A NEW FAMILY OF LIFETIME DISTRIBUTIONS IN TERMS OF CUMULATIVE HAZARD RATE FUNCTION

In the present paper, a new family of lifetime distributions is introduced according to cumulative hazard rate function, the well-known concept in survival analysis and reliability engineering. Some important properties of proposed model including survival function, quantile function, hazard function, order statistic and some results of stochastic ordering are obtained in general setting. An especial case of this new family is introduced by considering Weibull distribution as the parent distribution; in addition estimating unknown parameters of specialized model will be examined from the perspective of Bayesian and classic statistics. Moreover, three examples of real data sets: complete, right-censored and progressively type-I interval-censored data are studied; point and interval estimations of all parameters are obtained. Finally, the superiority of proposed model in terms of parent Weibull distribution over other fundamental statistical distributions is shown via complete real observations.


Introduction
The statistical distribution theory has been widely explored by researchers in recent years. Given the fact that the data from our surrounding environment follow various statistical models, is necessary to extract and develop appropriate highquality models.
Recently, Nadarajah and Haghighi (2011) have introduced a new model of lifetime distributions, which the researchers refer it as N H distribution. It is an extended form of exponential distribution and attracted the attention of some researchers. We refer the reader to (Lemonte (2013), Dey et al. (2017) and Kumar et al. (2017)). This model has a number of desirable features and is comprehensively studied by the authors. For example, whenever the data contains zero values, N H model can be a strong competitor for other well-known lifetime distributions such 2 OM ID KHARAZM I AND SHAHLA JAHANGARD as gamma, Weibull and generalized exponential distribution. The cumulative distribution function (cdf ) and probability distribution function (pdf ) related to N H model is given respectively as; where the parameters > 0 controls the shapes of the distribution and the parameter > 0 is the scale parameter. It is easy to see that the N H model has increasing, decreasing and constant hazard shapes.
In the present paper, we introduce a New family of Lifetime distributions based on the Cumulative Hazard rate quantity of a parent distribution G, so-called N LCH G distribution. One of our main motivation to introduce this new category of distributions is that, when the parent distribution G be exponential, the proposed model reduced to N H distribution.
The cumulative hazard rate function is a prominent concept in topics of survival analysis and reliability engineering and plays an important role in this area of science. Suppose that X be a random variable with density function f and cumulative distribution function F , then hazard rate and cumulative hazard rate functions are de…ned; and H(x) = log R(x) = e R x 0 h(t)dt ; respectively, where R(x) = 1 F (x) denotes the survival function of X ( Barlow and Proschan (1975)).
In the next, we …rst obtain the fundamental and statistical properties of N LCH G in general setting and then we propose an especial case of N LCH G model by considering Weibull distribution instead of the parent distribution G. It is referred as N LCH W eibull (or N LCH W ) distribution. We provide a comprehensive discussion about statistical and reliability properties of new N LCH W model. Furthermore, we consider Maximum likelihood, Bayesian and bootstrap estimation procedures in order to estimate the unknown parameters of the new model for complete, right-censored and progressively type-I interval-censored data sets. In the Bayesian discussion, we consider di¤erent types of symmetric and asymmetric loss functions such as squared error, absolute value, Linear Exponential (LIN EX) and generalized entropy to estimate three unknown parameters of N LCH W model. Since the parameter space for all three parameters is positive, we use gamma priors distributions. Bayesian %95 credible and highest posterior density (HP D) intervals (see Chen et al. (1999)) are provided for each parameter of proposed model. In addition, the asymptotic con…dence intervals and parametric and nonparametric bootstrap con…dence intervals are calculated in order to compare with corresponding Bayesian intervals.
The rest of the paper organized as follows. In the section 2, a new category of lifetime distributions is introduced based on the fundamental quantity H(x) and then the main statistical and reliability properties are obtained in general setting. In section 3, by considering the Weibull distribution as the base distribution, a new model is presented according to the general model discussed in section 1 and its prominent characteristics are studied. This new model refer as N LCH W distribution. In section 4, we examine the inferential procedures for estimation unknown parameters of the N LCH W model. In this Section, we provide discussions about three important estimation methods maximum likelihood, Bayesian and bootstrap. Here we use four well-known loss functions like squared error, absolute value, LINEX and generalized entropy. Application and numerical analysis of three real data sets (complete, right-censored and progressively type-I interval-censored) are presented in section 5. Finally, in section 6 the paper is concluded.

