1. 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 (1954), Proschan (1963), Barlow et al. (1963), Barlow and Marshall (1964)-(1965), Marshall and Proschan (1965), Cozzolino (1968), Dahiya and Gurland (1972), McNolty et al. (1980), Saunders and Myhre (1983), Nassar (1988), Gleser (1989), Gurland and Sethuraman (1994), Adamidis and Loukas (1998), Kus (2007), and Tahmasbi and Rezaei (2008).

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 (1993) 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 (1995). Bebbington et al. (2007) proposed a new two-parameter distribution which is a generalization of the Weibull. Recently, Gupta et al. (2008) introduced another member of the Weibull family, which is called as the 'flexible Weibull distribution'.

Recently, many new distributions, generalizing well-known distributions used to study lifetime data, have been introduced. Mudholkar and Srivastava (1993) presented a generalization of the Weibull distribution called the exponentiated (generalized)-Weibull distribution, GWD. The generalized exponential distribution, GED, introduced by Gupta and Kundu (1999). Nadarajah and Kotz (2006) introduced four exponentiated type distributions: the exponentiated gamma, exponentiated Weibull, exponentiated Gumbel. Sarhan and Kundu (2009) presented a generalization of the linear hazard rate distribution called the generalized linear hazard rate distribution, GLFRD. Sarhan et al. (2008) obtained Bayes and maximum likelihood estimates of the three parameters of the generalized linear hazard ratedistribution based on grouped and censored data. Recently, Sarhan (2009) 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 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 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 seems that the proposed model can be considered as a suitable candidate model in reliability analysis, biological systems, data modeling, and related fields.

2. The Maximum Survival Time Distribution

Let be a random sample from the generalized quadratic hazard rate distribution with Cumulative Density Function (cdf) and Z is a random variable from truncate at zero Poisson distribution with probability mass function as follows:

(1)

where

By assuming that the random variables and Z are independent and defining then, the marginal distribution of X, for is

(2)

with probability density function

(3)

where

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

(4)

One of the characteristic in reliability analysis is the hazard rate function (HRF) defined by

(5)

The hazard rate function is such that:

● if the hazard function is either increasing (if b > 0) or constant (if b = 0 and a >0);

● when the hazard function should be:

(1) increasing if b >0;

(2) upside-down bath-tub shaped if b <0; and

● if 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 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 and is defined as

(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 cannot be obtained in explicit forms. It can be obtained as infinite series expansion so, in general different moments of Also, we cannot get the quantile of in a closed form by using the equation. Thus, by using Equation (2), we find that

(7)

The median can be obtained from (7), when as follows

(8)

Moreover, the mode of can be obtained as a solution of the following nonlinear equation.

(9)

4.2. Moments

The following theorem 1 gives the moment of

Theorem 1. If has the moment is given as follows for

(10)

Proof

Substituting (3) into the above relation, we get

(11)

The series expansions of is

We get

(12)

Since for x, then by using the binomial series expansion of given by

We get

(13)

The series expansions of and are

(14)

and

(15)

Substituting (14) and (15) into (13), we get

(16)

The integral in (16) can be computed as follows

(17)

Substituting (17) into (16), we get (10). This completes the proof.

4.3. The Moment Generating Function

The moment generating function of the GQHRTPM distribution has the following form

(18)

(19)

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 be a simple random sample from with cumulative distribution function and probability density function as in (2) and (3), respectively. Let denote the order statistics obtained from this sample. In reliability literature, denote the lifetime of an system which consists of independent and identically components. Then the pdf of is given by

(20)

Using

(21)

then,

(22)

and

(23)

Substituting (22) and (23) into (20), we get

(24)

From (5) and (6), then

(25)

We defined the first order statistics , the last order statistics as and median order

5.1. Distribution of Minimum, Maximum and Median

Let 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

(26)

(27)

(28)

(29)

and

(30)

(31)

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 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 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.

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

(32)

The series expansions of is

(33)

We get

(34)

Since for x, then by using the binomial series expansion of and given by

(35)

and

(36)

The series expansions of and are

(37)

and

(38)

Substituting (35), (36), (37) and (38) into (34), we get

(39)

Thus, according to the definition of Rényi entropy we have

(40)

The Shannon entropy is defined by This is a special case derived from

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 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 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 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 between 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

(41)

where is the survival function.

The series expansions of is

(42)

In what seen this onwards, we use the binomial series expansion of and given by

(43)

and

(44)

Then,

(45)

The series expansions of are

(46)

and

(47)

Substituting (46) and (47) into (45), we get

(48)

Where is the upper incomplete gamma function.

The r^th order moment of the residual life of the GQHRTPM distribution is given by

(49)

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 = 1 in (49), is given in the following theorem.

Theorem 2. The MRL function of the GQHRTPM distribution with cdf (2) is given by

(50)

The second moment of the residual life function of the GQHRTPM distribution is

(51)

The variance of the residual life function of the GQHRTPM distribution can be obtained using

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 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

(52)

The r^th moment of the reversed residual life function can be obtained by the formula

(53)

Hence,

(54)

The series expansions of

We get

(55)

The mean and second moment of the reversed residual life of the GQHRTPM distribution can be obtained by setting r = 1 and 2 in (55). Also, using and 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 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

(56)

From the relationship between the Bonferroni curve and reversed residual life function of the GQHRTPM distribution, the Bonferroni curve of the GQHRTPM distribution is given by

(57)

where μ is the mean of the GQHRTPM distribution.

The scaled total time on test transform of a distribution function F is defined by

(58)

If denotes the cdf of the GQHRTPM distribution then

Hence,

(59)

The cumulative total time can be obtained by using formula and the Gini index can be derived from the relationship

9. Parameters Estimation

9.1. Maximum Likelihood Estimates

In this section we discuss the maximum likelihood estimators (MLE’s) and inference for the distribution. Let be a random sample of size from then the likelihood function can be written as

(60)

Substituting from (3) into (60), we get

(61)

The log-likelihood function becomes

(62)

Therefore, the normal equations are

The likelihood equations can be obtained by setting the first partial derivatives of the unknown parameters to zero's. That is, the likelihood equations are:

(63)

(64)

(65)

(66)

(67)

we get the following system of three nion-linear equations from (63, 64 and 65)

(68)

where

The normal equations do not have explicit solutions and they have to be obtained numerically. Therefore, the MLEs of can be obtained by solving system of three nion-linear equations with two non-linear equations.

9.2. Asymptotic Confidence Bounds

Since the MLEs of the unknown parameters 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 The simplest large sample approach is to assume that the are approximately multivariate normal with mean and covariance matrix , see Lawless (2003), where is the inverse of the observed information matrix

(69)

Thus,

(70)

The derivatives in are given as follows:

The above approach is used to derive the confidence intervals of the parameters as in the following forms

(71)

Here, is the upper percentile of the standard normal distribution. It should be mentioned here 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 the parameter space then the 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 for example 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

So, we can conclude that the GQHRTPM distribution is the most appropriate model for the data sets among the considered distributions.

11. 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 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 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 seems that the proposed model can be considered as a suitable candidate model in reliability analysis, biological systems, data modeling, and related fields.