International Journal of 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 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 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 '. 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 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 , 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 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 −𝜆𝜆 −! −𝜆𝜆 𝑘𝑘 fields. � −𝑒𝑒 � 𝑒𝑒 𝜆𝜆 where >Р 0𝑍𝑍. 𝑘𝑘 𝑘𝑘 𝑘𝑘 By assuming that the random variables and Z are independent𝜆𝜆 and defining = max{ 1, 2, … , } then, the 2. The Maximum Survival Time 𝑖𝑖 marginal distribution of X, for > 0, is 𝑌𝑌 Distribution 𝑧𝑧 1 𝑋𝑋 𝑌𝑌 𝑌𝑌 𝑌𝑌 ( ) = 1 ( ) 1 Let 1, 2, … … … , be a random sample from the 𝜆𝜆 −𝜆𝜆 − −𝜆𝜆 𝜆𝜆𝐹𝐹𝑌𝑌 𝑥𝑥 generalized quadratic hazard rate distribution with 𝑋𝑋 ( + 2+ 3) 𝑧𝑧 𝐹𝐹 𝑥𝑥 � − 𝑒𝑒 � 1 𝑒𝑒 1�𝑒𝑒 2− 3� 𝑌𝑌 𝑌𝑌 𝑌𝑌 𝑏𝑏 𝑐𝑐 𝛼𝛼 Cumulative Density Function (cdf) = 1 − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 1 (2) 𝜆𝜆� −𝑒𝑒 � α 𝜆𝜆 − bc23 −+ay y + y �𝑒𝑒 − � �𝑒𝑒 − � Fy( ) =1 − e23 ; , 0, > 0, with probability density function Y  1 ( + 2+ 3)  ( ) = 1 ( + + 2) 2 3 𝑎𝑎 𝑐𝑐 ≥ 𝛼𝛼 𝑏𝑏 ≥ 𝑏𝑏 𝑐𝑐 𝜆𝜆 − − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 2 and Z is a from truncate at zero 𝑋𝑋 ( + 2+ 3) 1 1 2 3 𝑓𝑓 𝑥𝑥 𝜆𝜆𝜆𝜆�𝑒𝑒( −+ �2+ 𝑎𝑎3) 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 𝛼𝛼 Poisson distribution with probability mass function as 1 2 3 𝑏𝑏 𝑐𝑐 . 𝑏𝑏 𝑐𝑐 𝛼𝛼− − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 (3) follows:− √𝑎𝑎𝑎𝑎 − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 𝜆𝜆� −𝑒𝑒 � where �, − 𝑒𝑒0, , > 0, � 2𝑒𝑒 .

𝑎𝑎 𝑐𝑐 ≥ 𝛼𝛼 𝜆𝜆 𝑏𝑏 ≥ − √𝑎𝑎𝑎𝑎

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 𝑅𝑅 𝑥𝑥 1 1 2 3 𝑏𝑏 𝑐𝑐 𝛼𝛼 ( ) = 1 ( ) = 1 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � , (4) 𝜆𝜆� −𝑒𝑒 � −𝜆𝜆 − −𝜆𝜆 𝑅𝑅 𝑥𝑥 − 𝐹𝐹 𝑥𝑥 � − 𝑒𝑒 � � − 𝑒𝑒 𝑒𝑒 � One of the characteristic in reliability analysis is the hazard rate function (HRF) defined by ( ) ( ) = ( ) 𝑓𝑓 𝑥𝑥 2 3 ℎ 𝑥𝑥 𝑅𝑅 𝑥𝑥 1 + + 2 3 1 2 3 + + 𝑏𝑏 𝑐𝑐 𝛼𝛼 1 2 3 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � + 2+ 3 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜆𝜆� −𝑒𝑒 � = ( + + 2) 2 3 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � , 𝑏𝑏 𝑐𝑐 (5) � − 𝑒𝑒 � 𝑒𝑒 + 2+ 3 −𝜆𝜆 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 1 2 3 𝑏𝑏 𝑐𝑐 𝛼𝛼 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝜆𝜆𝜆𝜆𝑒𝑒 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 𝜆𝜆� −𝑒𝑒 � −𝜆𝜆 −𝑒𝑒 𝑒𝑒 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: 𝛼𝛼 (1) increasing if b >0; 𝛼𝛼 (2) upside-down bath-tub shaped if b <0; and ● if < 1, then the hazard function will be: (1) decreasing if b = 0 or 𝛼𝛼 (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 𝑥𝑥 1 + 2+ 3 ∫ ℎ 𝑡𝑡 𝑑𝑑𝑑𝑑 2 3 2 3 1 2 3 2 + + + + 𝑏𝑏 𝑐𝑐 𝛼𝛼 ( + + ) 2 3 1 2 3 −�𝑎𝑎𝑎𝑎 𝑡𝑡 𝑡𝑡 � 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜆𝜆� −𝑒𝑒 � = 𝑥𝑥 −𝜆𝜆 −�𝑎𝑎𝑎𝑎 𝑡𝑡 𝑡𝑡 � −�𝑎𝑎𝑎𝑎 𝑡𝑡 𝑡𝑡 � + 2+ 3 0 𝛼𝛼 𝜆𝜆 𝑒𝑒 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑡𝑡 𝑒𝑒 �1 − 𝑒𝑒 2 3 � 𝑒𝑒 𝑏𝑏 𝑐𝑐 𝛼𝛼 � 1 −�𝑎𝑎𝑎𝑎 𝑡𝑡 𝑡𝑡 � 𝑑𝑑𝑑𝑑 𝜆𝜆� −𝑒𝑒 � −𝜆𝜆+ 2+ 3 1 2 3 − 𝑒𝑒 𝑒𝑒𝑏𝑏 𝑐𝑐 𝛼𝛼 = ln 1 ln 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � . (6) 𝜆𝜆� −𝑒𝑒 � −𝜆𝜆 −𝜆𝜆 � − 𝑒𝑒 � − � − 𝑒𝑒 𝑒𝑒 � 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 and It is observed as expected that the 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 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑥𝑥𝑞𝑞

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 𝑋𝑋 𝑞𝑞 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 1 𝐹𝐹 �𝑥𝑥 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆� − 𝑞𝑞 + 2 + 3 = ln ln 1 + 1 ln , 0 < < 1. (7) 2 3 𝑏𝑏 𝑐𝑐 𝜆𝜆 The median ( ) of GQHRTPM�𝑎𝑎𝑥𝑥𝑞𝑞 ( 𝑥𝑥,𝑞𝑞 , , 𝑥𝑥,𝑞𝑞 �) can𝛼𝛼 be� obtained� ��𝑒𝑒 from− � 𝑞𝑞(7), when�� − =𝜆𝜆�0.5, as𝑞𝑞 follows 1 + 2 + 3 = ln ln 1 0.5 + 1 ln . (8) 𝑚𝑚 𝑋𝑋 0.5 𝑎𝑎 2𝑏𝑏 0𝑐𝑐.5𝛼𝛼 𝜆𝜆3 0.5 𝑞𝑞 𝑏𝑏 𝑐𝑐 𝜆𝜆 Moreover, the mode of GQHRTPM�𝑎𝑎𝑥𝑥 ( , 𝑥𝑥, , , 𝑥𝑥) can� be 𝛼𝛼obtained� � � as�𝑒𝑒 a solution− � of the�� following− 𝜆𝜆� nonlinear equation. 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 ( ; , , , , ) = 0

𝑑𝑑 𝑋𝑋 1 ( + 2+ 3) 1 𝑓𝑓 𝑥𝑥 2𝑎𝑎 𝑏𝑏3 𝑐𝑐 𝛼𝛼 𝜆𝜆 2 3 1 2 3 2 𝑑𝑑𝑑𝑑( + + ) ( + + ) 𝑏𝑏 𝑐𝑐 𝛼𝛼 1 ( + + ) 2 3 1 2 3 − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 = 0 (9) 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜆𝜆� −𝑒𝑒 � 𝑑𝑑 𝜆𝜆 − − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 𝑑𝑑𝑑𝑑 �𝜆𝜆𝜆𝜆�𝑒𝑒 − � 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � 𝑒𝑒 � 4.2. Moments The following theorem 1 gives the of GQHRTPM ( , , , , ). Theorem 1. If has GQHRTPM ( , , 𝑡𝑡ℎ, , ) the moment of X, say , is given as follows for , 0, , > 0, 2 𝑟𝑟 𝑡𝑡ℎ 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼′ 𝜆𝜆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑟𝑟 𝜇𝜇𝑟𝑟 𝑎𝑎 𝑐𝑐 ≥ 𝛼𝛼 𝜆𝜆 𝑏𝑏 ≥ ( 1) + + +1 + 1 = − √𝑎𝑎𝑎𝑎 = = = =0 ! ! ! 2 3 𝑘𝑘+2𝑚𝑚 +𝑗𝑗3 ( +𝑛𝑛1) +𝑚𝑚 +𝑗𝑗2 +1 1 ′ ∞ − 𝛼𝛼 𝜆𝜆 𝑏𝑏 𝑐𝑐 𝑟𝑟 𝑛𝑛 𝑘𝑘 𝑚𝑚 𝑗𝑗 𝑚𝑚 𝑗𝑗 𝑟𝑟 (𝑚𝑚 +2𝑗𝑗 +3 +𝑟𝑟2)𝑚𝑚 𝑗𝑗 ( +𝜆𝜆2 +3𝑛𝑛+𝑛𝑛3) 𝛼𝛼 − 𝜇𝜇 ∑ ( + 2 +𝑛𝑛 3𝑚𝑚 +𝑗𝑗 1) +𝑎𝑎 𝑘𝑘 + �𝑒𝑒 − � � . � (10) ( +1) 2 ( +1)2 3 𝑘𝑘 𝑏𝑏 𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑐𝑐 𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 Proof �𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑘𝑘 𝑎𝑎 𝑘𝑘 𝑎𝑎 � ( ) ( ) = = 0 ′ 𝑟𝑟 ∞ 𝑟𝑟 Substituting (3) into the above relation, we get𝜇𝜇 𝑟𝑟 Е 𝑋𝑋 ∫ 𝑥𝑥 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 1 + 2+ 3 = 1 ( + + 2) 2 3 0 𝑏𝑏 𝑐𝑐 ′ 𝜆𝜆 − ∞ 𝑟𝑟 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝑟𝑟 + 2+ 3 1 2 3 𝜇𝜇 𝜆𝜆𝜆𝜆�𝑒𝑒 − �+ 2∫+ 𝑥𝑥3 𝑎𝑎 𝑏𝑏𝑏𝑏1 𝑐𝑐𝑥𝑥 𝑒𝑒 𝛼𝛼 1 2 3 𝑏𝑏 𝑐𝑐 , 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � (11) −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝜆𝜆� −𝑒𝑒 � ( + 2+ 3) 1 � 2− 𝑒𝑒3 � 𝑒𝑒 𝑑𝑑𝑑𝑑 The series expansions of 𝑏𝑏 𝑐𝑐 𝛼𝛼 is − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 𝜆𝜆� −𝑒𝑒 � 2 3 + 2+ 3 + + 2 3 1 2 3 𝑒𝑒 1 𝛼𝛼 𝑏𝑏 𝑐𝑐 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑐𝑐 − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � = =0 𝑛𝑛 � � 𝜆𝜆� −𝑒𝑒 � 𝜆𝜆 � −𝑒𝑒 ! � ∞ We get 𝑒𝑒 ∑𝑛𝑛 𝑛𝑛 1 +1 + 2+ 3 = 1 ( + + 2) 2 3 =0 𝑛𝑛 ! 0 𝑏𝑏 𝑐𝑐 − 𝜆𝜆 ∞ −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � ′ 𝜆𝜆 ∞ + 𝑟𝑟 1 𝑟𝑟 +𝑛𝑛 2+ 𝑛𝑛 3 𝜇𝜇 𝛼𝛼 � 𝑒𝑒1 − � ∑2 3 ∫ 𝑥𝑥 𝑎𝑎 , 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 𝑏𝑏 𝑐𝑐 𝑛𝑛𝑛𝑛 𝛼𝛼− (12) −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � + 1 + 2+ 3 � − 𝑒𝑒 � 𝑑𝑑𝑑𝑑 + 2+ 3 0 < 2 3 < 1 1 2 3 Since 𝑏𝑏 𝑐𝑐 for x, then by using the binomial series expansion of 𝑏𝑏 𝑐𝑐 𝑛𝑛𝑛𝑛 𝛼𝛼− given −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � by 𝑒𝑒 � − 𝑒𝑒 � + 1 + 2+ 3 + 1 + 2+ 3 1 2 3 = ( 1) 2 3 𝑏𝑏 𝑐𝑐 𝑛𝑛𝑛𝑛 𝛼𝛼− =0 𝑏𝑏 𝑐𝑐 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � ∞ 𝑘𝑘 −𝑘𝑘�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝑘𝑘 𝑛𝑛𝑛𝑛 𝛼𝛼 − We get � − 𝑒𝑒 � ∑ � � − 𝑒𝑒 𝑘𝑘 1 ( 1) +1 + 1 = 1 = =0 𝑘𝑘! 𝑛𝑛 ′ 𝜆𝜆 − ∞ − 𝜆𝜆 𝑟𝑟 𝑛𝑛 𝑘𝑘 ( +1) 𝑛𝑛𝑛𝑛+ 2𝛼𝛼+−3 𝜇𝜇 𝛼𝛼�𝑒𝑒 − ( �+ ∑ + 2 ) 𝑛𝑛 � 2 3 � , 0 𝑏𝑏 𝑘𝑘 𝑐𝑐 (13) ∞ 𝑟𝑟 − 𝑘𝑘 �𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � ∫ 𝑥𝑥 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 𝑑𝑑𝑑𝑑