New model and properties
In this section, …rst we introduce a new category of lifetime distributions and then we obtain main statistical and reliability properties of the proposed family in general setting.
and its cumulative distribution function (cdf ) is given by where, H(x) is cumulative hazard function of baseline distribution G(x) and h(x) = @H(x) @x .
The corresponding survival function of (1) is given as R(x; ; ) = e ( +H(x)) ; x > 0; > 0; > 0: Remark 2.2. Let = 1, then we get F (x; = 1; ) = G(x): In the following theorem we investigate the connection between N H and N LCH G models. Theorem 2.3. Suppose that the random variable X be a continuous random variable with cumulative hazard rate function H(x), and the random variable Y has N H distribution with parameter and . Then the transformed variable Z = H 1 ( Y ) has a density with pdf (1) as parameter = 1.  Proof. Using the method of distribution function we have; so the proof is completed.
Following this section, we get some fundamental properties of proposed model such as hazard rate function, survival function, quantile function and order statistic distribution. It is seen that all of these measures have closed expression in terms of quantity H(x).

Hazard Rate Function.
The hazard rate is a key concept in analysis of reliability and measuring the aging process. Knowing shape and behavior of the hazard rate in reliability theory, risk analysis, and so on, is very important. The hazard rate function of the N LCH G distribution is given as Proof. The proof is straightforward.
In the following lemma we provide a result about stochastic order in hazard function to compare proposed model and baseline distribution. First we recall the following de…nition. The random variable X is said to be less than variable Y in hazard rate order, X hr Y , if h X (x) h Y (x), for all x in the union of supports of X and Y , where h X (x)(h Y (x)) is the hazard rate of X(Y ). For more details see Shaked and Shanthikumar (2007). Lemma 2.6. Let X F and X G be two random variables corresponding with proposed model (1) and distribution G respectively, then under the condition Proof. The proof is straightforward.
A NEW FAM ILY OF LIFETIM E DISTRIBUTIONS 5 2.2. Random variate generation. One important quantity for each probabilistic model is to have the data generator function based on an explicit formula, because the simulation studies researchers are more satis…ed with the data generator functions of a given form. For generating random variables from the N LCH G distribution, we use the inverse transformation method. The quantile of order p of the N LCH G distribution is where H 1 (x) is inverse function of quantity H(x). Let U be a random variable generated from a uniform distribution on(0; 1), then is a random variable generated from the N LCH G distribution by the probability integral transform.

Order statistics.
Order statistics have applications in various directions such as statistical inference, reliability engineering, quality control and etc. Let X 1 ; X 2 ; : : : ; X n be a random sample from N LCH G distribution. Let X i:n denote the ith order statistic. Then the pdf of X i:n is given by 1 e ( +H(x)) i 1

NLCH-Weibull (NLCH-W) model
Without loss of generality let parameter = 1 and consider the Weibull distribution as a parent distribution with cdf function F (x; ; ) = 1 e x ; x > 0; > 0; > 0. By replacing this model in relation (3), the pdf of the N LCH W is given as and its cdf is given by Remark 3.1. If = 1, we attain the pdf of Weibull distribution and If = 1, we get N H distribution respectively. 6 OM ID KHARAZM I AND SHAHLA JAHANGARD 3.1. Density shape. It is easy to investigate that the shape of N LCH W is unimodal and and lim x!1 f (x) = 0: (9) Figure 1. The graphs of pdf (a) and hazard rate function (b, c and d) of the N LCH W distribution for some selected values of parameters.
In the next section, we consider the hazard shape of N LCH W distribution.

3.2.
Hazard rate function of NLCH-W distribution. The hazard rate func- Determining the behavior of the hazard rate is very important in various applications, especially in reliability theory. It can easily be shown that the proposed model (7) has a variety of hazard shapes. The hazard rate function allows for constant, monotonically increasing, monotonically decreasing, unimodal and bathtub shaped hazard rates. In summary, di¤erent types of hazard rates are as follows.
if > 1 and Some shapes of pdf and hazard function for the selected values of parameters is given in Figure 1.

Estimation procedures
Nowadays, three methods of maximum likelihood estimation, Bayesian and bootstrap procedures are of particular importance in the theory of statistical inference undoubtedly. In this section, we describe each of these methods separately for estimating the parameters , and of the N LCH W distribution. For all methods we consider the case when all three parameters are unknown.

4.1.
Maximum likelihood estimation. The maximum likelihood procedure is one of the most common methods for obtaining an estimator for an unknown parameter in classic statistical inference. The likelihood function is a function that written based on the mechanism of the observations occurrence. Here, the structure of the likelihood function is expressed for three modes of observations including complete data, right-censored and progressive interval-censored data sets. 4.1.1. Maximum likelihood estimation for complete data set. Let X 1 ; X 2 ; : : : ; X n be a random sample of size n from N LCH W distribution. The likelihood function is given for equation (7) by So, the log-likelihood function is written as The normal equations are derived by di¤erentiation of the log-likelihood function with respect to parameters ; and .
Setting these di¤erentiations equal to zero and solving for ; and , then we can obtain the maximum likelihood estimator M LE of parameters ; and .

