Reliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution

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1 International Journal of Statistics and Applications 5, 5(6): -6 DOI:.59/j.statistics Reliability Analysis Using the Generalized Quadratic M. A. El-Damcese, Dina A. Ramadan,* Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt Mathematics 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. Introduction In reliability, many phenomena are modeled by statistical distributions. The probability distribution of the time -to-failure of a device can be characterized by the failure rate or hazard function. There are some parametric models that have successfully served as population models for failure times arising from a wide range of products and failure mechanisms. The distributions with Decreasing Failure Rate (DFR) property are studied in the works of Lomax (954), Proschan (96), Barlow et al. (96), Barlow and Marshall (964)-(965), Marshall and Proschan (965), Cozzolino (968), Dahiya and Gurland (97), McNolty et al. (98), Saunders and Myhre (98), Nassar (988), Gleser (989), Gurland and Sethuraman (994), Adamidis and Loukas (998), Kus (7), and Tahmasbi and Rezaei (8). For modeling the reliability and survival data with Increasing Failure Rate (IFR) property or bathtub failure rate, numerous hazard functions are proposed by different researches that most of them are based on Weibull distribution. Muldholkar and Srivastava (99) proposed an Exponentiated Weibull family for analyzing bathtub failure-rate data. A model based on adding two Weibull distributions is presented by Xie and Lai (995). Beington et al. (7) proposed a new two-parameter distribution which is a generalization of the Weibull. Recently, Gupta et al. (8) introduced another member of the Weibull family, * Corresponding author: dina_ahmed88@yahoo.com (Dina A. Ramadan) Published online at Copyright 5 Scientific & Academic Publishing. All Rights Reserved which is called as the 'flexible Weibull distribution'. Recently, many new distributions, generalizing well-known distributions used to study lifetime data, have bn introduced. Mudholkar and Srivastava (99) presented a generalization of the Weibull distribution called the exponentiated (generalized)-weibull distribution, GWD. The generalized exponential distribution, GED, introduced by Gupta and Kundu (999). Nadarajah and Kotz (6) introduced four exponentiated type distributions: the exponentiated gamma, exponentiated Weibull, exponentiated Gumbel. Sarhan and Kundu (9) presented a generalization of the linear hazard rate distribution called the generalized linear hazard rate distribution, GLFRD. Sarhan et al. (8) obtained Bayes and maximum likelihood estimates of the thr parameters of the generalized linear hazard ratedistribution based on grouped and censored data. Recently, Sarhan (9) introduced a generalization of the quadratic hazard rate distribution called the generalized quadratic hazard rate distribution (GQHRD). This paper is organized as follows: a new IFR distribution is obtained for maximum survival time by mixing generalized quadratic hazard rate and geometric distribution. Various properties of the proposed distribution are discussed in Section, 4 and 5. Rényi and Shaon entropies of the GQHRTPM distribution are given in Section 6. Residual and reverse residual life functions of the GQHRTPM distribution are discussed in Section 7. Section 8 is devoted to the Bonferroni and Lorenz curves of the GQHRTPM distribution. The maximum likelihood estimation procedure is presented. Fitting the GQHRTPM model to real data set indicate the flexibility and capacity of the proposed distribution in data modeling. In view of the density and failure rate function shapes, it sms that the proposed model

2 International Journal of Statistics and Applications 5, 5(6): -6 can be considered as a suitable candidate model in reliability analysis, biological systems, data modeling, and related fields.. The Maximum Survival Time Distribution Let YY, YY,, YY zz be a random sample from the generalized quadratic hazard rate distribution with Cumulative Density Function (cdf) α b c ay y y FY ( y) = e ; aa, cc, >, aaaa and Z is a random variable from truncate at zero Poisson distribution with probability mass function as follows: Р(ZZ = ) = λλ λλ λλ ; =,, ()! where λλ >. By assuming that the random variables YY and Z are independent and defining XX = max{ YY, YY,, YY zz } then, the marginal distribution of X, for λλ >, is FF XX (xx) = λλ λλ λλff YY (xx) = λλ λλ (aaaa xx cc xx ) () with probability density function ff XX (xx) = λλλλ λλ (aa ccxx ) (aaaa xx cc xx ) (aaaa xx cc xx ) (aaaa λλ xx cc xx ). () where aa, cc,, λλ >, aaaa. Figure. Probability density function for GQHRTPMD from different values for a, b and c. Reliability Analysis The reliability function (R) of the Generalized Quadratic Hazard Rate Truncated Poisson Maximum distribution is denoted by RR(xx) also known as the survivor function and is defined as RR(xx) = FF(xx) = λλ λλ λλ aaaa xx cc xx, (4) One of the characteristic in reliability analysis is the hazard rate function (HRF) defined by h(xx) = ff(xx) RR(xx) = λλλλ λλ (aa ccxx ) aaaa xx cc aaaa xx xx cc xx λλ aaaa xx cc xx λλ λλ aaaa xx cc, (5) xx

