International Journal of Statistics and Applications 2015, 5(6): 302-316 DOI: 10.5923/j.statistics.20150506.06 Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution M. A. El-Damcese1, Dina A. Ramadan2,* 1Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt 2Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt Abstract In this paper, a new five-parameter lifetime distribution with failure rate is introduced for maximum reliability time in generalized linear hazard rate truncated poisson distribution. We obtain several properties of the new distribution such as its probability density function, its reliability and failure rate functions, quantiles and moments. Furthermore, estimation by maximum likelihood and inference are discussed. In the end, Application to real data set is given to show superior performance versus at least five of the known lifetime models. Keywords Failure rate, Generalized linear hazard rate truncate Poisson maximum distribution, Reliability, Order statistics, Residual life function, Maximum likelihood which is called as the 'flexible Weibull distribution'. 1. Introduction Recently, many new distributions, generalizing well-known distributions used to study lifetime data, have In reliability, many phenomena are modeled by statistical been introduced. Mudholkar and Srivastava (1993) distributions. The probability distribution of the time presented a generalization of the Weibull distribution called -to-failure of a device can be characterized by the failure rate the exponentiated (generalized)-Weibull distribution, GWD. or hazard function. The generalized exponential distribution, GED, introduced There are some parametric models that have successfully by Gupta and Kundu (1999). Nadarajah and Kotz (2006) served as population models for failure times arising from a introduced four exponentiated type distributions: the wide range of products and failure mechanisms. The exponentiated gamma, exponentiated Weibull, distributions with Decreasing Failure Rate (DFR) property exponentiated Gumbel. Sarhan and Kundu (2009) presented are studied in the works of Lomax (1954), Proschan (1963), a generalization of the linear hazard rate distribution called Barlow et al. (1963), Barlow and Marshall (1964)-(1965), the generalized linear hazard rate distribution, GLFRD. Marshall and Proschan (1965), Cozzolino (1968), Dahiya Sarhan et al. (2008) obtained Bayes and maximum and Gurland (1972), McNolty et al. (1980), Saunders and likelihood estimates of the three parameters of the Myhre (1983), Nassar (1988), Gleser (1989), Gurland and generalized linear hazard ratedistribution based on grouped Sethuraman (1994), Adamidis and Loukas (1998), Kus and censored data. Recently, Sarhan (2009) introduced a (2007), and Tahmasbi and Rezaei (2008). generalization of the quadratic hazard rate distribution called For modeling the reliability and survival data with the generalized quadratic hazard rate distribution (GQHRD). Increasing Failure Rate (IFR) property or bathtub failure rate, This paper is organized as follows: a new IFR distribution numerous hazard functions are proposed by different is obtained for maximum survival time by mixing researches that most of them are based on Weibull generalized quadratic hazard rate and geometric distribution. distribution. Muldholkar and Srivastava (1993) proposed an Various properties of the proposed distribution are discussed Exponentiated Weibull family for analyzing bathtub in Section 3, 4 and 5. Rényi and Shannon entropies of the failure-rate data. A model based on adding two Weibull GQHRTPM distribution are given in Section 6. Residual and distributions is presented by Xie and Lai (1995). Bebbington reverse residual life functions of the GQHRTPM distribution et al. (2007) proposed a new two-parameter distribution are discussed in Section 7. Section 8 is devoted to the which is a generalization of the Weibull. Recently, Gupta et Bonferroni and Lorenz curves of the GQHRTPM al. (2008) introduced another member of the Weibull family, distribution. The maximum likelihood estimation procedure is presented. Fitting the GQHRTPM model to real data set * Corresponding author: [email protected] (Dina A. Ramadan) indicate the flexibility and capacity of the proposed Published online at http://journal.sapub.org/statistics distribution in data modeling. In view of the density and Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved failure rate function shapes, it seems that the proposed model International Journal