that was recorded in Basan, Korea, from 194 to 211. The data are presented in Table 4. The data-set has been analysed by Jeong, Murshed, Am Seo, and Park (214 and was recently reported in Mansoor et al. (216. Results from the data fitting to the distributions are presented in Table 5. Table 4. Rainfall data Page 15 of 19

16 Table 5. Results from modelling the rainfall data Models Estimates s.e l AIC AICc W A MOGETE β λ ETE β λ Exp Fréchet β ŝ Exp Burr XII k ĉ The variance covariance matrix of the MOGETE distribution under the fitted Rainfall data is given by I 1 ( n Ω Discussion of results The density plots in Figure 2 (left panel depict some monotonic decreasing function of x for 2 and for > 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. 8. 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 Page 16 of 19

17 Figure 2. Possible shapes of the probability density function f (x(left and cumulative distribution function F(x(right of the MOGETE distribution for fixed parameter values of β and λ and selected values of parameter. Figure 3. Possible shapes of the survival/reliability function F(x(left and hazard rate function h(x(right of the MOGETE distribution for fixed parameter values of β and λ and selected values of parameter. Figure 4. Probability-Probability (P-P plots of the fitted distributions with the waiting times data. Page 17 of 19

18 Figure 5. Probability-Probability (P-P plots of the fitted distributions with the Rainfall data. 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 1 bank customers waiting times data and 15 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. Funding This work was supported by the University of Manchester, UK. Author details Idika E. Okorie 1 idika.okorie@manchester.ac.uk ORCID ID: Anthony C. Akpanta 2 ac_akpa@yahoo.com ORCID ID: Johnson Ohakwe 3 ohakwejj@fuotuoke.edu.ng ORCID ID: 1 School of Mathematics, University of Manchester, Manchester M13 9PL, UK. 2 Department of Statistics, Abia State University, Uturu, Abia State, Nigeria. 3 Faculty of Sciences, Department of Mathematics & Statistics, Federal University Otuoke, P.M.B 126 Yenagoa, Bayelsa, Bayelsa State, Nigeria. Citation information Cite this article as: Marshall-Olkin generalized Erlangtruncated exponential distribution: Properties and applications, Idika E. Okorie, Anthony C. Akpanta & Johnson Ohakwe, Cogent Mathematics (217, 4: References Akaike, H. (1981. Likelihood of a model and information criteria. Journal of Econometrics. 16, Alizadeh, M., Tahir, M. H., Cordeiro, G. M., Mansoor, M., Zubair, M., & Hamedani, G. G. (215. The Kumaraswamy Marshal- Olkin family of distributions. Journal of the Egyptian Mathematical Society, 23, Al-Saiari, A. Y., Baharith, L. A., & Mousa, S. A. (214. Marshall- Olkin extended Burr type XII distribution. International Journal of Statistics and Probability, 3, 78. Barreto-Souza, W., Lemonte, A. J., & Cordeiro, G. M. (213. General results for the Marshall and Olkin s family of distributions. Anais da Academia Brasileira de Cincias, 85, Bhati, D., Malik, M. A., & Vaman, H. J. (215. Lindley-Exponential distribution: Properties and applications. Metron, 73, Bowley, A. L. ( Elements of statistics (4th ed.. New York, NY: Charles Scribner. Chen, G., & Balakrishnan, N. (1995. A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, El-Alosey, A. R. (27. Random sum of new type Of mixture Of distribution. International Journal Of Statistics and Systems, 2, Ghitany, M. E. (25. Marshall--Olkin extended Pareto and its application. International Journal of Applied Mathematics, 18, Ghitany, M. E., Al-Awadhi, F. A., & Alkhalfan, L. A. (27. Marshall -Olkin extended Lomax distribution and its application to censored data. Communications in Statistics Theory and Methods, 36, Ghitany, M. E., Al-Hussaini, E. K., & Al-Jarallah, R. A. (25. Marshall--Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics, 32, Ghitany, M. E., Atieh, B., & Nadarajah, S. (28. Lindley distribution and its application. Mathematics and Computers in Simulation, 78, Gui, W. (213. Marshall-Olkin extended log-logistic distribution and its application in minification processes. Applied Mathematical Sciences, 7, Inlow, M. (21. A moment generating function proof of the Lindebergy central limit theorem. The American Statistician, 64, Jeong, B. Y., Murshed, M. S., Am Seo, Y., & Park, J. S. (214. A three-parameter kappa distribution with hydrologic application: A generalized gumbel distribution. Stochastic Environmental Research and Risk Assessment, 28, Jose, K. K. (211. Marshall-Olkin family of distributions and their applications in reliability theory, time series Page 18 of 19

19 modeling and stress-strength analysis. Proceeding 58th World Statistics Congress of the International Statistical Institute, Dublin (Session CPS5, 21st-26th August (pp Jose, K. K., & Krishna, E. (211, October. Marshall-Olkin extended uniform distribution. ProbStat Forum, 4, Mansoor, M., Tahir, M. H., Alzaatreh, A., Cordeiro, G. M., Zubair, M., & Ghazali, S. S. (216. An extended Fréchet distribution: Properties and applications. Journal of Data Science, 14, Marshall, A. W., & Olkin, I. (1997. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84, McLeish, D. (214. Simulating random variables using moment-generating functions and the saddlepoint approximation. Journal of Statistical Computation and Simulation, 84, Meintanis, S. G. (21. Testing skew normality via the moment generating function. Mathematical Methods of Statistics, 19, Merovci, F., & Elbatal, I. (213. Transmuted Lindley-geometric distribution and its applications. arxiv preprint arxiv: Moors, J. J. A. (1986. The meaning of kurtosis: Darlington reexamined. The American Statistician, 4, Nadarajah, S. (28. Marshall and Olkins distributions. Acta Applicandae Mathematicae, 13, Oluyede, B., Foya, S., Warahena-Liyanage, G., & Huang, S. (216. The log-logistic Weibull distribution with applications to lifetime data. Austrian Journal of Statistics, 45, Pogány, T. K., Saboor, A., & Provost, S. (215. The Marshall- Olkin exponential Weibull distribution. Hacettepe Journal of Mathematics and Statistics. RistiĆ, M. M., Jose, K. K., & Ancy, J. (27. A Marshall--Olkin gamma distribution and minification process. Stress and Anxiety Research Society, 11, RistiĆ, M. M., & Kundu, D. (215. Marshall-Olkin generalized exponential distribution. METRON, 73, Sandhya, E., & Prasanth, C. B. (214. Marshall-Olkin discrete uniform distribution. Journal of Probability, 214, 1 1. Song, K. S. (21. Rènyi information, loglikelihood and an intrinsic distribution measure. Journal Of Statistical Planning And Inference, 93, Sugiura, N. (1978. Further analysts of the data by akaike s information criterion and the finite corrections: Further analysts of the data by akaike s. Communications in Statistics-Theory and Methods, 7, Villa, E. R., & Escobar, L. A. (26. Using moment generating functions to derive mixture distributions. The American Statistician, 6, doi:1.1198/3136x9819 von Waldenfels, W. (1987. Proof of an algebraic central limit theorem by moment generating functions (pp Berlin Heidelberg: Springer. 217 The Author(s. This open access article is distributed under a Creative Commons Attribution (CC-BY 4. license. You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics (ISSN: is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog with, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at Page 19 of 19