3 4 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic The hazard rate function is such that: if =, the hazard function is either increasing (if b > ) or constant (if b = and a >); when >, the hazard function should be: () increasing if b >; () upside-down bath-tub shaped if b <; and if <, then the hazard function will be: () decreasing if b = or () bath-tub shaped if b. Figure. 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. The cumulative hazard function of the Generalized Quadratic Hazard Rate Truncated Poisson Maximum distribution is denoted by H(x) and is defined as xx H(x) = h(tt)dddd xx = λλ λλ (aa cctt ) aaaa tt cc tt aaaa tt cc tt aaaa λλ tt cc tt aaaa λλ λλ tt cc dddd tt = ln λλ ln λλ λλ aaaa xx cc xx. (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.. The Median and Mode It is observed as expected that the mean of GQHRTPM(aa,, cc,, λλ) caot be obtained in explicit forms. It can be obtained as infinite series expansion so, in general different moments of GQHRTPM(aa,, cc,, λλ). Also, we caot get the quantile xx qq

4 International Journal of Statistics and Applications 5, 5(6): -6 5 of GQHRTPM(aa,, cc,, λλ) in a closed form by using the equation FF XX xx qq ; aa,, cc,, λλ qq =. Thus, by using Equation (), we find that aaxx qq xx qq cc xx qq = ln ln λλ qq ln λλ, < qq <. (7) The median mm(xx) of GQHRTPM(aa,, cc,, λλ) can be obtained from (7), when qq =.5, as follows aaxx.5 xx.5 cc xx.5 = ln ln λλ.5 ln λλ. (8) Moreover, the mode of GQHRTPM(aa,, cc,, λλ) can be obtained as a solution of the following nonlinear equation. dd dddd ff XX (xx; aa,, cc,, λλ) = dd dddd λλλλλλ (aa ccxx ) (aaaa xx cc xx ) (aaaa xx cc xx ) (aaaa λλ xx cc xx ) = (9) 4.. Moments The following theorem gives the rr tth moment of GQHRTPM (aa,, cc,, λλ). Theorem. If has GQHRTPM (aa,, cc,, λλ) the moment rr tth of X, say μμ rr, is given as follows for aa, cc,, λλ >, aaaa ( ) μμ rr = mm jj λλ mm cc jj ==mm =jj =! mm! jj! mm jj aa rrmm jj () rr mm jj λλ Proof Substituting () into the above relation, we get ΓΓ(rrmmjj ) cc ΓΓ(rrmmjj ) ΓΓ(rr mm jj ). () ()aa () aa μμ rr = λλλλ λλ μμ rr = Е(XX rr ) = xx rr xx rr ff(xx)dddd (aa ccxx ) aaaa xx cc xx aaaa xx cc xx aaaa λλ xx cc xx dddd, () The series expansions of λλ (aaaa xx cc xx ) is We get aaaa λλ xx cc xx λλ = λλ aaaa xx cc xx =! μμ rr = λλ = xx rr (aa ccxx ) aaaa xx cc xx! aaaa xx cc xx dddd, () Since < aaaa xx cc xx < for x, then by using the binomial series expansion of aaaa xx cc xx by aaaa xx cc xx = ( ) aaaa xx cc xx = We get μμ rr = λλ xx rr ( ) λλ ==! given (aa ccxx ) ()aaaa xx cc xx dddd, ()