306 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution

( +1) 2 ( +1) 3 2 3 The series expansions of 𝑏𝑏 and 𝑐𝑐 are − 𝑘𝑘 𝑥𝑥 − 𝑘𝑘 𝑥𝑥 ( 1) ( +1) 2 𝑒𝑒 𝑒𝑒 ( +1) 2 2 𝑚𝑚 2 = 𝑚𝑚 𝑏𝑏 , (14) 𝑏𝑏 =0 − � 𝑘𝑘 ! 𝑥𝑥 � − 𝑘𝑘 𝑥𝑥 ∞ and 𝑒𝑒 ∑𝑚𝑚 𝑚𝑚

( 1) ( +1) 3 ( +1) 3 3 3 𝑗𝑗 = =0 𝑗𝑗 𝑐𝑐 , (15) 𝑐𝑐 − � 𝑘𝑘 ! 𝑥𝑥 � − 𝑘𝑘 𝑥𝑥 ∞ Substituting (14) and (15) into (13), we get 𝑒𝑒 ∑𝑗𝑗 𝑗𝑗 1 ( 1) + + +1( +1) + + 1 = 1 = = = =0 𝑘𝑘 𝑚𝑚 𝑗𝑗! 𝑛𝑛! ! 2 3 𝑚𝑚 𝑗𝑗 𝑚𝑚 𝑗𝑗 ′ 𝜆𝜆 − ∞ − 𝜆𝜆 𝑘𝑘 𝑏𝑏 𝑐𝑐 ( ) 𝑚𝑚 𝑗𝑗 𝑟𝑟 +2 +3 (𝑛𝑛 𝑘𝑘 𝑚𝑚 𝑗𝑗 2) 𝑛𝑛+1𝑚𝑚 𝑗𝑗 𝑛𝑛𝑛𝑛 𝛼𝛼 − 𝜇𝜇 𝛼𝛼0�𝑒𝑒 − � ∑ + + , � � (16) ∞ 𝑟𝑟 𝑚𝑚 𝑗𝑗 − 𝑘𝑘 𝑎𝑎𝑎𝑎 𝑘𝑘 The integral in (16) can be computed∫ 𝑥𝑥 as follows𝑎𝑎 𝑏𝑏 𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 𝑑𝑑𝑑𝑑 ( 2 ) +2 +3 ( +1) = 0 + + ∞ 𝑟𝑟 𝑚𝑚 𝑗𝑗 − 𝑘𝑘 𝑎𝑎𝑎𝑎 = +2 +3 ( +1) + +2 +3 +1 ( +1) + +2 +3 +2 ( +1) 𝐼𝐼 ∫ 0 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑥𝑥 𝑒𝑒 0 𝑑𝑑𝑑𝑑 0 ∞ ∞ ∞ ( +𝑟𝑟2 +𝑚𝑚3 +𝑗𝑗1) − 𝑘𝑘 𝑎𝑎(𝑎𝑎 +2 +3 +2) 𝑟𝑟 𝑚𝑚( +𝑗𝑗2 +3−+𝑘𝑘3) 𝑎𝑎𝑎𝑎 𝑟𝑟 𝑚𝑚 𝑗𝑗 − 𝑘𝑘 𝑎𝑎𝑎𝑎 = + + , (17) 𝑎𝑎((∫+1)𝑥𝑥 ) +2 +3 +𝑒𝑒1 (( +1) 𝑑𝑑) 𝑑𝑑+2 +𝑏𝑏3∫+2 𝑥𝑥 (( +1) ) +2𝑒𝑒 +3 +3 𝑑𝑑𝑑𝑑 𝑐𝑐 ∫ 𝑥𝑥 𝑒𝑒 𝑑𝑑𝑑𝑑 𝑎𝑎 𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑏𝑏 𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑐𝑐 𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑟𝑟 𝑚𝑚 𝑗𝑗 Substituting (17)𝑘𝑘 into𝑎𝑎 (16), we get𝑘𝑘 (10).𝑎𝑎 This completes𝑘𝑘 the𝑎𝑎 proof.

4.3. The Moment Generating Function The moment generating function ( ) of the GQHRTPM distribution ( , , , , ) has the following form

𝑋𝑋 ( ) = ( ) = ( ) 𝑀𝑀 𝑡𝑡 0 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑡𝑡𝑡𝑡 ∞ 𝑡𝑡𝑡𝑡 𝑋𝑋 = ( ) = . (18) 𝑀𝑀 𝑡𝑡 Е0 𝑒𝑒 =0 ∫𝑟𝑟 ! 𝑟𝑟 𝑒𝑒 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 =0 𝑟𝑟! ∞ ∞ 𝑡𝑡 𝑥𝑥 ∞ 𝑡𝑡 ′ + 𝑟𝑟1 ( 1) + + 𝑟𝑟 +1 𝑟𝑟 ( ) = ∫ ∑ 𝑟𝑟 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 ∑ 𝑟𝑟 𝜇𝜇 = = = = =0 ! ! ! ! 2 3 𝑘𝑘 +𝑚𝑚2 𝑗𝑗+3 (𝑛𝑛 +1)𝑚𝑚+ 𝑗𝑗 +𝑟𝑟2 +1 1 ∞ − 𝛼𝛼 𝜆𝜆 𝑏𝑏 𝑐𝑐 𝑡𝑡 𝑋𝑋 𝑟𝑟 𝑛𝑛 𝑘𝑘 𝑚𝑚 𝑗𝑗 𝑛𝑛𝑛𝑛 𝛼𝛼 − ( +2 +3 +2𝑚𝑚) 𝑗𝑗 𝑟𝑟 (𝑚𝑚+2 𝑗𝑗 +3 +𝑟𝑟3)𝑚𝑚 𝑗𝑗 𝜆𝜆 𝑀𝑀 𝑡𝑡 ∑ ( + 2 + 3� + 1) + � 𝑟𝑟 𝑛𝑛 𝑚𝑚 𝑗𝑗 +𝑎𝑎 𝑘𝑘 � 𝑒𝑒 − � (19) 𝑘𝑘 ( +1) 2 ( +1)2 3 𝑏𝑏 𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑐𝑐 𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 �𝛤𝛤 𝑟𝑟 𝑚𝑚 𝑗𝑗 𝑘𝑘 𝑎𝑎 𝑘𝑘 𝑎𝑎 � 5. Order Statistics The order statistics have many applications in reliability and life testing. The order statistics arise in the study of reliability of a system. Let 1, 2, … … , be a simple random sample from GQHRTPM( , , , , , ) with cumulative distribution function and probability density function as in (2) and (3), respectively. Let (1: ), (2: ), … . , ( : ) denote the order statistics 𝑋𝑋 𝑋𝑋 𝑋𝑋𝑛𝑛 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑥𝑥 system which ـــ of ـــ out ـــ (obtained from this sample. In reliability literature, denote the lifetime of an ( i+1 ( : ) 𝑋𝑋 𝑛𝑛 𝑋𝑋 𝑛𝑛 𝑋𝑋 𝑛𝑛 𝑛𝑛 1 consists of independent and identically components.𝑖𝑖 𝑛𝑛 Then the pdf of ( : ), ] is given by 𝑋𝑋 1 𝑛𝑛 − 𝑛𝑛 ( ) = [ ( )] 1[1 𝑖𝑖 𝑛𝑛 ( )] ( ), (20) 𝑛𝑛 : ( , +1) 𝑋𝑋 ≤ 𝑖𝑖 ≤ 𝑛𝑛 𝑖𝑖− 𝑛𝑛−𝑖𝑖 Using 𝑓𝑓𝑖𝑖 𝑛𝑛 𝑥𝑥 В 𝑖𝑖 𝑛𝑛−𝑖𝑖 𝐹𝐹 𝑥𝑥 − 𝐹𝐹 𝑥𝑥 𝑓𝑓 𝑥𝑥 ( ) = 1 ( ) , ( ) = ( ) ( ), (21) then, −𝐻𝐻 𝑥𝑥 −𝐻𝐻 𝑥𝑥 𝐹𝐹 𝑥𝑥 − 𝑒𝑒 𝑓𝑓 𝑥𝑥 ℎ 𝑥𝑥 𝑒𝑒 [1 ( )] = ( ) ( ) (22) and 𝑛𝑛−𝑖𝑖 − 𝑛𝑛−𝑖𝑖 𝐻𝐻 𝑥𝑥 − 𝐹𝐹 𝑥𝑥 𝑒𝑒 1 ( ) 1 1 1 ( ) [ ( )] = 1 = = ( 1) . (23) 𝑖𝑖− −𝐻𝐻 𝑥𝑥 𝑖𝑖− 𝑖𝑖− 𝑙𝑙 −𝑙𝑙𝑙𝑙 𝑥𝑥 𝑙𝑙 𝑜𝑜 𝑖𝑖 − Substituting (22) and (23) into (20), we𝐹𝐹 𝑥𝑥get � − 𝑒𝑒 � ∑ � � − 𝑒𝑒 𝑙𝑙 1 1 ( ) = 1 ( 1) ( ) ( + +1 ) ( ) (24) : ( , +1) = 𝑖𝑖− 𝑙𝑙 − 𝑛𝑛 𝑙𝑙 −𝑖𝑖 𝐻𝐻 𝑥𝑥 𝑖𝑖 𝑛𝑛 𝑙𝑙 𝑜𝑜 𝑖𝑖 − From (5) and (6), then 𝑓𝑓 𝑥𝑥 В 𝑖𝑖 𝑛𝑛−𝑖𝑖 ∑ � � − ℎ 𝑥𝑥 𝑒𝑒 𝑙𝑙

International Journal of Statistics and Applications 2015, 5(6): 302-316 307

1 1 ( + 2+ 3) ( ) = 1 ( 1) ( + + 2) 2 3 : ( , +1) = 𝑏𝑏 𝑐𝑐 𝑖𝑖− 𝑙𝑙 −𝜆𝜆 − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 𝑖𝑖 𝑛𝑛 𝑙𝑙 𝑜𝑜 𝑖𝑖 − 𝑓𝑓 𝑥𝑥 В 𝑖𝑖 𝑛𝑛−𝑖𝑖 ∑ � � − 𝜆𝜆𝜆𝜆𝑒𝑒 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 + 2+ 3 2 3 1 2 3 1 +𝑙𝑙 + 𝑏𝑏 𝑐𝑐 𝛼𝛼 2 3 1 2 3 ( + +1 ) ln 1 ln 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � + + 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆� −𝑒𝑒 � 1 2 3 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � ⎛ ⎛ ⎞⎞ 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜆𝜆� −𝑒𝑒 � −𝜆𝜆 −𝜆𝜆 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � − 𝑛𝑛 𝑙𝑙 −𝑖𝑖 ⎜ � −𝑒𝑒 �− ⎜ −𝑒𝑒 𝑒𝑒 ⎟⎟. (25) � − 𝑒𝑒 � 𝑒𝑒 + 2+ 3 1 2 3 𝑏𝑏 𝑐𝑐 𝛼𝛼 ⎝ ⎝ ⎠⎠ 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝜆𝜆� −𝑒𝑒 � 𝑒𝑒 −𝜆𝜆 We defined the first order statistics−𝑒𝑒 𝑒𝑒 (1) = ( 1, 2, … … , ), the last order statistics as ( ) = ( 1, 2, … … , ) and median order +1. 𝑋𝑋 𝑀𝑀𝑀𝑀𝑀𝑀 𝑋𝑋 𝑋𝑋 𝑋𝑋𝑛𝑛 𝑋𝑋 𝑛𝑛 𝑀𝑀𝑀𝑀𝑀𝑀 𝑋𝑋 𝑋𝑋 𝑋𝑋𝑛𝑛 5.1. Distribution 𝑋𝑋of𝑚𝑚 Minimum, Maximum and Median