of Statistics and Applications 2015, 5(6): 302-316 303 1 can be considered as a suitable candidate model in reliability 1 ( = ) = ; = 1,2, … (1) analysis, biological systems, data modeling, and related − −! − fields. � − � where >Р 0. By assuming that the random variables and Z are independent and defining = max{ 1, 2, … , } then, the 2. The Maximum Survival Time marginal distribution of X, for > 0, is Distribution 1 ( ) ( ) = 1 1 Let 1, 2, … … … , be a random sample from the − − − generalized quadratic hazard rate distribution with ( + 2+ 3) � − � 1 1� 2− 3� Cumulative Density Function (cdf) = 1 − 1 (2) � − � α − bc23 −+ay y + y � − � � − � Fy( ) =1 − e23 ; , 0, > 0, with probability density function Y 1 ( + 2+ 3) ( ) = 1 ( + + 2) 2 3 ≥ ≥ − − 2 and Z is a random variable from truncate at zero ( + 2+ 3) 1 1 2 3 �( −+ �2+ 3) Poisson distribution with probability mass function as 1 2 3 . − − (3) follows:− √ − � − � where �, − 0, , > 0, � 2 . ≥ ≥ − √ Figure 1. Probability density function for GQHRTPMD from different values for a, b and c 3. Reliability Analysis The reliability function (R) of the Generalized Quadratic Hazard Rate Truncated Poisson Maximum distribution is denoted by ( ) also known as the survivor function and is defined as + 2+ 3 1 1 2 3 ( ) = 1 ( ) = 1 1 −� � , (4) � − � − − − − � − � � − � One of the characteristic in reliability analysis is the hazard rate function (HRF) defined by ( ) ( ) = ( ) 2 3 ℎ 1 + + 2 3 1 2 3 + + 1 2 3 −� � + 2+ 3 − � − � = ( + + 2) 2 3 −� � , (5) � − � + 2+ 3 − −� � 1 2 3 1 −� � � − � − − 304 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution The hazard rate function is such that: ● if = 1, the hazard function is either increasing (if b > 0) or constant (if b = 0 and a >0); ● when > 1, the hazard function should be: (1) increasing if b >0; (2) upside-down bath-tub shaped if b <0; and ● if < 1, then the hazard function will be: (1) decreasing if b = 0 or (2) bath-tub shaped if b ≠ 0. Figure 2. The hazard rate function (HRF) for GQHRTPM from different values for a,b and c It is important to note that the units for h(x) is the probability of failure per unit of time, distance or cycles. These failure rates are defined with different choices of parameters in Figure 2. The cumulative hazard function of the Generalized Quadratic Hazard Rate Truncated Poisson Maximum distribution is denoted by H(x) and is defined as ( ) ( ) H x = 0 1 + 2+ 3 ∫ ℎ 2 3 2 3 1 2 3 2 + + + + ( + + ) 2 3 1 2 3 −� � − � − � = − −� � −� � + 2+ 3 0 �1 − 2 3 � � 1 −� � � − � −+ 2+ 3 1 2 3 − = ln 1 ln 1 −� � . (6) � − � − − � − � − � − � It is important to note that the units for H (x) is the cumulative probability of failure per unit of time, distance or cycles. We can show that. For all choice of parameters the distribution has the increasing patterns of cumulative instantaneous failure rates. 4. Statistical Analysis 4.1. The Median and Mode It is observed as expected that the mean of GQHRTPM( , , , , ) cannot be obtained in explicit forms. It can be obtained as infinite series expansion so, in general different moments of GQHRTPM( , , , , ). Also, we cannot get the quantile International Journal of Statistics and Applications 2015, 5(6): 302-316 305 of GQHRTPM( , , , , ) in a closed form by using the equation ; , , , , = 0. Thus, by using Equation (2), we find that � � − 2 3 1 + + = ln ln 1 + 1 ln , 0 < < 1. (7) 2 3 The median ( ) of GQHRTPM� ( , , , , �) can be� obtained� �� from− � (7), when�� − =�0.5, as follows 2 3 1 + + = ln ln 1 0.5 + 1 ln . (8) 0.5 2 0.5 3 0.5 Moreover, the mode of GQHRTPM� ( , , , , ) can� be obtained� � � as� a solution− � of the�� following− � nonlinear equation. ( ; , , , , ) = 0 1 ( + 2+ 3) 1 2 3 2 3 1 2 3 2 ( + + ) ( + + ) 1 ( + + ) 2 3 1 2 3 − = 0 (9) − � − � − − − �� − � � − � � 4.2. Moments The following theorem 1 gives the moment of GQHRTPM ( , , , , ). Theorem 1. If has GQHRTPM ( , , ℎ, , ) the moment of X, say , is given as follows for , 0, , > 0, 2 ℎ ′ ≥ ≥ ( 1) + + +1 + 1 = − √ = = = =0 ! ! ! 2 3 +2 +3 ( +1) + +2 +1 1 ′ ∞ − − ∑ ( +2 +3 +2) (�+2− �+3 +3) ( + 2 + 3 + 1) + + � . � (10) ( +1) 2 ( +1)2 3 Proof � � ( ) ( ) = = 0 ′ ∞ Substituting (3) into the above relation, we get Е ∫ 1 + 2+ 3 = 1 ( + + 2) 2 3 0 ′ − ∞ −� � + 2+ 3 1 2 3 � − �+ 2∫+ 3 1 1 2 3 , − −� � (11) −� � � − � ( + 2+ 3) 1 � 2− 3 � The series expansions of is − � − � 2 3 + 2+ 3 + + 2 3 1 2 3 1 − −� � = =0 � � � − � � − ! � ∞ We get ∑ 1 +1 2 3 2 + + = 1 ( + + ) 2 3 =0 ! 0 − ∞ −� � ′ ∞ + 1 � � ∑+ 2+ 3 1 − 2 3 ∫ , − (12) −� � + 1 +