5 6 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic and The series expansions of () xx and ()cc xx are Substituting (4) and (5) into (), we get ( ) mm () mm xx () xx = mm=, (4) mm! ( ) jj () cc xx jj ()cc xx = jj =, (5) μμ rr = λλ ( ) mm jj λλ () mm jj mm cc jj ==mm =jj =! mm! jj! mm jj rr mm jj xx (aa ccxx ) ()aaaa dddd, (6) The integral in (6) can be computed as follows II = (aa ccxx ) xx rr mmjj ()aaaa dddd = aa xx rr mm jj ()aaaa dddd xx rr mm jj ()aaaa dddd cc xx rr mm jj ()aaaa dddd = aa ΓΓ(rr mm jj ) ΓΓ(rrmmjj ) cc ΓΓ(rrmm jj ) (()aa) rrmm jj (()aa) rrmm jj (()aa) rrmm jj, (7) Substituting (7) into (6), we get (). This completes the proof. 4.. The Moment Generating Function The moment generating function MM XX (tt) of the GQHRTPM distribution (aa,, cc,, λλ) has the following form MM XX (tt) = Е( tttt ) = MM XX (tt) = rr===mm=jj = tt rr xx rr jj! tttt ff(xx) dddd = rr= ff(xx) dddd = rr= μμ rr. (8) rr! tt rr rr! ( ) mm jj λλ mm cc jj tt rr rr!! mm! jj! mm jj aa rrmm jj () rr mm jj λλ ΓΓ(rrmmjj ) cc ΓΓ(rrmmjj ) ΓΓ(rr mm jj ) (9) ()aa () 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 XX, XX,, XX be a simple random sample from GQHRTPM(aa,, cc,, λλ, xx) with cumulative distribution function and probability density function as in () and (), respectively. Let XX (:), XX (:),., XX (:) denote the order statistics obtained from this sample. In reliability literature, XX (:) denote the lifetime of an ( i) out of system which consists of independent and identically components. Then the pdf of XX (: ), ] is given by then, and Using ff : (xx) = В(, ) [FF(xx)] [ FF(xx)] ff(xx), () FF(xx) = HH(xx), ff(xx) = h(xx) HH(xx), () [ FF(xx)] = ( )HH(xx) () [FF(xx)] = HH(xx) = ll=oo ( ) ll ll llll(xx). () Substituting () and () into (), we get ff : (xx) = ( ) В(, ) ll=oo ll ll h(xx) (ll )HH(xx) (4) From (5) and (6), then

6 International Journal of Statistics and Applications 5, 5(6): -6 7 ff : (xx) = ( ) В(, ) ll ll λλλλ λλ (aa ccxx ) (aaaa xx cc xx ) ll=oo aaaa xx cc xx λλ aaaa xx cc xx (ll ) ln λλ ln λλ λλ aaaa xx cc xx λλ λλ aaaa xx cc xx We defined the first order statistics XX () = MMMMMM(XX, XX,, XX ), the last order statistics as XX () = MMMMMM(XX, XX,, XX ) and median order XX mm.. (5) 5.. Distribution of Minimum, Maximum and Median Let XX, XX,, XX 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 ff : (xx) = [ FF(xx)] ff(xx) = h(xx)e n H(x) (6) and ff : (xx) = λλ aa ccxx aaaa xx cc xx aaaa xx cc xx λλ aaaa xx cc xx λλ n ln λλ ln λλ λλ aaaa xx cc xx λλ aaaa xx cc xx e. (7) ff : (xx) = [FF(xx)] ff(xx) = ll=oo ( ) ll ll h(xx) (ll)hh(xx), (8) ff : (xx) = ( ) ll ll λλλλ λλ (aa ccxx ) aaaa xx cc xx ll=oo aaaa xx cc xx λλ aaaa xx cc xx λλ λλ aaaa xx cc xx ff mm :mm (xx) = (mm)! [FF(xx)] mm [ FF(xx)] mm ff(xx) = mm!mm! ff mm :mm (xx) = (mm)! mm!mm! mm mm ll ( )ll λλλλ aa ccxx (aaaa ll=oo aaaa xx cc xx (ll) ln λλ ln λλ λλ aaaa xx cc xx (mm )! mm!mm!. (9) mm mm ll=oo ( )ll ll (mm ll)hh(xx) ff(xx), () xx cc xx λλ (aaaa xx cc xx ) ) λλ (mm ll) ln λλ ln λλ λλ aaaa xx cc xx. () 6. Rényi and Shaon Entropies If X is a random variable having an absolutely continuous cumulative distribution function FF(xx) and probability distribution function ff(xx) then the basic uncertainty measure for distribution F (called the entropy of F) is defined as HH(xx) = EE[ log(ff(xx))]. Statistical entropy is a probabilistic measure of uncertainty or ignorance about the outcome of a random experiment, and is a measure of a reduction in that uncertainty. Numerous entropy and information indices, among them the Rényi entropy, have bn developed and used in various disciplines and contexts. Information theoretic principles and methods have become integral parts of probability and statistics and have bn applied in various branches of statistics and related fields.