Let 1, 2, … … , be independently identically distributed order random variables from the GQHRTPM distribution having first, last and median order probability density function are given by the following 𝑛𝑛 𝑋𝑋 𝑋𝑋 𝑋𝑋 1 n H(x) 1: ( ) = [1 ( )] ( ) = ( )e (26) 𝑛𝑛− − 1 + 2+ 3 + 2+ 3 𝑓𝑓 𝑛𝑛 𝑥𝑥 𝑛𝑛 − 𝐹𝐹+ 𝑥𝑥+ 2 𝑓𝑓 𝑥𝑥 2 𝑛𝑛3 ℎ 𝑥𝑥1 2 3 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 1: ( ) = −𝜆𝜆 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝑛𝑛𝑛𝑛𝑛𝑛 𝑒𝑒 �𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 �𝑒𝑒 + �2+− 𝑒𝑒3 � 1 2 3 𝑛𝑛 𝑏𝑏 𝑐𝑐 𝛼𝛼 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝑓𝑓 𝑥𝑥 𝜆𝜆� −𝑒𝑒 � −𝜆𝜆 + 2+ 3 −𝑒𝑒 𝑒𝑒 1 2 3 𝑏𝑏 𝑐𝑐 𝛼𝛼 n ln 1 ln 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � + 2+ 3 𝜆𝜆� −𝑒𝑒 � 1 2 3 ⎛ −𝜆𝜆 ⎛ −𝜆𝜆 ⎞⎞ 𝑏𝑏 𝑐𝑐 𝛼𝛼 − � −𝑒𝑒 �− −𝑒𝑒 𝑒𝑒 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � e ⎜ ⎜ ⎟⎟. (27) 𝜆𝜆� −𝑒𝑒 � 1 ⎝1 1 ⎝ ( +1) ( ) ⎠⎠ : ( ) = 𝑒𝑒[ ( )] ( ) = = ( 1) ( ) , (28) 𝑛𝑛− 𝑛𝑛− 𝑙𝑙 − 𝑙𝑙 𝐻𝐻 𝑥𝑥 𝑛𝑛 𝑛𝑛 1 𝑙𝑙 𝑜𝑜 𝑛𝑛 − + 2+ 3 𝑓𝑓( ) 𝑥𝑥= 𝑛𝑛 𝐹𝐹1 𝑥𝑥 𝑓𝑓( 𝑥𝑥1) 𝑛𝑛 ∑ ( �+ +� −2) ℎ 𝑥𝑥 2𝑒𝑒 3 : = 𝑙𝑙 𝑏𝑏 𝑐𝑐 𝑛𝑛− 𝑙𝑙 −𝜆𝜆 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝑛𝑛 𝑛𝑛 𝑙𝑙 𝑜𝑜 𝑛𝑛 − 𝑓𝑓 𝑥𝑥 𝑛𝑛 ∑ � � − 𝜆𝜆𝜆𝜆𝑒𝑒 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 𝑒𝑒 + 2+ 3 1 2 3 𝑙𝑙 + 2+ 3 𝛼𝛼 1 1 2 3 ( +1) ln 1 ln 1 𝑏𝑏 𝑐𝑐 + 2+ 3 𝛼𝛼 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 2 3 𝑏𝑏 𝑐𝑐 𝜆𝜆� −𝑒𝑒 � 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � ⎛ −𝜆𝜆 ⎛ −𝜆𝜆 ⎞⎞ 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜆𝜆� −𝑒𝑒 � −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � − 𝑙𝑙 ⎜ � −𝑒𝑒 �− ⎜ −𝑒𝑒 𝑒𝑒 ⎟⎟. (29) � − 𝑒𝑒 � 𝑒𝑒 + 2+ 3 1 2 3 𝑏𝑏 𝑐𝑐 𝛼𝛼 ⎝ ⎝ ⎠⎠ 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 𝜆𝜆� −𝑒𝑒 � 𝑒𝑒 −𝜆𝜆 and −𝑒𝑒 𝑒𝑒 (2 +1)! (2 +1)! ( ) = [ ( )] [1 ( )] ( ) = ( 1) ( + ) ( ) ( ), (30) +1:2 +1 ! ! ! ! = 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑙𝑙 − 𝑚𝑚 𝑙𝑙 𝐻𝐻 𝑥𝑥 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑙𝑙 𝑜𝑜 ( + 2+ 3) 𝑓𝑓 𝑥𝑥 𝐹𝐹 𝑥𝑥 − 𝐹𝐹 𝑥𝑥 𝑓𝑓 𝑥𝑥 2 3∑ 1 � � −2 3𝑒𝑒 𝑓𝑓 𝑥𝑥 2 ( + + ) 𝑙𝑙 𝑏𝑏 𝑐𝑐 𝛼𝛼 (2 +1)! ( 1) + + 2 3 − 𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 ( ) = 𝑏𝑏 𝑐𝑐 𝜆𝜆� −𝑒𝑒 � +1:2 +1 ! ! = 𝑙𝑙 − 𝑎𝑎𝑎𝑎 𝑥𝑥 1 𝑥𝑥 𝑚𝑚 𝑚𝑚 − 𝜆𝜆𝜆𝜆 �𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 �𝑒𝑒 𝑒𝑒 𝑚𝑚 𝑚𝑚 𝑙𝑙 𝑜𝑜 𝑚𝑚 𝜆𝜆 𝑓𝑓 𝑥𝑥 𝑚𝑚 𝑚𝑚 ∑ � � �𝑒𝑒 − � + 2+ 3 𝑙𝑙 1 2 3 𝑏𝑏 𝑐𝑐 𝛼𝛼 ( + ) ln 1 ln 1 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 1 𝜆𝜆� −𝑒𝑒 � + 2+ 3 ⎛ −𝜆𝜆 ⎛ −𝜆𝜆 ⎞⎞ 1 2 3 − 𝑚𝑚 𝑙𝑙 ⎜ � −𝑒𝑒 �− ⎜ −𝑒𝑒 𝑒𝑒 ⎟⎟. 𝑏𝑏 𝑐𝑐 𝛼𝛼− (31) −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � ⎝ ⎝ ⎠⎠ � − 𝑒𝑒 � 𝑒𝑒 6. Rényi and Shannon Entropies If X is a random variable having an absolutely continuous cumulative distribution function ( ) and probability distribution function ( ) then the basic uncertainty measure for distribution F (called the entropy of F) is defined as ( ) = [ log( ( ))]. Statistical entropy is a probabilistic measure of uncertainty or ignorance about𝐹𝐹 𝑥𝑥 the outcome of a random , and𝑓𝑓 𝑥𝑥is a measure of a reduction in that uncertainty. Numerous entropy and information indices, among 𝐻𝐻them𝑥𝑥 the𝐸𝐸 Rényi− entropy,𝑓𝑓 𝑋𝑋 have been developed and used in various disciplines and contexts. Information theoretic principles and methods have become integral parts of probability and statistics and have been applied in various branches of statistics and related fields.