4.1.2.
Maximum likelihood estimation for right-censored data set. Let (X 1 ; 1 ); (X 2 ; 2 ); : : : ; (X n ; n ) be a right-censored random sample of size n from N LCH W distribution. Where i is a censoring indicator variable, that is, i = 1 for an observed survival time and i = 0 for a right-censored survival time. In the case N LCH W distribution the likelihood function and the corresponding log-likelihood are given as respectively. Analogous above results the normal equations can be derived in the case right-censored sample data.

4.1.3.
Maximum likelihood estimation for progressively type-I interval-censored data set. Let n items to be applied on a life testing simultaneously at time t = 0 and suppose that m pre-speci…ed times t 1 < t 2 < ::: < t m , where t m is scheduled time to terminate the experiment, be determined . At the ith inspection time, t i , the number, X i , of failures within (t i 1; t i ] is recorded and R i surviving items are randomly removed from the life testing for i = 1; 2; :::; m. Therefore, a progressively type-I interval-censored sample can be denoted as S = (X i ; R i ; t i ) and sample size is n = P m i=1 (X i + R i ). The likelihood function of density (1) based on progressively type-I interval-censored sample S = (X i ; R i ; t i ); i = 1; 2; :::; n is given as The log-likelihood function is given as; Also, here we can derive normal equations for corresponding log-likelihood function similar complete and right-censored samples data. In practical due to the nonlinearity of corresponding normal equations in three cases that discussed above, we use numerical algorithms to extract MLEs estimators.

4.2.
Bootstrap estimation. The uncertainty in parameters of the …tted distribution can be estimated by parametric (re-sampling from the …tted distribution) or non-parametric (re-sampling with replacement from the original data set) bootstraps re-sampling Efron and Tibshirani (1994). These two parametric and nonparametric bootstrap procedures for complete data set are described as follows. Non-parametric bootstrap procedure: (1) Generate a bootstrap sample fX 1 ; : : : ; X m g , with replacement from original data set. Obtain the bootstrap estimate of with MLE procedure, say b using the bootstrap sample.
In case of the N LCH W distribution, the non-parametric bootstrap estimators (NPBs) of ; and , say^ N P B ;^ N P B and^ N P B , respectively. Analogous algorithms can be expressed for bootstrap estimation of right-censored sample data.

Bayesian estimation.
Bayesian inference procedure for censored data have been taken into consideration by many statistical researchers, especially researchers in the …eld of survival analysis and reliability engineering. In this section, a complete sample data and two widely used types of censored observations, right-censored and progressively type-I interval-censored observations are analyzed through Bayesian point of view. We assume that the parameters , and of N LCH W distribution have independent prior distributions as In the Bayesian estimation, according to that we do not know the actual value of the parameter, we may be adversely a¤ected by loss when we choose an estimator. This loss can be measured by a function of the parameter and corresponding estimator.
Generalized entropy loss function and Bayesian estimator For more details see Calabria and Pulcini (1996).
In the next, we provide the posterior probability distributions in three modes: complete, right-censored and progressively type-I interval-censored data sets. Let we de…ne the function ' as Hence, we get joint posterior density of parameters , and for complete sample data by combining the likelihood function (10) and joint prior density (13). Therefore, the joint posterior density function is given by where K is given as Furthermore, by using likelihood functions (11), (12) and joint prior distribution (13) the joint posterior probability distribution functions for right-censored (x; ) and progressively type-I interval-censored sample data (S = (X i ; R i ; t i ); i = 1; 2; :::; n) presented respectively with (a; ; jx; ) = M '( ; ; ) where M is given as OM ID KHARAZM I AND SHAHLA JAHANGARD and Z is given as Here we interested in obtaining Bayesian estimators for three sample data sets (complete, right-censored and type-I progressive interval-censored data sets) under the four loss functions described above. As it is observed, there are no closed forms for the Bayes estimators. It is possible to simulated posterior sample data sets by using Gibbs sampling method and Metropolis-Hasting algorithm. Thus, by applying MCMC algorithm the corresponding Bayes estimators, Bayesian credible and HPD intervals are calculated.

Application of NLCH-W distribution on the real datasets
This section aims to show applications of the N LCH W model under the methods discussed in the section 4 via real data examples. In order to achieve this target, we consider three real data sets to illustrate the application of proposed distribution in real world and the superiority of this model to some other useful classic models. Furthermore, in this section, we provide Bayesian and bootstrap analysis of parameter estimation of N LCH W model for three real data sets. The following data sets contain three modes of real world observations: complete, right-censored and progressively type-I interval-censored.