7 8 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic Entropy has bn used in various situations in science and enginring. 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 II RR (rr) = rr log[ ffrr (yy)dddd], for rr > and rr. ff rr (yy)dddd = λλλλ λλ rr (aa ccyy ) rr rraaaa yy cc yy aaaa yy cc yy rrrr rr aaaa rrrr yy cc yy dddd, () The series expansions of rrrr aaaa yy cc yy is We get ff rr (yy)dddd = rrrr aaaa yy cc yy λλ rr rr λλ rr! rr λλ aaaa yy cc yy = =, () = (aa ccyy ) rr rraaaa yy cc yy! aaaa yy cc yy (rr) rr dddd, (4) Since < aaaa yy cc yy < for x, then by using the binomial series expansion of aaaa yy cc yy (rr) rr and (aa ccyy ) rr given by aaaa yy cc yy (rr) rr = rrrr rr ( ) aaaa yy cc yy =, (5) and and (aa ccyy ) rr = rr mm = mm aarr mm ( ccyy ) mm = rr mm=ll= mm mm aarr mm ll mm ll cc ll yy mmll. (6) The series expansions of (rr) yy and (rr )cc yy are ( ) (rr )! (rr ) yy = = yy, (7) ( )jj (rr ) jj cc jj (rr )cc yy = jj = yy jj. (8) Substituting (5), (6), (7) and (8) into (4), we get ff rr (yy)dddd = λλ rr rrrr rr rr mm mm ll ( )jj rr λλ rr aa rr mm mm ll cc lljj (rr) jj ==mm =ll==jj =!! jj! jj yy mmlljj = (rr)aaaa dddd, λλ rr ==mm =ll==jj = rrrr rr Thus, according to the definition of Rényi entropy we have II RR (rr) = log rr jj! jj rr mm mm ll ( )jj rr λλ rr aa rr mm mm ll cc lljj (rr) jj!! jj! jj rrrr rr rr λλ rr mm mm ll ( ) jj rr λλ rr mm ll cc lljj ==mm=ll==jj =!! jj! jj aa mm lljj rrmm (rr) mm lljj ΓΓ(mm ll jj ) The Shaon entropy is defined by EE[ log(ff(yy))]. This is a special case derived from lim rr II RR (rr). ΓΓ(mmlljj ) (rr)aa mm lljj. (9). (4) 7. Residual Life Function of the GQHRTPM Distribution Given that a component survives up to time t, 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 survival analysis, actuarial science, 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

8 International Journal of Statistics and Applications 5, 5(6): -6 9 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 (98)). The r th 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 betwn the behaviors of the two functions of a distribution was studied by many authors such The r th order moment of the residual life of the GQHRTPM distribution is given by the general formula where RR(tt) = FF(tt), is the survival function. The series expansions of λλ aaaa yy cc yy mm rr (tt) = EE[(YY tt) rr YY > tt] = is λλ aaaa yy cc yy RR(tt) (yy tt tt)rr λλ aaaa yy cc yy ff(yy) dddd, (4) = =. (4) In what sn this onwards, we use the binomial series expansion of aaaa yy cc yy () and (yy tt) rr given by aaaa yy cc yy () = rr ( ) aaaa yy cc yy =, (4) and and Then, mm rr (tt) = RR(tt) rr ==ss= rr ss ( )rr ss tt rr ss λλ! λλ The series expansions of () yy and ()cc yy are Substituting (46) and (47) into (45), we get RR(tt) ==ss==jj = tt! (yy tt) rr = rr ss= ss ( )rr ss tt rr ss yy ss. (44) (aayy ss yy ss ccyy ss ) ()aaaa yy cc yy dddd, (45) tt ( ) ()! () yy = = yy, (46) ( ) jj () jj cc jj ()cc yy = jj = yy jj, (47) jj! jj mm rr (tt) = rr rr ss ( )rrjj ss tt rr ss λλ cc jj () jj! jj!! jj λλ (aayy ssjj yy ssjj ccyy ssjj ) ()aaaa dddd, (48) xx Where ΓΓ(ss, xx) = tt ss tt dddd = (ss )! xx ss xx = is the upper incomplete gamma function.! The r th order moment of the residual life of the GQHRTPM distribution is given by mm rr (tt) = rr RR(tt) ==ss==jj = rr ss ( ) rrjj ss tt rr ss λλ cc jj! jj!! jj aa ssjj λλ () ssjj ΓΓ(ssjj,()aaaa ) cc ΓΓ(ssjj,()aaaa ) ΓΓ(ss jj, ( )aaaa). (49) aa () aa () where R(t) (the survival function of Y ) is given in (4). For the GQHRTPM distribution the MRL function which is obtained by setting r = in (49), is given in the following theorem. Theorem. The MRL function of the GQHRTPM distribution with cdf () is given by mm (tt) = RR(tt) ==ss==jj = ss ( ) jj ss tt ss λλ cc jj! jj!! jj aa ssjj λλ () ssjj ΓΓ(ss jj, ( )aaaa) ΓΓ(ssjj,()aaaa ) aa () cc ΓΓ(ssjj,()aaaa ) aa (). (5)