308 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution

Entropy has been used in various situations in science and engineering. The entropy of a random variable Y is a measure of variation of the uncertainty. For a random variable with the pdf f, the Rényi entropy is defined by 1 ( ) = log[ ( ) ], for > 0 and 0. 1 𝑟𝑟 𝑅𝑅 + 2+ 3 𝐼𝐼 𝑟𝑟 −𝑟𝑟 ∫ 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑑𝑑 𝑟𝑟 ( ) = 𝑟𝑟 ≠ ( + + 2) 2 3 0 1 𝑟𝑟 0 𝑏𝑏 𝑐𝑐 ∞ 𝑟𝑟 𝜆𝜆𝜆𝜆 ∞ 𝑟𝑟 −𝑟𝑟�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � 𝜆𝜆 + 2+ 3 ∫ 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑑𝑑 �𝑒𝑒 − � ∫ 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑦𝑦 𝑒𝑒 2 3 + 2+ 3 1 𝛼𝛼 1 2 3 𝑏𝑏 𝑐𝑐 , 𝑏𝑏 𝑐𝑐 𝑟𝑟𝑟𝑟 −𝑟𝑟 −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � (32) −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � 𝑟𝑟𝑟𝑟 � −𝑒𝑒 � + 2+ � 3 − 𝑒𝑒 � 𝑒𝑒 𝑑𝑑𝑑𝑑 The series expansions of 1 2 3 is 𝑏𝑏 𝑐𝑐 𝛼𝛼 −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � 𝑟𝑟𝑟𝑟 � −𝑒𝑒 � 2 3 + 2+ 3 + + 2 3 1 2 3 𝑒𝑒 1 𝛼𝛼 𝑏𝑏 𝑐𝑐 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑐𝑐 − 𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � = =0 𝑛𝑛 𝑛𝑛 � � , (33) 𝑟𝑟𝑟𝑟 � −𝑒𝑒 � 𝑟𝑟 𝜆𝜆 � −𝑒𝑒 ! � ∞ We get 𝑒𝑒 ∑𝑛𝑛 𝑛𝑛 + 2 3 2 + + ( ) = =0 ( + + ) 2 3 0 1 𝑟𝑟 𝑛𝑛 𝑛𝑛! 𝑟𝑟 0 𝑏𝑏 𝑐𝑐 ∞ 𝑟𝑟 𝛼𝛼 ∞ 𝑟𝑟 𝜆𝜆 ∞ 𝑟𝑟 −𝑟𝑟�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � 𝜆𝜆 𝑛𝑛 ( + ) 𝑒𝑒 − ∑ + 2𝑛𝑛+ 3 ∫ 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑑𝑑 � 1 � 2 3 ∫ 𝑎𝑎 𝑏𝑏𝑏𝑏 , 𝑐𝑐𝑦𝑦 𝑒𝑒 𝑏𝑏 𝑐𝑐 𝑟𝑟 𝑛𝑛 𝛼𝛼−𝑟𝑟 (34) −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � ( + ) + 2+ 3 � − 𝑒𝑒 � 𝑑𝑑𝑑𝑑 + 2+ 3 0 < 2 3 < 1 1 2 3 Since 𝑏𝑏 𝑐𝑐 for x, then by using the binomial series expansion of 𝑏𝑏 𝑐𝑐 𝑟𝑟 𝑛𝑛 𝛼𝛼−𝑟𝑟 and −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � ( + + 2) given by 𝑒𝑒 � − 𝑒𝑒 � 𝑟𝑟 ( + ) + 2+ 3 + + 2+ 3 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑦𝑦 1 2 3 = ( 1) 2 3 , 𝑏𝑏 𝑐𝑐 𝑟𝑟 𝑛𝑛 𝛼𝛼−𝑟𝑟 =0 𝑏𝑏 𝑐𝑐 (35) −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � −𝑘𝑘�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � ∞ 𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟 − 𝑟𝑟 𝑘𝑘 and � − 𝑒𝑒 � ∑𝑘𝑘 � � − 𝑒𝑒 𝑘𝑘 2 2 + ( + + ) = =0 ( + ) = = =0 . (36) 𝑟𝑟 ∞ 𝑟𝑟−𝑚𝑚 𝑚𝑚 ∞ 𝑟𝑟−𝑚𝑚 𝑚𝑚−𝑙𝑙 𝑙𝑙 𝑚𝑚 𝑙𝑙 ( + ) 2 𝑚𝑚 ( 𝑟𝑟+ ) 3 𝑚𝑚 𝑙𝑙 𝑟𝑟 𝑚𝑚 𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑦𝑦 2 ∑ � � 𝑎𝑎3 𝑏𝑏𝑏𝑏 𝑐𝑐𝑦𝑦 ∑ � � � � 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑦𝑦 The series expansions of 𝑏𝑏 and 𝑚𝑚 𝑐𝑐 are 𝑚𝑚 𝑙𝑙 − 𝑟𝑟 𝑘𝑘 𝑦𝑦 − 𝑟𝑟 𝑘𝑘 𝑦𝑦 ( + ) 2 ( 1) ( + ) 2 = 2 , (37) 𝑒𝑒 𝑒𝑒 𝑏𝑏 =0 𝑖𝑖 ! 2 𝑖𝑖 𝑖𝑖 − 𝑟𝑟 𝑘𝑘 𝑦𝑦 ∞ − 𝑟𝑟 𝑘𝑘 𝑏𝑏 𝑖𝑖 𝑖𝑖 and 𝑒𝑒 ∑𝑖𝑖 𝑖𝑖 𝑦𝑦 ( + ) 3 ( 1) ( + ) 3 = 3 . (38) 𝑐𝑐 =0 𝑗𝑗 ! 3 𝑗𝑗 𝑗𝑗 − 𝑟𝑟 𝑘𝑘 𝑦𝑦 ∞ − 𝑟𝑟 𝑘𝑘 𝑐𝑐 𝑗𝑗 𝑗𝑗 Substituting (35), (36), (37) and (38) into𝑒𝑒 (34), we get∑ 𝑗𝑗 𝑗𝑗 𝑦𝑦 + ( 1) + + + + + ( + ) + ( ) ======0 0 1 𝑟𝑟 𝑘𝑘 𝑖𝑖 𝑗𝑗 𝑛𝑛 𝑛𝑛 𝑟𝑟! !𝑟𝑟 −!𝑚𝑚 2 𝑚𝑚3 −𝑙𝑙 𝑖𝑖 𝑙𝑙 𝑗𝑗 𝑖𝑖 𝑗𝑗 ∞ 𝑟𝑟 𝛼𝛼 ∞ − 𝑟𝑟 𝜆𝜆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑟𝑟 𝑘𝑘 + +2 +3 (𝜆𝜆 + ) 𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟 − 𝑟𝑟 𝑟𝑟 𝑚𝑚 𝑖𝑖 𝑗𝑗 ∫ 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑑𝑑 �𝑒𝑒 − � ∑𝑛𝑛 ,𝑘𝑘 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 � � � � � � 𝑛𝑛 𝑖𝑖 𝑗𝑗 0 𝑘𝑘 𝑚𝑚 𝑙𝑙 ∞ 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 − 𝑟𝑟 𝑘𝑘 𝑎𝑎𝑎𝑎 + ( 1) + + + + + ( + ) + ( + +2 +3 +1) ∫= 𝑦𝑦 =𝑒𝑒 = = = =𝑑𝑑=𝑑𝑑0 + +2 +3 . (39) 1 𝑟𝑟 𝑘𝑘 𝑖𝑖 𝑗𝑗 𝑛𝑛 𝑛𝑛 𝑟𝑟! !𝑟𝑟 −!𝑚𝑚 2 𝑚𝑚3 −𝑙𝑙 𝑖𝑖 𝑙𝑙 𝑗𝑗 𝑖𝑖 𝑗𝑗 ( + ) 𝛼𝛼 ∞ − 𝑟𝑟 𝜆𝜆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑟𝑟 𝑘𝑘 𝛤𝛤 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 𝜆𝜆 𝑛𝑛 𝑘𝑘 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟 − 𝑟𝑟 𝑟𝑟 𝑚𝑚 𝑖𝑖 𝑗𝑗 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 �𝑒𝑒 − � ∑ � � � � � � 𝑛𝑛 𝑖𝑖 𝑗𝑗 � 𝑟𝑟 𝑘𝑘 𝑎𝑎� Thus, according to the definition of Rényi 𝑘𝑘entropy we𝑚𝑚 have𝑙𝑙 + ( 1) + + + + + 1 ======0 𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟 −𝑟𝑟 𝑟𝑟 𝑚𝑚+ +2 +3𝑘𝑘 𝑖𝑖 +𝑗𝑗 𝑛𝑛 𝑛𝑛 𝑟𝑟 +𝑚𝑚 +−𝑙𝑙+2𝑖𝑖 𝑙𝑙 𝑗𝑗 ( ) = log 1 𝑟𝑟 � ! ! ! 2� � 3 �� � − 𝑟𝑟 (𝜆𝜆 + ) 𝑏𝑏 𝑐𝑐 . (40) 1 𝛼𝛼 ∞ 𝑘𝑘 𝑚𝑚 𝑙𝑙 𝜆𝜆 𝑛𝑛 𝑘𝑘 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 ( + + 2𝑖𝑖 +𝑗𝑗 3𝑚𝑚 +𝑙𝑙 1𝑖𝑖) 𝑗𝑗−𝑟𝑟 𝑚𝑚 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 𝑅𝑅 𝑒𝑒 − ∑ 𝑛𝑛 𝑖𝑖 𝑗𝑗 𝑎𝑎 𝑟𝑟 𝑘𝑘 𝐼𝐼 𝑟𝑟 −𝑟𝑟 �� � � The Shannon entropy is defined by [ log( ( ))]. This is a special case derived from lim ( ). 𝛤𝛤 𝑚𝑚 𝑙𝑙 𝑖𝑖 𝑗𝑗 1 𝐸𝐸 − 𝑓𝑓 𝑌𝑌 𝑟𝑟→ 𝐼𝐼𝑅𝑅 𝑟𝑟 7. Residual Life Function of the GQHRTPM Distribution Given that a component survives up to time t ≥ 0, the residual life is the period beyond t until the time of failure and defined by the conditional random variable X −t|X > t. The mean residual life (MRL) function is an important function in , , economics and other social sciences and reliability for characterizing lifetime. Although the shape of the failure rate function plays an important role in repair and replacement strategies, the MRL function is more relevant as the latter summarizes the entire residual life function, whereas the former considers only the risk of instantaneous failure. In

International Journal of Statistics and Applications 2015, 5(6): 302-316 309

reliability, it is well known that the MRL function and ratio of two consecutive moments of residual life determine the distribution uniquely (Gupta and Gupta (1983)). The rth order moment of the residual life of the GQHRTPM distribution is given by the general formula MRL function as well as failure rate function is very important, since each of them can be used to determine a unique corresponding lifetime distribution. Lifetimes can exhibit IMRL (increasing MRL) or DMRL (decreasing MRL). MRL functions that first decreases (increases) and then increases (decreases) are usually called bathtub-shaped (upside-down bathtub), BMRL (UMRL). The relationship between the behaviors of the two functions of a distribution was studied by many authors such The rth order moment of the residual life of the GQHRTPM distribution is given by the general formula 1 ( ) = [( ) > ] = ( ) ( ) , ( ) (41) 𝑟𝑟 ∞ 𝑟𝑟 where ( ) = 1 ( ), is the survival𝑚𝑚 function.𝑟𝑟 𝑡𝑡 𝐸𝐸 𝑌𝑌 − 𝑡𝑡 𝑌𝑌 𝑡𝑡 𝑅𝑅 𝑡𝑡 ∫𝑡𝑡 𝑦𝑦 − 𝑡𝑡 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑑𝑑 + 2+ 3 The series expansions of 1 2 3 is 𝑅𝑅 𝑡𝑡 − 𝐹𝐹 𝑡𝑡 𝑏𝑏 𝑐𝑐 𝛼𝛼 −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � 𝜆𝜆� −𝑒𝑒 � 2 3 + 2+ 3 + + 2 3 1 2 3 𝑒𝑒 1 𝛼𝛼 𝑏𝑏 𝑐𝑐 𝑛𝑛𝑛𝑛 𝑏𝑏 𝑐𝑐 − 𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � = =0 𝑛𝑛 � � . (42) 𝜆𝜆� −𝑒𝑒 � 𝜆𝜆 � −𝑒𝑒 ! � ∞ 𝑛𝑛 ( +1) 1 ∑ 𝑛𝑛 + 2+ 3 𝑒𝑒 1 2 3 ( ) In what seen this onwards, we use the binomial series expansion of 𝑏𝑏 𝑐𝑐 𝑛𝑛 𝛼𝛼− and given −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � by 𝑟𝑟 � − 𝑒𝑒 � 𝑦𝑦 − 𝑡𝑡 ( +1) 1 + 2+ 3 + + 2+ 3 1 2 3 = ( 1) 2 3 , 𝑏𝑏 𝑐𝑐 𝑛𝑛 𝛼𝛼− =0 𝑏𝑏 𝑐𝑐 (43) −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � −𝑘𝑘�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � ∞ 𝑛𝑛𝑛𝑛 𝛼𝛼 − 𝑟𝑟 𝑘𝑘 and � − 𝑒𝑒 � ∑𝑘𝑘 � � − 𝑒𝑒 𝑘𝑘 ( ) = =0 ( 1) . (44) 𝑟𝑟 ∞ 𝑟𝑟 𝑟𝑟−𝑠𝑠 𝑟𝑟−𝑠𝑠 𝑠𝑠 Then, 𝑦𝑦 − 𝑡𝑡 ∑𝑠𝑠 � � − 𝑡𝑡 𝑦𝑦 𝑠𝑠 1 + ( 1) + +1 ( +1) + 2+ 3 ( ) = ( + +1 + +2) 2 3 , (45) ( ) = = =0 𝑘𝑘 𝑟𝑟!− 𝑠𝑠 1𝑟𝑟−𝑠𝑠 𝑛𝑛 𝑏𝑏 𝑐𝑐 ∞ − 𝛼𝛼 𝑡𝑡 𝜆𝜆 ∞ 𝑠𝑠 𝑠𝑠 𝑠𝑠 − 𝑘𝑘 �𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � 𝑟𝑟 𝜆𝜆 𝑟𝑟 𝑅𝑅 𝑡𝑡 𝑛𝑛 𝑘𝑘 𝑠𝑠 𝑛𝑛(𝑛𝑛 +1)𝛼𝛼 −2 𝑟𝑟 ( +1) 𝑛𝑛3 �𝑒𝑒 − � 𝑡𝑡 𝑚𝑚 𝑡𝑡 ∑ � 2 � � � 3 ∫ 𝑎𝑎𝑦𝑦 𝑏𝑏𝑦𝑦 𝑐𝑐𝑦𝑦 𝑒𝑒 𝑑𝑑𝑑𝑑 The series expansions of 𝑘𝑘𝑏𝑏 and 𝑠𝑠 𝑐𝑐 are − 𝑘𝑘 𝑦𝑦 − 𝑘𝑘 𝑦𝑦 ( +1) 2 ( 1) ( +1) 2 = 2 , (46) 𝑒𝑒 𝑒𝑒 𝑏𝑏 =0 𝑖𝑖 ! 2 𝑖𝑖 𝑖𝑖 − 𝑘𝑘 𝑦𝑦 ∞ − 𝑘𝑘 𝑏𝑏 𝑖𝑖 𝑖𝑖 and 𝑒𝑒 ∑𝑖𝑖 𝑖𝑖 𝑦𝑦 ( +1) 3 ( 1) ( +1) 3 = 3 , (47) 𝑐𝑐 =0 𝑗𝑗 ! 3 𝑗𝑗 𝑗𝑗 − 𝑘𝑘 𝑦𝑦 ∞ − 𝑘𝑘 𝑐𝑐 𝑗𝑗 𝑗𝑗 Substituting (46) and (47) into (45), we get 𝑒𝑒 ∑𝑗𝑗 𝑗𝑗 𝑦𝑦 1 + ( 1) + + + +1 ( +1) + ( ) = ( ) = = = = =0 𝑘𝑘 𝑟𝑟 𝑖𝑖 𝑗𝑗!− 𝑠𝑠! ! 2𝑟𝑟− 3𝑠𝑠 𝑛𝑛 1𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 ∞ − 𝛼𝛼 𝑡𝑡 𝜆𝜆 𝑏𝑏 𝑐𝑐 𝑘𝑘 𝑛𝑛𝑛𝑛 𝛼𝛼 − 𝑟𝑟 𝑟𝑟 𝑖𝑖 𝑗𝑗 𝜆𝜆 𝑚𝑚𝑟𝑟 𝑡𝑡 𝑅𝑅 𝑡𝑡 (∑𝑛𝑛 𝑘𝑘+2𝑠𝑠+3𝑖𝑖 𝑗𝑗+ � +2 +3 +1 +� � �+2 +3 +2)𝑛𝑛 𝑗𝑗 (𝑖𝑖 +1) �𝑒𝑒 −, � (48) 𝑘𝑘 𝑠𝑠 ∞ 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑠𝑠 𝑖𝑖 𝑗𝑗 − 𝑘𝑘 𝑎𝑎𝑎𝑎 Where ( , ) = 1 ∫𝑡𝑡 𝑎𝑎𝑦𝑦= ( 1)! 𝑏𝑏𝑦𝑦 1 is 𝑐𝑐the𝑦𝑦 upper incomplete𝑒𝑒 gamma𝑑𝑑𝑑𝑑 function. =0 𝑘𝑘! ∞ 𝑥𝑥 The rth order moment of𝑠𝑠− the −residual𝑡𝑡 life of the −GQHR𝑥𝑥 𝑠𝑠−TPM distribution is given by 𝛤𝛤 𝑠𝑠 𝑥𝑥 ∫𝑥𝑥 𝑡𝑡 𝑒𝑒 𝑑𝑑𝑑𝑑 𝑠𝑠 − 𝑒𝑒 ∑𝑘𝑘 𝑘𝑘 1 + ( 1) + + + +1 ( ) = ( ) = = = = =0 ! ! ! 2 3𝑘𝑘 𝑟𝑟 +𝑖𝑖 2𝑗𝑗+−3𝑠𝑠 𝑟𝑟−𝑠𝑠1 𝑛𝑛 ( +1)𝑖𝑖 +𝑗𝑗 +2 +1 ∞ − 𝛼𝛼 𝑡𝑡 𝜆𝜆 𝑏𝑏 𝑐𝑐 𝑟𝑟 𝑛𝑛 𝑘𝑘 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑛𝑛𝑛𝑛 𝛼𝛼 − 𝑟𝑟 𝑟𝑟 ( +2 +𝑖𝑖3 𝑗𝑗+1,𝑠𝑠( +𝑖𝑖1) 𝑗𝑗 ) 𝜆𝜆 ( +2 𝑠𝑠+3𝑖𝑖 +𝑗𝑗1,( +1) ) 𝑚𝑚 𝑡𝑡 𝑅𝑅 𝑡𝑡 (∑ + 2 + 3 +�1, ( + 1) �)�+� 𝑛𝑛 𝑗𝑗 𝑖𝑖 𝑎𝑎 �𝑒𝑒 +− � 𝑘𝑘 . (49) 𝑘𝑘 𝑠𝑠 2( +1) 3( +1)2 𝑏𝑏 𝛤𝛤 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑎𝑎𝑎𝑎 𝑐𝑐 𝛤𝛤 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑎𝑎𝑎𝑎 where R(t) (the �𝛤𝛤 of𝑠𝑠 Y ) 𝑖𝑖is given𝑗𝑗 in (𝑘𝑘4). 𝑎𝑎𝑎𝑎 𝑎𝑎 𝑘𝑘 𝑎𝑎 𝑘𝑘 � For the GQHRTPM distribution the MRL function which is obtained by setting r = 1 in (49), is given in the following theorem. Theorem 2. The MRL function of the GQHRTPM distribution with cdf (2) is given by 1 + 1 1 ( 1) + + +1 +1 +1 ( ) = 1 ( ) = = = = =0 ! ! ! 2𝑘𝑘 3 𝑖𝑖 𝑗𝑗−+𝑠𝑠 2 +3 −𝑠𝑠 1 𝑛𝑛 ( +1𝑖𝑖) +𝑗𝑗 +2 ∞ − 𝛼𝛼 𝑡𝑡 𝜆𝜆 𝑏𝑏 𝑐𝑐 𝑛𝑛 𝑘𝑘 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑛𝑛𝑛𝑛 𝛼𝛼 − ( +2 +3𝑖𝑖 +𝑗𝑗1,( 𝑠𝑠+1𝑖𝑖) 𝑗𝑗) 𝜆𝜆 ( +2 +𝑠𝑠3 𝑖𝑖+1𝑗𝑗,( +1) ) 𝑚𝑚 𝑡𝑡 𝑅𝑅 𝑡𝑡( ∑+ 2 + 3 + 1�, ( + 1) )�+� � 𝑛𝑛 𝑗𝑗 𝑖𝑖 𝑎𝑎 �𝑒𝑒+− � 𝑘𝑘 . (50) 𝑘𝑘 𝑠𝑠 2( +1) 3( +1)2 𝑏𝑏 𝛤𝛤 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑎𝑎𝑎𝑎 𝑐𝑐 𝛤𝛤 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑎𝑎𝑎𝑎 �𝛤𝛤 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑎𝑎𝑎𝑎 𝑎𝑎 𝑘𝑘 𝑎𝑎 𝑘𝑘 � 310 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution

The second moment of the residual life function of the GQHRTPM distribution is 1 + 2 2 ( 1) + + +2 +2 +1 ( ) = 2 ( ) = = = = =0 ! ! ! 2𝑘𝑘 3 𝑖𝑖 𝑗𝑗−+𝑠𝑠 2 +3 −𝑠𝑠 1 𝑛𝑛 ( +1𝑖𝑖) +𝑗𝑗 +2 ∞ − 𝛼𝛼 𝑡𝑡 𝜆𝜆 𝑏𝑏 𝑐𝑐 𝑛𝑛 𝑘𝑘 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑛𝑛𝑛𝑛 𝛼𝛼 − ( +2 +𝑖𝑖3 𝑗𝑗+1,𝑠𝑠( +𝑖𝑖1) 𝑗𝑗 ) 𝜆𝜆 ( +2 𝑠𝑠+3𝑖𝑖 +𝑗𝑗1,( +1) ) 𝑚𝑚 𝑡𝑡 𝑅𝑅 𝑡𝑡 (∑ + 2 + 3 +�1, ( + 1) �)�+� 𝑛𝑛 𝑗𝑗 𝑖𝑖 𝑎𝑎 �𝑒𝑒 +− � 𝑘𝑘 . (51) 𝑘𝑘 𝑠𝑠 2( +1) 3( +1)2 𝑏𝑏 𝛤𝛤 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑎𝑎𝑎𝑎 𝑐𝑐 𝛤𝛤 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑎𝑎𝑎𝑎 The of the residual life�𝛤𝛤 𝑠𝑠function𝑖𝑖 of𝑗𝑗 the GQHR𝑘𝑘 TPM𝑎𝑎𝑎𝑎 distribution𝑎𝑎 can𝑘𝑘 be obtained using𝑎𝑎 𝑘𝑘 1( ) and � 2( ). On the other hand, we analogously discuss the reversed residual life and some of its properties. The reversed residual life can be defined as the conditional random variable t − X|X ≤ t which denotes the time elapsed from the𝑚𝑚 failure𝑡𝑡 of𝑚𝑚 a component𝑡𝑡 given that its life is less than or equal to t. This random variable may also be called the inactivity time (or time since failure); for more details one may see Kundu and Nanda (2010) and Nanda et al. (2003). Also, in reliability, the mean reversed residual life (MRRL) and ratio of two consecutive moments of reversed residual life characterize the distribution uniquely. Using (2) and (3), the reversed failure (or reversed hazard) rate function of the GQHRTPM is given by

2 3 1 + + 2 3 2 3 1 2 3 2 + + + + 𝑏𝑏 𝑐𝑐 𝛼𝛼 + + 2 3 1 2 3 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � ( ) 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜆𝜆� −𝑒𝑒 � ( ) = = −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � , (52) ( ) 𝜆𝜆𝜆𝜆 �𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑥𝑥 �𝑒𝑒 � − 𝑒𝑒 + 2+ 3 � 𝑒𝑒 𝑓𝑓 𝑦𝑦 1 2 3 𝑌𝑌 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝐹𝐹 𝑦𝑦 −�𝑎𝑎𝑎𝑎 𝑥𝑥 𝑥𝑥 � 1 𝑟𝑟ℎ 𝑦𝑦 ⎡ 𝜆𝜆� −𝑒𝑒 � ⎤ ⎢ ⎥ ⎢𝑒𝑒 − ⎥ th ⎢ ⎥ The r moment of the reversed residual life function can be⎣ obtained by the formula⎦ 1 µ ( ) = [( ) ] = ( ) ( ) , ( ) 0 (53) 𝑟𝑟 𝑡𝑡 𝑟𝑟 Hence, 𝑟𝑟 𝑡𝑡 𝐸𝐸 𝑌𝑌 − 𝑡𝑡 𝑌𝑌 ≤ 𝑡𝑡 𝐹𝐹 𝑡𝑡 ∫ 𝑡𝑡 − 𝑦𝑦 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑑𝑑 1 + 1 ( 1) + + + +1 ( +1) + µ ( ) = ( ) = = = = =0 𝑘𝑘 𝑠𝑠 𝑖𝑖 !𝑗𝑗 ! ! 𝑟𝑟 2−𝑠𝑠 3 𝑛𝑛 𝑖𝑖 1𝑗𝑗 𝑖𝑖 𝑗𝑗 ∞ − 𝛼𝛼 𝑡𝑡 𝜆𝜆 𝑏𝑏 𝑐𝑐 𝑘𝑘 𝑟𝑟 𝑛𝑛 𝑘𝑘 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑛𝑛𝑛𝑛 𝛼𝛼 − 𝑟𝑟 𝑖𝑖 𝑗𝑗 𝜆𝜆 𝑡𝑡 𝐹𝐹 𝑡𝑡 (∑ +2 +3 + � +2 +3 +1 +� � �+2 +3 +2)𝑛𝑛 𝑗𝑗 (𝑖𝑖 +1) �𝑒𝑒 −, � 0 𝑠𝑠 (54) 𝑡𝑡 𝑘𝑘 ( +1) 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑠𝑠 𝑖𝑖 𝑗𝑗 − 𝑘𝑘 𝑎𝑎𝑎𝑎 The series expansions of ∫ 𝑎𝑎𝑦𝑦 , 𝑏𝑏𝑦𝑦 𝑐𝑐𝑦𝑦 𝑒𝑒 𝑑𝑑𝑑𝑑 − 𝑘𝑘 𝑎𝑎𝑎𝑎 ( 1) ( +1) ( +1) = , 𝑒𝑒 =0 𝑚𝑚 ! 𝑚𝑚 𝑚𝑚 − 𝑘𝑘 𝑎𝑎𝑎𝑎 ∞ − � 𝑘𝑘 𝑎𝑎� 𝑦𝑦 We get 𝑒𝑒 ∑𝑚𝑚 𝑚𝑚 1 + ( 1) + + + + +1 ( +1) + + µ ( ) = ( ) ======0 𝑘𝑘 𝑠𝑠 𝑖𝑖 𝑗𝑗 𝑚𝑚! ! 𝑟𝑟! −!𝑠𝑠 2𝑛𝑛 3 𝑚𝑚 𝑖𝑖 1 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑚𝑚 ∞ − 𝛼𝛼 𝑡𝑡 𝜆𝜆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑘𝑘 𝑟𝑟 +𝑛𝑛 +𝑘𝑘2 +𝑠𝑠3 𝑖𝑖+1𝑗𝑗 𝑚𝑚 + 𝑛𝑛+𝑛𝑛2 +3 𝛼𝛼+2− 𝑟𝑟 +𝑟𝑟 +2 +3 +3 𝑖𝑖 𝑗𝑗 𝜆𝜆 𝑡𝑡 𝐹𝐹 𝑡𝑡 ∑ + � + � � � . 𝑚𝑚 𝑛𝑛 𝑗𝑗 𝑖𝑖 �𝑒𝑒 − � (55) +𝑠𝑠 𝑚𝑚+2 +𝑖𝑖3 +𝑗𝑗 1 +𝑠𝑠 𝑚𝑚+2 +𝑖𝑖 3 𝑘𝑘𝑗𝑗+2 +𝑠𝑠 𝑠𝑠𝑚𝑚+2 +𝑖𝑖 3 𝑗𝑗+3 𝑎𝑎 𝑡𝑡 𝑏𝑏 𝑡𝑡 𝑐𝑐 𝑡𝑡 The mean and second moment� 𝑠𝑠 𝑚𝑚 of𝑖𝑖 the𝑗𝑗 reversed𝑠𝑠 𝑚𝑚 residual𝑖𝑖 𝑗𝑗 life𝑠𝑠 of𝑚𝑚 the𝑖𝑖 GQHR𝑗𝑗 � TPM distribution can be obtained by setting r = 1 and 2 in (55). Also, using µ1( ) and µ2( ) one can obtain the variance of the reversed residual life function of the GQHRTPM distribution. 𝑡𝑡 𝑡𝑡 8. Bonferroni and Lorenz Curves Study of income inequality has gained a lot of importance over the last many years. Lorenz curve and the associated Gini index are undoubtedly the most popular indices of income inequality. However, there are certain measures which despite possessing interesting characteristics have not been used often for measuring inequality. Bonferroni curve and scaled total time on test transform are two such measures, which have the advantage of being represented graphically in the unit square and can also be related to the Lorenz curve and Gini ratio (Giorgi (1988)). These two measures have some applications in reliability and life testing as well (Giorgi and Crescenzi (2001)). The Bonferroni and Lorenz curves and Gini index have many applications not only in economics to study income and poverty, but also in other fields like reliability, medicine and insurance. For a random variable X with cdf F(.), the Bonferroni curve is given by 1 [ ( )] = ( ) . (56) µ ( ) 0 𝑥𝑥 From the relationship between the Bonferroni 𝐵𝐵curve𝐹𝐹 𝐹𝐹 𝑥𝑥and reversed𝐹𝐹 𝑥𝑥 ∫ residual𝑦𝑦 𝑓𝑓 𝑦𝑦 life𝑑𝑑𝑑𝑑 function of the GQHRTPM distribution, the Bonferroni curve of the GQHRTPM distribution is given by