Complete data set: Failure times of 84 Aircraft Windshield
We consider the data of service times for a particular model windshield. These data were recently studied by Ramos et al. (2013) Progressively type-I interval-censored data set: 112 patients with plasma cell myeloma Table 1 contains a typical progressively type-I interval-censored data that devoted to 112 patients with plasma cell myeloma treated at the National Cancer Institute (see Carbone et al. (1967)).   Table 2 includes the MLEs of parameters, log-likelihood and Akaike information criterion (AIC) for N LCH W distribution and the mentioned above distributions in the case complete real data set. The results of Table 2 shows that, the N LCH W  Figure 2. Histogram and …tted density plots, the plots of empirical and …tted cdf s, P P plots and Q Q plots for the complete data set.
distribution provides the best …t for the complete data set as it has lower AIC statistic than the other competitor models. The histogram of data set, …tted pdf of the N LCH W distribution and …tted pdf s of other competitor distributions for the real data set are plotted in Figure 2. Also, the plots of empirical and …tted cdf s functions, P P plots and Q Q plots for the N LCH W and other …tted distributions are displayed in Figure 2. These plots also support the results in Table  2.
In the rest of this subsection, we provide Bayesian and Bootstrap estimation results. It is clear from the equation (15) that there is no closed form for the Bayesian estimators under the four loss functions described in subsection 4.3, so we suggest using an MCMC procedure based on 1000 replicates to compute Bayesian estimators. The corresponding Bayesian point and interval estimation provided in Table 3. The posterior samples extracted by using Gibbs sampling technique. Moreover, we provide the posterior summary plots. These plots con…rm that the sampling process is of prime quality and convergence is occurred.
Also, here we obtain point and %95 con…dence interval estimation of parameters of the N LCH W distribution by parametric and non-parametric bootstrap methods for complete real data set. We provide results of bootstrap estimation based on 10000 bootstrap replicates in Table 3. It is interesting to look at the joint distribution of the bootstrapped values in a scatter plot in order to understand the potential structural correlation between parameters (see Figures 3 and 4).   of N LCH W in Table 2, it can be seen that, in general the estimation results are , and . Also, Figure 5 relates to the Bayesian analysis process, including history (Trace plot), autocorrelation function(acf ) and histogram of three parameters samples drown from posterior distribution (15). These plots show that convergence was reached, no autocorrelation problems were encountered and the density of the posterior is extracted. 5.2. MLE, bootstrap and Bayesian estimation in case right-censored data set. Here, we provide the MLE, non-parametric bootstrap and Bayesian estimation of , and , the parameters of N LCH W distribution for right-censored data set that given at the beginning of section 5. In order to compare di¤erent estimation results, we also provide interval estimation (%95 asymptotic con…dence, quantile bootstrap, %95 credible and HPD intervals) of parameters under the three estimation procedures that considered in section 4. Table 4 shows the corresponding results for right-censored data set. In addition, the plots of empirical and theoretical cdf s and diagrams of the Bayesian analysis process are provided in Figure 6 and 18 OM ID KHARAZM I AND SHAHLA JAHANGARD Figure 7, respectively. Associated Bayesian procedure plots show that convergence was reached and no autocorrelation problems there exist. Also, Figure 6 represent the estimated lower and upper bound of cumulative probability.

MLE and
Bayesian estimation in the case progressively type-I intervalcensored data set. Analogous two previous subsections, here we provide a summary of numerical analysis of progressively type-I interval-censored data set based on the Bayesian and maximum likelihood methods described in section 5. Table 5 is devoted to estimation of parameters. This table provides the Bayesian estimators and %95 credible and HPD intervals for each parameter of proposed N LCH W model. In addition, the maximum likelihood estimators are calculated in order to compare with corresponding Bayesian estimators under the di¤erent loss functions. Plots of history, acf plots and histogram of posterior samples of each parameter of proposed distribution provided in Figures 8. These …gures show that the simulation processes of Gibbs algorithm has good quality and convergence is occurred.

Conclusion
In this article, a new model of lifetime distributions is introduced and main properties of it are obtained. One of the interesting and important properties of proposed family is that, it results the Nadarajah and Haghighi (2011) famous  distribution, as an especial case, when the parent distribution is exponential. An especial example of this family is introduced by considering Weibull distribution as the base distribution. We also show that the proposed distribution has variability of hazard rate shapes such as increasing, decreasing, bathtub shape and upsidedown bathtub shapes. Classic and Bayesian inferences for three cases of real data such as complete, right-censored and progressively type-I interval-censored data sets are investigated. Bayesian estimators under the four well-known loss functions are presented. Numerical results of maximum likelihood, Bayesian and bootstrap procedures for each set of real data are presented in separate tables. From a practical point of view, the distribution introduced in this study was shown to be better than some common statistical distributions for some real data sets applied as an example.