9 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic The second moment of the residual life function of the GQHRTPM distribution is mm (tt) = RR(tt) ==ss==jj = ss ( ) jj ss tt ss λλ cc jj ΓΓ(ss jj, ( )aaaa)! jj!! jj aa ssjj λλ () ssjj ΓΓ(ssjj,()aaaa ) aa () cc ΓΓ(ssjj,()aaaa ). (5) aa () The variance of the residual life function of the GQHRTPM distribution can be obtained using mm (tt) and mm (tt). 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 the failure of a component 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 s Kundu and Nanda () and Nanda et al. (). Also, in reliability, the mean reversed residual life (MRRL) and ratio of two consecutive moments of reversed residual life characterize the distribution uniquely. Using () and (), the reversed failure (or reversed hazard) rate function of the GQHRTPM is given by rrh YY (yy) = ff(yy) λλλλ aa ccxx aaaa xx cc xx aaaa xx cc xx λλ aaaa xx cc xx = FF(yy) λλ aaaa xx cc xx, (5) The r th moment of the reversed residual life function can be obtained by the formula µ rr (tt) = EE[(YY tt) rr YY tt] = tt (tt FF(tt) yy)rr ff(yy) dddd, (5) Hence, µ rr (tt) = FF(tt) ==ss==jj = rr ss ( )ssjj tt rr ss λλ cc jj () jj! jj!! jj λλ tt (aayy ssjj yy ssjj ccyy ssjj ) ()aaaa dddd, (54) The series expansions of ()aaaa, We get µ rr (tt) = FF(tt) ==ss==jj =mm = ()aaaa = rr ( ) mm ()aa mm yy mm mm=, mm! rr ss ( )ssjj mm tt rr ss λλ aa mm cc jj () jj mm mm!! jj!! jj λλ aa tt ssmm jj tt ssmm jj cc tt ssmm jj. (55) ssmm jj ssmm jj ssmm jj The mean and second moment of the reversed residual life of the GQHRTPM distribution can be obtained by setting r = and in (55). Also, using µ (tt) and µ (tt) one can obtain the variance of the reversed residual life function of the GQHRTPM distribution. 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 bn 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 (988)). These two measures have some applications in reliability and life testing as well (Giorgi and Crescenzi ()). 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 BB FF [FF(xx)] = xx yy ff(yy) dddd. (56) µ FF(xx) From the relationship betwn the Bonferroni curve and reversed residual life function of the GQHRTPM distribution, the Bonferroni curve of the GQHRTPM distribution is given by

