Further Results on Dynamic Additive Hazard Rate Model

In the past, the proportional and additive hazard rate models have been investigated in the works. Nanda andDas (2011) introduced and studied the dynamic proportional (reversed) hazard ratemodel. In this paper we study the dynamic additive hazard ratemodel, and investigate its aging properties for different aging classes. The closure of the model under some stochastic orders has also been investigated. Some examples are also given to illustrate different aging properties and stochastic comparisons of the model.


Introduction
It is common practice in statistical analysis that covariates are often introduced to account for factors that increase the heterogeneity of a population.When the effect of a factor under study has a multiplicative (or additive) effect on the baseline hazard function, we have a proportional (or an additive) hazard model.The latter category of model is preferred in any situation.For example, in tumorigenicity cases, where the dose effect on tumor risk is of interest, the excess risk becomes an important factor.Clinical trials that seek the effectiveness of treatments often experience lag times of treatment effectiveness after which treatment procedures will be in full effect.
In reliability and survival analysis, devices or systems always operate in a changing environment.The conditions under which systems operate can be harsher or gentler in modeling lifetime of the devices or systems.The most known Cox [1] model is that the changing conditions are assumed to act multiplicatively on the baseline hazard rate.This model has been widely used in many experiments where the time to systems' failure depends on a group of covariates, which may be regarded as different treatments, operating conditions, heterogeneous environments, and so forth.P. L. Gupta and R. C. Gupta [2] studied the relation between the conditional and unconditional failure rates in mixtures when the distributions in the mixture follow the proportional hazard rate.For further research, one may see Cox and Oakes [3], Kumar and Westberg [4], Dupuy [5], Lau [6], Zhao and Zhou [7], X. Li and Z. Li [8], and Yu [9].R. C. Gupta and R. D. Gupta [10] proposed and studied the proportional reversed hazard model to analyze failure time data.For more details on this model, see Gupta and Wu [11], X. Li and Z. Li [12], and so forth.
Recently, Nanda and Das [13] introduced the dynamic proportional hazard rate (DPHR) model and the dynamic proportional reversed hazard rate (DPRHR) model and studied their properties for different aging classes.The closure of the models under different stochastic orders has also been studied.

Mathematical Problems in Engineering
Assume that  and  are the lifetimes of two systems with corresponding hazard rate functions ℎ  () and ℎ  () for  ≥ 0. Let () =   (); the model (with time-dependent covariates) in (1) would reduce to the form which is named as dynamic additive hazard rate (DAHR) model.Sometimes the hazard rate functions of  and  may not be additive over the whole interval [0, ∞), but they may be additive differently from different intervals.Specifically, they may be related as for  = 1, 2, . .., and  0 = 0, where   ( = 1, 2, . ..) are some constants.When the intervals [ −1 ,   )( = 1, 2, . ..) become smaller and smaller, a model as in (2) will be naturally obtained.
In order to guarantee that ℎ  () is a hazard rate function of a nonnegative random variable , the following lemma is given.
In Section 2 of the paper, we discuss some aging properties of the DAHR model.In Section 3, the closure of DAHR model under different stochastic orderings is studied.Some examples are given to illustrate the results concerned in Sections 2 and 3.
Throughout the paper, assume that all random variables under consideration have 0 as the common left end point of their supports, and the terms increasing and decreasing stand for monotone nondecreasing and monotone nonincreasing, respectively.

Aging Properties of DAHR Model
At first we introduce some concepts of aging notions that will be useful in the section.Recall that a random variable  is said to be (a) increasing in failure rate (IFR) [ , for all  ≥ 0. For more discussions on properties of aging notions, readers may refer to Barlow and Proschan [19], Müller and Styan [20], and so forth.
In the following we give some aging closure properties between the random variables  and  under some conditions of ().Some results are obvious and hence their proofs are omitted.Proposition 2. If the random variable  is IFR (DFR) and, for  ≥ 0, () is increasing (decreasing), then the random variable  is IFR (DFR).
In the following, we give two examples related to this proposition.Example 3 is an application of the proposition.Example 4 indicates that the condition of () is sufficient but not a necessary one.
Example 3. Let  be a random variable having Weibull distribution with hazard rate function ℎ  () = 2,  ≥ 0. Take () =  for  ≥ 0. It is obvious that () satisfies all the conditions of Lemma 1. Obviously, if  is IFR and () is increasing in , hence  is IFR.Proof.For  ≥ 0, let Note that  is IFRA (DFRA) and () is increasing (decreasing) implying that Hence the desired result follows directly.
Example 3 can be regarded as an application of the above proposition.Example 6 below indicates that the condition of () is sufficient but not a necessary one for the monotone property of .Example 6.Let  be a random variable having Weibull distribution with hazard rate function ℎ  () = 2,  ≥ 0. Take () = − for  ≥ 0. It is obvious that () satisfies all the conditions of Lemma 1. Obviously,  is IFRA and  is IFRA.However, () is decreasing in  ≥ 0. Proposition 7. If the random variable  is NBU (NWU) and () is increasing (decreasing) in  ≥ 0, then the random variable  is NBU (NWU).
Proof.We only give the proof for the case of NBU.In order to prove that  is NBU, it is sufficient to prove that, for all  ≥ 0 and  ≥ 0, It is equivalent to That is, Note that  is NBU which implies that That is, From the fact that () is increasing and ( 11), ( 9) holds, and hence the desired result follows.
Example 3 is an application of the above proposition.
The following example indicates that the condition of () is sufficient but not a necessary one for the NBU property of .
Example 8. Assume that  is a random variable having exponential distribution with mean 1/2.It is clear that  is NBU.Let () = (1 + )/(1 +  2 ) for  ≥ 0. By some computations, we have It can be verified that (, ) is nonnegative for ,  ≥ 0 (see also Figure 1).From ( 9), we conclude that  is NBU.However, it is easily obtained that () is increasing in [0, √ 2 − 1) but decreasing in ( √ 2 − 1, +∞).Proof.We only give the proof for the case of NBAFR.It is noted that  is NBAFR which is equivalent to that, for all  ≥ 0, (∫ Remark 11.Example 3 is an application of Propositions 9 and 10.Example 6 can be regarded as a counterexample, which shows that the condition () ≥ 0 is a sufficient but not a necessary one in Propositions 9 and 10.

Stochastic Comparisons of DAHR Model
Firstly let us recall the concepts of some stochastic orders that are closely related to the main results in this section.A random variable  is said to be larger than another random variable  in (a) aging intensity ordering (denoted by ≥  ), if  In the following we give some sufficient (and necessary) conditions of stochastic ordering between random variables  and .Some results are obvious and hence their proofs are omitted.
The following example indicates that the condition of the monotone property of the ()/ℎ  () is sufficient but not a necessary one for the aging intensity ordering between  and .