International Journal of Statistics and Applications 2015, 5(6): 302-316 311

1 + 1 ( 1) + + + +1 ( +1) + + [ ( )] = µ ( ) = = = = =0 𝑘𝑘 𝑖𝑖 𝑗𝑗 !𝑚𝑚 ! !𝑛𝑛 ! 2 𝑚𝑚3 𝑖𝑖 𝑗𝑗 1 𝑖𝑖 𝑗𝑗 𝑚𝑚 ∞ − 𝛼𝛼𝜆𝜆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑘𝑘 𝐹𝐹 +2𝑛𝑛+3𝑘𝑘+2𝑖𝑖 𝑗𝑗 𝑚𝑚 +2𝑛𝑛+𝑛𝑛3 +3 𝛼𝛼 − +2 +3 +4 𝑖𝑖 𝑗𝑗 𝜆𝜆 𝐵𝐵 𝐹𝐹 𝑥𝑥 𝐹𝐹 𝑥𝑥 ∑ + � + � . 𝑚𝑚 𝑛𝑛 𝑗𝑗 𝑖𝑖 � 𝑒𝑒 − � (57) +𝑚𝑚2 +𝑖𝑖3 +𝑗𝑗 2 +𝑚𝑚2 +𝑖𝑖3 +𝑖𝑖 3 𝑘𝑘 +𝑚𝑚2 +𝑖𝑖3 +𝑖𝑖 4 𝑎𝑎 𝑥𝑥 𝑏𝑏 𝑥𝑥 𝑐𝑐 𝑥𝑥 where μ is the mean of the GQHR�TPM𝑚𝑚 𝑖𝑖 distribution.𝑗𝑗 𝑚𝑚 𝑖𝑖 𝑗𝑗 𝑚𝑚 𝑖𝑖 𝑖𝑖 � The scaled total time on test transform of a distribution function F is defined by 1 [ ( )] = ( ) . (58) µ 0 𝑡𝑡 If ( ) denotes the cdf of the GQHRTPM distribution𝑆𝑆𝐹𝐹 𝐹𝐹 𝑡𝑡 then ∫ 𝑅𝑅 𝑦𝑦 𝑑𝑑𝑑𝑑

+ 2+ 3 𝐹𝐹 𝑡𝑡 1 1 2 3 [ ( )] 𝑏𝑏 𝑐𝑐 𝛼𝛼 = 0 −�𝑎𝑎𝑎𝑎 𝑦𝑦 𝑦𝑦 � , µ 1 𝜆𝜆� −𝑒𝑒 � 𝑡𝑡 𝜆𝜆 𝜆𝜆 𝑆𝑆𝐹𝐹 𝐹𝐹 𝑡𝑡 �𝑒𝑒 − � ∫ �𝑒𝑒 − 𝑒𝑒 � 𝑑𝑑𝑑𝑑 Hence, 1 ( 1) + + + + + +2 +3 +1 [ ( )] = . (59) µ 1 = = = = =0 𝑘𝑘 𝑖𝑖 𝑗𝑗! 𝑚𝑚! ! 𝑛𝑛! 2𝑚𝑚 3𝑖𝑖 (𝑗𝑗 +𝑖𝑖 2𝑗𝑗 +𝑚𝑚3 +𝑚𝑚1) 𝑖𝑖 𝑗𝑗 𝜆𝜆 ∞ − 𝜆𝜆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑘𝑘 𝑡𝑡 𝜆𝜆 𝑛𝑛𝑛𝑛 1 𝑖𝑖 𝑗𝑗 𝑆𝑆𝐹𝐹 𝐹𝐹 𝑡𝑡 �𝑒𝑒 − � �𝑡𝑡𝑡𝑡 − ∑𝑛𝑛 𝑘𝑘 𝑖𝑖 𝑗𝑗 𝑚𝑚 � �= 𝑚𝑚[ 𝑛𝑛( 𝑗𝑗)]𝑖𝑖 ( ) 𝑚𝑚 𝑖𝑖 𝑗𝑗 � The cumulative total time can be obtained by using formula 𝑘𝑘 0 and the Gini index can be derived from the relationship = 1 . 𝐶𝐶𝐹𝐹 ∫ 𝑆𝑆𝐹𝐹 𝐹𝐹 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 𝐺𝐺 − 𝐶𝐶𝐹𝐹 9. Parameters Estimation

9.1. Maximum Likelihood Estimates In this section we discuss the maximum likelihood estimators (MLE’s) and inference for the GQHRTPM( , , , , , ) distribution. Let 1, 2, … … , be a random sample of size from GQHRTPM( , , , , , ) then the can be written as 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑥𝑥 𝑋𝑋 𝑋𝑋 𝑋𝑋𝑛𝑛 𝑛𝑛 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑥𝑥 = =1 ( ; , , , , ) (60) Substituting from (3) into (60), we get 𝑛𝑛 𝐿𝐿 ∏𝑖𝑖 𝑓𝑓 𝑥𝑥𝑖𝑖 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 1 + 2+ 3 = 1 ( + + 2) 1 2 3 =1 =1 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝑛𝑛 𝑛𝑛 𝜆𝜆 −𝑛𝑛 𝑛𝑛 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 + 2+ 3 𝐿𝐿 𝜆𝜆 𝛼𝛼 �𝑒𝑒 − �2 ∏3 𝑎𝑎 𝑏𝑏1 𝑥𝑥 𝑐𝑐𝑥𝑥2 ∏3 � − 𝑒𝑒 � + + =1 𝛼𝛼 =1 2 3 𝑏𝑏 𝑐𝑐 . 𝑏𝑏 𝑐𝑐 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � (61) 𝑛𝑛 𝜆𝜆 ∑𝑖𝑖 � −𝑒𝑒 � − ∑𝑖𝑖 �𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � The log-likelihood function becomes 𝑒𝑒 𝑒𝑒 ln = + 2+ 3 ln + ln ln 1 + ln( + + 2) + 2 + 3 + 1 2 3 𝐿𝐿 =1 =1 2 3 =1 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝑛𝑛 𝑛𝑛 𝑏𝑏 𝑐𝑐 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � + 2+ 𝑖𝑖 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 +𝑛𝑛( 𝜆𝜆 1)𝑛𝑛 𝛼𝛼ln− 𝑛𝑛1 �𝑒𝑒 − �2 ∑3 , 𝑎𝑎 𝑏𝑏𝑥𝑥 𝑐𝑐𝑥𝑥 − ∑ �𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � 𝜆𝜆 ∑ � − 𝑒𝑒 � =1 𝑏𝑏 𝑐𝑐 (62) 𝑛𝑛 −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � Therefore,𝛼𝛼 − ∑ the𝑖𝑖 normal� − equations𝑒𝑒 are � ln = 𝜕𝜕 𝐿𝐿 1 + 2+ 3 1 + 2+ 3 + 2+ 3 2 3 𝜕𝜕𝜕𝜕 2 3 2 3 =1 2 =1 + =1 1 + ( 1) =1 𝑏𝑏 𝑐𝑐 , + + 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 𝑛𝑛 𝑛𝑛 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑛𝑛 1𝑥𝑥𝑖𝑖𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ �𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 � ∑ ∑ ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � ln − 𝑥𝑥 𝜆𝜆𝜆𝜆 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � 𝛼𝛼 − � −𝑒𝑒 � = 𝜕𝜕 𝐿𝐿 2 3 2 1 2 + + + 2+ 3 + 2+ 3 ( 1) 2 3 𝜕𝜕𝜕𝜕 2 2 3 2 3 =1 2 =1 + =1 1 + =1 𝑏𝑏 𝑐𝑐 , + + 2 2 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 2 −�𝑎𝑎𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 𝑛𝑛 𝑥𝑥𝑖𝑖 𝑛𝑛 𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ �𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 � ∑ ∑ ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � ln − 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � � −𝑒𝑒 � = 𝜕𝜕 𝐿𝐿 2 3 2 3 1 3 + + + 2+ 3 + 2+ 3 ( 1) 2 3 𝜕𝜕𝜕𝜕 3 2 3 2 3 =1 2 =1 + =1 1 + =1 𝑏𝑏 𝑐𝑐 , + + 3 3 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 3 −�𝑎𝑎𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 𝑛𝑛 𝑥𝑥𝑖𝑖 𝑛𝑛 𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 �𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ∑ − ∑ ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � ∑ � −𝑒𝑒 𝑖𝑖 𝑖𝑖 �

312 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution

ln + 2+ 3 + 2+ 3 + 2+ 3 = + 1 2 3 ln 1 2 3 + ln 1 2 3 , =1 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝑏𝑏 𝑐𝑐 =1 𝑏𝑏 𝑐𝑐 𝜕𝜕 𝐿𝐿 𝑛𝑛 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝜕𝜕ln𝜕𝜕 𝛼𝛼 ∑𝑖𝑖 � + �2+ 3� � ∑𝑖𝑖 � � = 𝜆𝜆 + − 𝑒𝑒 1 2 3 −. 𝑒𝑒 − 𝑒𝑒 𝜆𝜆1 =1 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜕𝜕 𝐿𝐿 𝑛𝑛 𝑛𝑛𝑒𝑒 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝜆𝜆 The𝜕𝜕𝜕𝜕 likelihood𝜆𝜆 − �𝑒𝑒 − equations� ∑𝑖𝑖 �can− be𝑒𝑒 obtained by setting� the first partial derivatives of ln w.r.t. the unknown parameters to zero's. That is, the likelihood equations are: 𝐿𝐿 1 + 2+ 3 1 + 2+ 3 + 2+ 3 2 3 2 3 2 3 =1 2 =1 + =1 1 + ( 1) =1 𝑏𝑏 𝑐𝑐 = 0, + + 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 𝑛𝑛 𝑛𝑛 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑛𝑛 1𝑥𝑥𝑖𝑖𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ �𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 � ∑ ∑ ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � − 𝑥𝑥 𝜆𝜆𝜆𝜆 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � 𝛼𝛼 − � −𝑒𝑒 � (63) 2 3 2 1 2 + + + 2+ 3 + 2+ 3 ( 1) 2 3 2 2 3 2 3 =1 2 =1 + =1 1 + =1 𝑏𝑏 𝑐𝑐 = 0,(64) + + 2 2 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 2 −�𝑎𝑎𝑥𝑥𝑖𝑖+ 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 𝑛𝑛 𝑥𝑥𝑖𝑖 𝑛𝑛 𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ �𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 � − ∑ ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � � −𝑒𝑒 2 3 � 2 3 1 3 + + + 2+ 3 + 2+ 3 ( 1) 2 3 3 2 3 2 3 =1 2 =1 + =1 1 + =1 𝑏𝑏 𝑐𝑐 (65) + + 3 3 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 3 −�𝑎𝑎𝑥𝑥𝑖𝑖+ 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 𝑛𝑛 𝑥𝑥𝑖𝑖 𝑛𝑛 𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 �𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ∑ − ∑ ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � ∑ � −𝑒𝑒 𝑖𝑖 𝑖𝑖 � + 2+ 3 + 2+ 3 + 2+ 3 + 1 2 3 ln 1 2 3 + ln 1 2 3 = 0, =1 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝑏𝑏 𝑐𝑐 =1 𝑏𝑏 𝑐𝑐 (66) 𝑛𝑛 𝑛𝑛 −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑛𝑛 −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑖𝑖 𝑖𝑖 𝛼𝛼 𝜆𝜆 ∑ � − 𝑒𝑒 + � 2+ �3 − 𝑒𝑒 � ∑ � − 𝑒𝑒 � + 1 2 3 = 0. (67) 𝜆𝜆1 =1 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝑛𝑛 𝑛𝑛𝑒𝑒 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝜆𝜆 we𝜆𝜆 −get�𝑒𝑒 the− �following∑𝑖𝑖 � system− 𝑒𝑒 of three nion�-linear equations from (63, 64 and 65)

1 1 + 2+ 3 + 2+ 3 + 2+ 3 ( 1) 2 3 2 3 2 3 𝑏𝑏 𝑐𝑐 =1 𝑗𝑗− 2 =1 𝑗𝑗 + =1 1 + =1 𝑗𝑗 −�𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � = 0. (68) + + 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝑖𝑖 + 𝑖𝑖2+ 𝑖𝑖3 𝑛𝑛 𝑥𝑥𝑖𝑖 𝑛𝑛 𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 𝑗𝑗 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ �𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 � − ∑ 𝑗𝑗 𝑗𝑗 ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � 𝑗𝑗 ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � where = 1,2,3. � −𝑒𝑒 � The normal equations do not have explicit solutions and they have to be obtained numerically. Therefore, the MLEs of , , , 𝑗𝑗 and can be obtained by solving system of three nion-linear equations with two non-linear equations.