10 International Journal of Statistics and Applications 5, 5(6): -6 BB FF [FF(xx)] = µ FF(xx) ===jj =mm= aa xx mm jj mm jj xx mm mm jj where μ is the mean of the GQHRTPM distribution. The scaled total time on test transform of a distribution function F is defined by If FF(tt) denotes the cdf of the GQHRTPM distribution then Hence, ( ) jj mm λλ aa mm cc jj () jj mm mm!! jj!! jj λλ cc xx mm 4. (57) mm 4 tt SS FF [FF(tt)] = RR(yy) dddd. (58) µ SS FF [FF(tt)] = tt µ λλ λλ λλ SS FF [FF(tt)] = µ λλ ttttλλ ===jj =mm= aaaa yy cc yy dddd, ( ) jj mm λλ aa mm cc jj jj mm tt mm jj mm!! jj!! jj (mm jj ) The cumulative total time can be obtained by using formula CC FF = SS FF [FF(tt)] ff(tt)dddd from the relationship GG = CC FF.. (59) and the Gini index can be derived 9. Parameters Estimation 9.. Maximum Likelihood Estimates In this section we discuss the maximum likelihood estimators (MLE s) and inference for the GQHRTPM(aa,, cc,, λλ, xx) distribution. Let XX, XX,, XX be a random sample of size from GQHRTPM(aa,, cc,, λλ, xx) then the likelihood function can be written as Substituting from () into (6), we get The log-likelihood function becomes ln LL = LL = ff(xx ; aa,, cc,, λλ) (6) LL = λλ λλ (aa xx ccxx ) aaxx xx cc xx aaxx xx cc xx λλ aaxx xx cc xx. (6) ln λλ ln ln λλ ln(aa xx ccxx ) aaxx xx cc xx λλ aaxx xx cc xx ( ) ln aa xx xx cc xx, (6) Therefore, the normal equations are ln LL = xx λλλλ xx aaxx xx cc xx aaxx xx cc xx ( ) ln LL aaxx ccxx = xx xx λλλλ xx aa xx xx cc xx ln LL aaxx ccxx = xx xx λλλλ xx aa xx xx cc xx aaxx ccxx aaxx xx cc xx aaxx xx cc xx ( ) ( ) xx aaxx xx cc xx, aaxx xx cc xx aaxx xx xx cc xx, aaxx xx cc xx aaxx xx xx cc xx, aaxx xx cc xx

11 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic ln LL = λλ aaxx xx cc xx ln aa xx xx cc xx ln aaxx xx cc xx ln LL = λλ λλ λλ aaxx xx cc xx., The likelihood equations can be obtained by setting the first partial derivatives of ln LL w.r.t. the unknown parameters to zero's. That is, the likelihood equations are: xx λλλλ xx aaxx xx cc xx aaxx ccxx xx xx λλλλ xx aaxx xx cc xx aaxx ccxx xx xx λλλλ xx aaxx xx cc xx aaxx ccxx aaxx xx cc xx aaxx xx cc xx aaxx xx cc xx xx aaxx xx cc xx ( ) =, ( ) ( ) aaxx xx cc xx xx aaxx xx cc xx (6) =,(64) aaxx xx cc xx xx aaxx xx cc xx (65) aaxx xx cc xx λλ aa xx xx cc xx ln aaxx xx cc xx ln aa xx xx cc xx =, (66) λλ aaxx xx cc xx =. (67) λλ λλ we get the following system of thr nion-linear equations from (6, 64 and 65) xx jj xx aaxx ccxx jj λλλλ jj jj xx jj aaxx xx cc xx aaxx xx cc xx ( ) jj xx jj aaxx xx cc xx =. (68) aaxx xx cc xx where jj =,,. The normal equations do not have explicit solutions and they have to be obtained numerically. Therefore, the MLEs of aa,, cc, and λλ can be obtained by solving system of thr nion-linear equations with two non-linear equations. 9.. Asymptotic Confidence Bounds Since the MLEs of the unknown parameters aa,, cc and λλ caot 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 aa >, >, cc >, > and λλ >. The simplest large sample approach is to assume that the MLE aa,, cc,, λλ are approximately multivariate normal with mean (aa,, cc,, λλ) and covariance matrix II, s Lawless (), where II is the inverse of the observed information matrix Thus, II = The derivatives in II are given as follows: aa cc VVVVVV(aa) CCCCCC(, aa) CCCCCC(aa, ) VVVVVV() CCCCCC(aa, cc ) CCCCCC(, cc ) II = CCCCCC(cc, aa) CCCCCC(cc, ) VVVVVV(cc ) CCCCCC(, aa) CCCCCC(, ) CCCCCC(, cc ) CCCCCC(λλ, aa) CCCCCC(λλ, ) CCCCCC(λλ, cc ) λλ CCCCCC(aa, ) CCCCCC(, ) CCCCCC(cc, ) VVVVVV() CCCCCC(λλ, aa) CCCCCC(aa, λλ ) CCCCCC(, λλ ) CCCCCC(cc, λλ ) CCCCCC(, λλ ) VVVVVV(λλ ) (69) (7)

12 International Journal of Statistics and Applications 5, 5(6): -6 aa ( ) ( ) 4 cc ( ) 9 λλ = λλλλ xx aaxx xx cc xx aaxx ccxx aaxx xx xx cc xx aaxx xx cc xx, xx = λλλλ xx 4 4 aaxx xx cc xx aaxx ccxx 4 aaxx xx xx cc xx aaxx xx cc xx, xx 4 = λλλλ xx 9 6 aaxx xx cc xx = aaxx ccxx 6 aaxx xx xx cc xx aaxx xx cc xx, = λλ = λλ λλ λλ, xx aaxx ccxx aaxx xx cc xx λλλλ xx aaxx xx cc xx ( ) = aaxx xx xx cc xx aaxx xx cc xx, xx λλλλ xx 4 aaxx xx cc xx ( ) aaxx ccxx 4 aaxx xx xx cc xx aaxx xx cc xx, = xx aaxx aaxx xx cc xx = xx aaxx = xx cc xx λλ xx aaxx xx cc xx xx cc xx λλλλ xx 6 5 aaxx xx cc xx ( ) 6 xx aaxx ccxx 5 aaxx xx xx cc xx aaxx xx cc xx, aaxx xx cc xx aa xx xx cc xx aaxx xx cc xx aaxx xx cc xx ln aaxx xx cc xx, aa xx xx cc xx aa xx xx cc xx ( ) aaxx ( ) aaxx xx cc xx aaxx xx cc xx ( ) aaxx xx cc xx aaxx xx cc xx ( ) aaxx xx cc xx aaxx xx cc xx xx cc xx aaxx xx cc xx ( ) aaxx xx cc xx aaxx xx cc xx aaxx xx cc xx ln aa xx xx cc xx, aaxx xx cc xx, aa xx xx cc xx ( ) aaxx xx cc xx aaxx xx cc xx = xx aaxx xx cc xx λλ xx aaxx xx cc xx aaxx xx cc xx ln aaxx xx cc xx,

13 4 M. A. El-Damcese et al.: Reliability Analysis Using the Generalized Quadratic = xx aaxx xx cc xx aa xx xx cc xx, = xx aaxx xx cc xx λλ xx aaxx xx cc xx aaxx xx cc xx = xx aaxx xx cc xx aa xx xx cc xx aaxx xx cc xx ln aaxx xx cc xx, = aa xx xx cc xx ln aaxx xx cc xx. The above approach is used to derive the ( γγ)% confidence intervals of the parameters aa,, cc, and λλ as in the following forms, aa ± ZZ γγ VVVVVV(aa), ± ZZ γγ VVVVVV, cc ± ZZ γγ VVVVVV(cc ), ± ZZ γγ VVVVVV(), λλ ± ZZ γγ VVVVVVλλ (7) Here, ZZ γγ is the upper (γγ ) th percentile of the standard normal distribution. It should be mentioned here as it was pointed by a refer that if we do not make the assumption that the true parameter vector (aa,, cc,, λλ) is an interior point of the parameter space then the asymptotic normality results will not hold. If any of the true parameter value is, then the asymptotic distribution of the maximum likelihood estimators is a mixture distribution, s for example Self and Liang (987) in this coection. In that case obtaining the asymptotic confidence intervals becomes quite difficult and it is not pursued here.. Application Here, we illustrate applicability of the GQHRTPM distribution using real data sets from Smith and Naylor (987) represents the strengths of.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. We can s that the smallest AIC and BIC are obtained for the GQHRTPM distribution. Table. Estimated parameters, AIC and BIC for the real data set Distribution Parameter AIC BIC EP(a,λλ) a =.7, λ = EG (a,λλ) a =.666, λ = PE (a,λλ) a =.6566, λ = GEP(a,, λλ) GQHRTPM(a,, cc,, λλ) a =.94, α =.956, λ = a =.5, b = 5.45, c =.5, α =.5, λ = So, we can conclude that the GQHRTPM distribution is the most appropriate model for the data sets among the considered distributions.. Conclusions We introduce a new five-parameter distribution called the generalized quadratic hazard rate truncated poisson (GQHRTPM) distribution. This distribution contains several lifetime sub-models such as: EP, EG, PE and GEP. Various properties of the proposed distribution are discussed in Section, 4 and 5. Rényi and Shaon entropies of the GQHRTPM distribution are given in Section 6. Residual and reverse residual life functions of the GQHRTPM distribution are discussed in Section 7. Section 8 is devoted to the Bonferroni and Lorenz curves of the GQHRTPM distribution. The maximum likelihood estimation procedure is presented. Fitting the GQHRTPM model to real data set

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