9.2𝑎𝑎 𝑏𝑏. Asymptotic𝑐𝑐 𝛼𝛼 𝜆𝜆 Confidence Bounds Since the MLEs of the unknown parameters , , and cannot be obtained in closed forms, it is not easy to derive the exact distributions of the MLEs. In this section, we derive the asymptotic confidence intervals of these parameters when > 0, > 0, > 0, > 0 and > 0. The simplest𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 large sample𝜆𝜆 approach is to assume that the MLE , , , , are approximately multivariate normal with mean ( , , , , ) and matrix 1, see Lawless (2003), where 1 is 0 � ̂0 the𝑎𝑎 inverse𝑏𝑏 of the𝑐𝑐 observed𝛼𝛼 information𝜆𝜆 matrix − �𝑎𝑎� 𝑏𝑏 𝑐𝑐̂ 𝛼𝛼� 𝜆𝜆�− 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 𝐼𝐼1 𝐼𝐼 2 ln 2 ln 2 ln 2 ln 2 ln

2 − 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 2 ln 2 ln 2 ln 2 ln 2 ln 𝜕𝜕𝑎𝑎 𝜕𝜕𝜕𝜕𝜕𝜕2𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕2 𝐿𝐿 𝜕𝜕2 𝐿𝐿 𝜕𝜕2 𝐿𝐿 𝜕𝜕2 𝐿𝐿 𝜕𝜕2 𝐿𝐿 1 ⎛ ln ln ln ln ln ⎞ 0 = 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝑏𝑏 𝜕𝜕𝜕𝜕𝜕𝜕2𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 (69) ⎜𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿⎟ − ⎜ 2 ln 2 ln 2 ln 2 ln 2 ln ⎟ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝑐𝑐 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝐼𝐼 ⎜ 2 ⎟ 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 ⎜ 2 ln 2 ln 2 ln 2 ln 2 ln ⎟ ⎜ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝛼𝛼 𝜕𝜕𝜕𝜕𝜕𝜕2𝜕𝜕 ⎟ 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 𝜕𝜕 𝐿𝐿 Thus, ⎝ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜆𝜆 ⎠ ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( , ) 1 𝑉𝑉𝑉𝑉𝑉𝑉 𝑎𝑎� 𝐶𝐶𝐶𝐶𝐶𝐶 𝑎𝑎� 𝑏𝑏� 𝐶𝐶𝐶𝐶𝐶𝐶 𝑎𝑎� 𝑐𝑐̂ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑎𝑎� 𝛼𝛼� 𝐶𝐶𝐶𝐶𝐶𝐶 𝑎𝑎� 𝜆𝜆̂ 0 = ( , ) ( , ) ( ) ( , ) ( , ) (70) 𝐶𝐶𝐶𝐶𝐶𝐶 𝑏𝑏� 𝑎𝑎� 𝑉𝑉𝑉𝑉𝑉𝑉 𝑏𝑏� 𝐶𝐶𝐶𝐶𝐶𝐶 𝑏𝑏� 𝑐𝑐̂ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑏𝑏� 𝛼𝛼� 𝐶𝐶𝐶𝐶𝐶𝐶 𝑏𝑏� 𝜆𝜆̂ − ⎛ ( , ) ( , ) ( , ) ( ) ( , )⎞ 𝐼𝐼 ⎜𝐶𝐶𝐶𝐶𝐶𝐶 𝑐𝑐̂ 𝑎𝑎� 𝐶𝐶𝐶𝐶𝐶𝐶 𝑐𝑐 𝑏𝑏� 𝑉𝑉𝑉𝑉𝑉𝑉 𝑐𝑐̂ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑐𝑐̂ 𝛼𝛼� 𝐶𝐶𝐶𝐶𝐶𝐶 𝑐𝑐̂ 𝜆𝜆̂ ⎟ ⎜ ( , ) ( , ) ( , ) ( , ) ( ) ⎟ 𝐶𝐶𝐶𝐶𝐶𝐶 𝛼𝛼� 𝑎𝑎� 𝐶𝐶𝐶𝐶𝐶𝐶 𝛼𝛼� 𝑏𝑏� 𝐶𝐶𝐶𝐶𝐶𝐶 𝛼𝛼� 𝑐𝑐̂ 𝑉𝑉𝑉𝑉𝑉𝑉 𝛼𝛼� 𝐶𝐶𝐶𝐶𝐶𝐶 𝛼𝛼� 𝜆𝜆̂ The derivatives in are given as follows: 0 ⎝𝐶𝐶𝐶𝐶𝐶𝐶 𝜆𝜆̂ 𝑎𝑎� 𝐶𝐶𝐶𝐶𝐶𝐶 𝜆𝜆̂ 𝑏𝑏� 𝐶𝐶𝐶𝐶𝐶𝐶 𝜆𝜆̂ 𝑐𝑐̂ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜆𝜆̂ 𝑎𝑎� 𝑉𝑉𝑉𝑉𝑉𝑉 𝜆𝜆̂ ⎠ 𝐼𝐼

International Journal of Statistics and Applications 2015, 5(6): 302-316 313

2 3 2 2 3 2 3 1 + + ln 1 2 + + + + ( 1) 2 3 2 3 2 3 𝑏𝑏 𝑐𝑐 2 = =1 2 =1 1 1 + + 2 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥+𝑖𝑖 2𝑥𝑥+𝑖𝑖 3𝑥𝑥𝑖𝑖 � 𝜕𝜕 𝐿𝐿 𝑛𝑛 − 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼1− 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝜕𝜕𝑎𝑎 + 2+ 3 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ∑ �𝑎𝑎 2𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 2� −3𝜆𝜆𝜆𝜆 ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 − � − 𝑏𝑏 𝑐𝑐 ( 1) =1 −�𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � 2, 𝑖𝑖+ 2𝑖𝑖+ 3𝑖𝑖 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝛼𝛼 − ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � � −𝑒𝑒 � 2 3 2 2 2 3 2 3 1 + + ln 4 + + + + ( 1) 2 3 2 3 2 3 𝑏𝑏 𝑐𝑐 2 = =1 2 + =1 1 1 + + 2 4 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥+𝑖𝑖 2𝑥𝑥+𝑖𝑖 3𝑥𝑥𝑖𝑖 � 𝜕𝜕 𝐿𝐿 𝑛𝑛 −𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼1− 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝜕𝜕𝑏𝑏 + 2+ 3 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ( 1) ∑ 4�𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 2𝑐𝑐𝑥𝑥𝑖𝑖 �3 ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 − � − 𝑏𝑏 𝑐𝑐 =1 −�𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � 2, 4 𝑖𝑖+ 2𝑖𝑖+ 3𝑖𝑖 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � � −𝑒𝑒 � 2 3 2 4 2 3 2 3 1 + + ln 6 + + + + ( 1) 2 3 2 3 2 3 𝑏𝑏 𝑐𝑐 2 = =1 2 + =1 1 1 + + 2 9 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥+𝑖𝑖 2𝑥𝑥+𝑖𝑖 3𝑥𝑥𝑖𝑖 � 𝜕𝜕 𝐿𝐿 𝑛𝑛 −𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼1− 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝜕𝜕𝑐𝑐 + 2+ 3 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ( 1) ∑ 6�𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 2𝑐𝑐𝑥𝑥𝑖𝑖 �3 ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 − � − 𝑏𝑏 𝑐𝑐 =1 −�𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � 2, 9 𝑖𝑖+ 2𝑖𝑖+ 3𝑖𝑖 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 � 2 2 ln + 2+ 3 + 2+ 3 = + 1 2 3 ln 1 2 3 , 2 2 =1 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝑏𝑏 𝑐𝑐 𝜕𝜕 𝐿𝐿 −𝑛𝑛 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 2 𝜕𝜕𝛼𝛼ln 𝛼𝛼 ∑ 𝑖𝑖 � � � � = 𝜆𝜆 2 , − 𝑒𝑒 − 𝑒𝑒 2 2 𝜆𝜆 𝜕𝜕 𝐿𝐿 −𝑛𝑛 𝑛𝑛 𝑒𝑒1 2 ln 𝜆𝜆 𝜕𝜕𝜆𝜆 = 𝜆𝜆 − �𝑒𝑒 − � 𝜕𝜕 𝐿𝐿 1 + 2+ 3 + 2+ 3 + 2+ 3 ( 1) 2 3 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 3 2 3 2 3 =1 2 + =1 1 𝑏𝑏 𝑐𝑐 1 + + 2 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥+𝑖𝑖 2𝑥𝑥+𝑖𝑖 3𝑥𝑥𝑖𝑖 � 𝑛𝑛 −𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼1− 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 2 3𝑖𝑖 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥3𝑖𝑖 + + −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ∑( 1) ∑2 3𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 − � − 𝑏𝑏 𝑐𝑐 =1 −�𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � 2, 2 𝑖𝑖+ 2𝑖𝑖+ 3𝑖𝑖 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 2 ln � −𝑒𝑒 � = 𝜕𝜕 𝐿𝐿 2 3 2 1 + + + 2+ 3 + 2+ 3 ( 1) 2 3 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 4 2 3 2 3 =1 2 + =1 1 𝑏𝑏 𝑐𝑐 1 + + 3 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥+𝑖𝑖 2𝑥𝑥+𝑖𝑖 3𝑥𝑥𝑖𝑖 � 𝑛𝑛 −𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼1− 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 + 2+ 3 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ∑( 1) 4 ∑2 3𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 − � − 𝑏𝑏 𝑐𝑐 =1 −�𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � 2, 3 𝑖𝑖+ 2𝑖𝑖+ 3𝑖𝑖 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 � + 2+ 3 2 ln 2 3 𝑏𝑏 𝑐𝑐 = 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � + 2+ 3 =1 1 𝑖𝑖 2 3 𝜕𝜕 𝐿𝐿 𝑥𝑥 𝑒𝑒 𝑏𝑏 𝑐𝑐 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 1 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑖𝑖 + 2+ 3 + 2+ 3 + 2+ 3 + 𝑛𝑛 2 3 1 2 3 ln 1 2 3 + 1 , − 𝑒𝑒 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝑏𝑏 𝑐𝑐 =1 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑖𝑖 2 𝜆𝜆 � 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 1 � �𝛼𝛼 � − 𝑒𝑒 � � ln 𝑖𝑖 + 2+ 3 + 2+ 3 = 2 3 1 2 3 , =1 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜕𝜕 𝐿𝐿 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 2 ln 𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 = 𝛼𝛼 ∑𝑖𝑖 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � 𝜕𝜕 𝐿𝐿 2 3 3 1 + + + 2+ 3 + 2+ 3 ( 1) 2 3 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 5 2 3 2 3 =1 2 + =1 1 𝑏𝑏 𝑐𝑐 1 + + 6 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− −�𝑎𝑎𝑥𝑥+𝑖𝑖 2𝑥𝑥+𝑖𝑖 3𝑥𝑥𝑖𝑖 � 𝑛𝑛 −𝑥𝑥𝑖𝑖 𝜆𝜆𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝛼𝛼1− 𝑒𝑒 2 3 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏𝑥𝑥𝑖𝑖 𝑐𝑐𝑥𝑥𝑖𝑖 + 2+ 3 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ∑( 1) 5 ∑2 3𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � � −𝑒𝑒 𝑖𝑖 𝑖𝑖 − � − 𝑏𝑏 𝑐𝑐 =1 −�𝑎𝑎𝑥𝑥 𝑥𝑥 𝑥𝑥 � 2, 6 𝑖𝑖+ 2𝑖𝑖+ 3𝑖𝑖 𝛼𝛼− 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝑖𝑖 𝑏𝑏 𝑐𝑐 ∑ −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � � −𝑒𝑒 2 � 3 2 + + 1 2 ln 1 2 3 + 2+ 3 + 2+ 3 + 2+ 3 2 2 3 2 3 2 3 = =1 𝑏𝑏 𝑐𝑐 + =1 1 ln 1 + 1 , 2 −�𝑎𝑎𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 2 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝑏𝑏 𝑐𝑐 𝜕𝜕 𝐿𝐿 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝑖𝑖 𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥 � ∑ −𝑒𝑒 𝑖𝑖 𝑖𝑖 ∑ 𝑥𝑥 𝑒𝑒 � − 𝑒𝑒 � �𝛼𝛼 � − 𝑒𝑒 � �

314 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution

1 2 ln + 2+ 3 + 2+ 3 = 2 2 3 1 2 3 , 2 =1 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜕𝜕 𝐿𝐿 𝛼𝛼 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 2 3 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑖𝑖 3𝑖𝑖 + + 1 2 ln 1 ∑ 𝑥𝑥 𝑒𝑒 2 3 � − 𝑒𝑒 + 2+� 3 + 2+ 3 + 2+ 3 3 2 3 2 3 2 3 = =1 𝑏𝑏 𝑐𝑐 + =1 1 ln 1 + 1 , 3 −�𝑎𝑎𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖2+ 𝑥𝑥𝑖𝑖3� 3 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝑏𝑏 𝑐𝑐 𝜕𝜕 𝐿𝐿 𝑛𝑛 1𝑥𝑥𝑖𝑖 𝑒𝑒 2 3 𝜆𝜆 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝑖𝑖 𝑏𝑏 𝑐𝑐 𝑖𝑖 𝑖𝑖 𝜕𝜕2𝜕𝜕𝜕𝜕𝜕𝜕 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥 2 𝑥𝑥 � 3 2 3 1 ln ∑ −𝑒𝑒 + 𝑖𝑖 +𝑖𝑖 ∑ 𝑥𝑥 𝑒𝑒 + + � − 𝑒𝑒 � �𝛼𝛼 � − 𝑒𝑒 � � = 3 2 3 1 2 3 , 3 =1 𝑏𝑏 𝑐𝑐 𝑏𝑏 𝑐𝑐 𝛼𝛼− 𝜕𝜕 𝐿𝐿 𝛼𝛼 𝑛𝑛 −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � 𝜕𝜕2𝜕𝜕ln𝜕𝜕𝜕𝜕 ∑𝑖𝑖 𝑖𝑖 + 2+ �3 +� 2+ 3 = 𝑥𝑥1𝑒𝑒 2 3 − 𝑒𝑒ln 1 2 3 . 3 =1 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝑏𝑏 𝑐𝑐 𝜕𝜕 𝐿𝐿 𝛼𝛼 𝑛𝑛 −�𝑎𝑎 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � −�𝑎𝑎𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 � The𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 above∑ approach𝑖𝑖 � − 𝑒𝑒is used to derive� the �100−(1𝑒𝑒 )% confidence� intervals of the parameters , , , and as in the following forms − 𝛾𝛾 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 ± 2 ( ) , ± 2 , ± 2 ( ) , ± 2 ( ) , ± 2 (71)

𝛾𝛾⁄ th 𝛾𝛾⁄ 𝛾𝛾⁄ 𝛾𝛾⁄ 𝛾𝛾⁄ Here, 𝑎𝑎� 2 𝑍𝑍is the� 𝑉𝑉upper𝑉𝑉𝑉𝑉 𝑎𝑎� ( 𝑏𝑏�2) 𝑍𝑍�𝑉𝑉𝑉𝑉𝑉𝑉 �of𝑏𝑏�� the𝑐𝑐 ̂ standard𝑍𝑍 � 𝑉𝑉normal𝑉𝑉𝑉𝑉 𝑐𝑐̂ distribution.𝛼𝛼� 𝑍𝑍 � 𝑉𝑉It𝑉𝑉𝑉𝑉 should𝛼𝛼� 𝜆𝜆bê mentioned𝑍𝑍 �𝑉𝑉𝑉𝑉𝑉𝑉 �here𝜆𝜆̂� as it was pointed by a referee that if we do not make the assumption that the true parameter vector ( , , , , ) is an interior point of 𝛾𝛾⁄ the parameter𝑍𝑍 space then the𝛾𝛾 ⁄asymptotic normality results will not hold. If any of the true parameter value is 0, then the asymptotic distribution of the maximum likelihood estimators is a mixture distribution, see for𝑎𝑎 𝑏𝑏 example𝑐𝑐 𝛼𝛼 𝜆𝜆 Self and Liang (1987) in this connection. In that case obtaining the asymptotic confidence intervals becomes quite difficult and it is not pursued here.

10. Application Here, we illustrate applicability of the GQHRTPM distribution using real data sets from Smith and Naylor (1987) represents the strengths of 1.5cm glass fibres, measured at the National Physical Laboratory, England and they are taken. We compare the fit of the GQHRTPM distribution with those of the Exponential Poisson (EP), Generalized Exponential Poisson (GEP) and the Exponentiated Exponential Poisson (EEP) distributions. For each distribution, the unknown parameters are estimated by the method of maximum likelihood. The maximum likelihood estimates and the corresponding AIC and BIC values are shown in Tables 1. We can see that the smallest AIC and BIC are obtained for the GQHRTPM distribution.

Table 1. Estimated parameters, AIC and BIC for the real data set

Distribution Parameter AIC BIC

EP(a, ) a = 0.0007, = 983.5960 181.7 186.0

EG (a,𝜆𝜆 ) a� = 0.6636, �λ = 1 × 10 7 181.7 186.0 − PE (a,𝜆𝜆) a� = 2.6566, �λ = 34.5193 65.0 69.3

𝜆𝜆 a� = 0.0094, �λ = 31.9563, GEP(a, , ) 69.1 75.5 � = 280.6076�α 𝛼𝛼 𝜆𝜆 aλ� = 2.05, b = 5.451,

GQHRTPM(a, , , , ) c� = 1.035,� =1.053− , 55.91 66.63

𝑏𝑏 𝑐𝑐 𝛼𝛼 𝜆𝜆 � = 0.05025�α

λ� So, we can conclude that the GQHRTPM distribution is lifetime sub-models such as: EP, EG, PE and GEP. Various the most appropriate model for the data sets among the properties of the proposed distribution are discussed in considered distributions. Section 3, 4 and 5. Rényi and Shannon entropies of the GQHRTPM distribution are given in Section 6. Residual and reverse residual life functions of the GQHRTPM distribution 11. Conclusions are discussed in Section 7. Section 8 is devoted to the We introduce a new five-parameter distribution called the Bonferroni and Lorenz curves of the GQHRTPM generalized quadratic hazard rate truncated poisson distribution. The maximum likelihood estimation procedure (GQHRTPM) distribution. This distribution contains several is presented. Fitting the GQHRTPM model to real data set

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indicate the flexibility and capacity of the proposed [15] Kundu, C. and Nanda, A.K. (2010). "Some reliability distribution in data modeling. In view of the density and properties of the inactivity time". Communications in Statistics-Theory and Methods 39, 899-911. failure rate function shapes, it seems that the proposed model can be considered as a suitable candidate model in reliability [16] Kus, C. (2007). "A new lifetime distribution". Computational analysis, biological systems, data modeling, and related Statistics and Data Analysis, 51(9), 4497-4509. fields. [17] Lawless, J. F. (2003). "Statistical Models and Methods for Lifetime Data". John Wiley and Sons, New York. [18] Lomax, K. S. (1954). "Business failure: another example of the analysis of failure data". Journal of the Americal REFERENCES Statistical Association, 49, 847-852. [19] Marshall, A.W. and Proschan, F. (1965). "Maximum [1] Adamidis, K. and Loukas, S. (1998). "A lifetime distribution likelihood estimates for distributions with monotone failure with decreasing failure rate". Statistics and Probability Letters, rate". Annals of , 36, 69-77. 39, 35-42. [20] McNolty, F., Doyle, J. and Hansen, E. (1980). "Properties of [2] Barlow, R. E. and Marshall, A. W. (1964). "Bounds for the mixed exponential failure process". Technometrics, 22, distribution with monotone hazard rate I and II". Annals of 555-565. Mathematical Statistics, 35, 1234-1274. [21] Mi, J. (1995). "Bathtub failure rate and upside-down bathtub [3] Barlow, R. E. and Marshall, A. W. (1965). "Tables of bounds mean residual life". IEEE Transactions on Reliability 44, for distribution with monotone hazard rate". Journal of the 388-391. Americal Statistical Association, 60, 872-890. [22] Mudholkar, G. S. and Srivastava, D.K. (1993). [4] Barlow, R. E., Marshall, A. W. and Proschan, P. (1963). "Exponentiated Weibull family for analysing bathtub failure "Properties of probability distributions with monotone hazard rate data". IEEE Transactions on Reliability, 42(2), 299–302. rate". Annals of Mathematical Statistics, 34, 375-389. [23] Nadarajah, S. and Kotz, S. (2006). "The exponentiated type [5] Bebbington, M. S., Lai, C. D. and Zitikis, R. (2007). "A distributions". Acta Applicandae Mathematicae, 92, 97-111. flexible Weibull extension". and System Safety, 92(6), 719–26. [24] Nanda, A. K., Singh, H., Misra, N. and Paul, P. (2003). "Reliability properties of reversed residual lifetime". [6] Cozzolino, J. M. (1968). "Probabilistic models of decreasing Communications in Statistics-Theory and Methods, 32, failure rare processes". Naval Research Logistics Quarterly, 2031-2042. 15, 361-374. [25] Nassar, M. M. (1988). "Two properties of mixtures of [7] Dahiya, R. C. and Gurland, J. (1972). " tests exponential distributions". IEEE Transactions on Reliability, for the gamma and exponential distributions". Technometrics, 37(4), 383-385. 14, 791-801. [26] Park, K. S. (1985). "Effect of burn-in on mean residual life". [8] Ghitany, M.E. (1998) "On a recent generalization of gamma IEEE Transactions on Reliability, 34, 522-523. distribution". Communications in Statistics-Theory and Methods, 27, 223-233. [27] Proschan, F. (1963). "Theoretical explanation of observed decreasing failure rate". Technometrics, 5, 375-383. [9] Giorgi, G. M. (1998). "Concentration index, Bonferroni". Encyclopedia of Statistical Sciences, vol. 2, Wiley, New York, [28] Sarhan, A. M. (2009)." Generalized quadratic hazard rate pp. 141-146. distribution". Inter. International Journal of Applied Mathematics and Statistics, 14(S09), 94-109. [10] Giorgi, G. M., Crescenzi, M. (2001). "A look at the Bonferroni inequality measure in a reliability framework". [29] Sarhan, A. M. and Kundu, D. (2009)."Generalized linear Statistica LXL, 4, 571-583. failure rate distribution". Communication in Statistics-Theory and Methods, In Press. [11] Gleser, L. J. (1989). "The gamma distribution as a mixture of exponential distributions". Journal of the Americal Statistical [30] Sarhan, A. M., Tadj, L. and Al-Malki, S. (2008). "Estimation Association, 43, 115-117. of the parameters of the generalized linear failure rate distribution". Bulletin of Statistics and Economics, 2, 52-63. [12] Gupta, R. D. and Kundu, D. (1999). "Generalized exponential distribution". Austral. & New Zealand Journal Statistics, [31] Saunders, S. C. and Myhre, J. M. (1983). "Maximum 41(2), 173-188. likelihood estimation for two-parameter decreasing hazard rate distributions using censored data". Journal of the [13] Gupta, P.L. and Gupta, R.C. (1983). "On the moments of Americal Statistical Association, 78, 664-673. residual life in reliability and some characterization results". Communications in Statistics-Theory and Methods 12, [32] Self, S. G. and Liang, K. Y. (1987). "Asymptotic properties of 449-461. maximum likelihood estimators and likelihood ratio tests under nonstandard conditions", Journal of the Americal [14] Gurland, J. and Sethuraman, J. (1994). "Reversal of Statistical Association, 82, 605 - 610. increasing failure rates when pooling failure data". Technometrics, 36, 416-418. [33] Smith, R. L., Naylor, J. C. (1987). "A comparison of

316 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution

maximum likelihood and Bayesian estimators for the [35] Tang, L. C., Lu, Y. and Chew, E.P. (1999) "Mean residual life three-parameter Weibull distribution". Applied Statistical, 36, distributions". IEEE Transactions on Reliability, 48, 68-73. 358-369. [36] Xie, M. and Lai, C. D. (1995). "Reliability analysis using an [34] Tahmasbi, R. and Rezaei, S. (2008). "A two-parameter additive Weibull model with bathtub-shaped failure rate lifetime distribution with decreasing failure rate". function". Reliability Engineering System Safety, 52, 87–93. Computational Statistics and Data Analysis, 52, 3889-3901.