Abstract
In the past, the proportional and additive hazard rate models have been investigated in the works. Nanda and Das (2011) introduced and studied the dynamic proportional (reversed) hazard rate model. In this paper we study the dynamic additive hazard rate model, 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.
1. 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.
Aranda-Ordaz [14] first dealt with an additive hazard model where is a baseline hazard rate and a time-dependent covariate vector , representing the changes in the operating conditions, and is a vector of parameters. For more details, one may see Cox and Oakes [3], Thomas [15], Breslow and Day [16], Finkelstein and Esaulova [17], Lim and Zhang [18], and so forth.
Assume that and are the lifetimes of two systems with corresponding hazard rate functions and for . 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 , but they may be additive differently from different intervals. Specifically, they may be related as for , and , where are some constants. When the intervals become smaller and smaller, a model as in (2) will be naturally obtained.
In order to guarantee that is a hazard rate function of a nonnegative random variable , the following lemma is given.
Lemma 1. Assume that and are defined before. Then, for , is a hazard rate function if and only if the following conditions hold:(i), for all ;(ii); (iii)if , then for some .
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 as the common left end point of their supports, and the terms increasing and decreasing stand for monotone nondecreasing and monotone nonincreasing, respectively.
2. 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) [decreasing in failure rate (DFR)] if is increasing [decreasing] in ; (b) increasing in failure rate in average (IFRA) [decreasing in failure rate in average (DFRA)] if is increasing [decreasing] in ; (c) new better than used (NBU) [new worse than used (NWU)] if , for all ; (d) new better than used in failure rate (NBUFR) [new worse than used in failure rate (NWUFR)] if , for all ; (e) new better than used in failure rate average (NBAFR) [new worse than used in failure rate average (NWAFR)] if , for all . 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 , 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 , . Take for . It is obvious that satisfies all the conditions of Lemma 1. Obviously, if is IFR and is increasing in , hence is IFR.
Example 4. Let be a random variable having Weibull distribution with hazard rate function , . Let for . It can be verified that is increasing in , and hence is IFR. However, is decreasing in but increasing in .
Proposition 5. If the random variable is IFRA (DFRA) and is increasing (decreasing) in , then the random variable is IFRA (DFRA).
Proof. For , 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 , . Take for . It is obvious that satisfies all the conditions of Lemma 1. Obviously, is IFRA and is IFRA. However, is decreasing in .
Proposition 7. If the random variable is NBU (NWU) and is increasing (decreasing) in , 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 and ,
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 for . By some computations, we have It can be verified that is nonnegative for (see also Figure 1). From (9), we conclude that is NBU. However, it is easily obtained that is increasing in but decreasing in .
Proposition 9. If the random variable is NBUFR (NWUFR) and for , then the random variable is NBUFR (NWUFR).
Proposition 10. If the random variable is NBAFR (NWAFR) and for , then the random variable is NBAFR (NWAFR).
Proof. We only give the proof for the case of NBAFR. It is noted that is NBAFR which is equivalent to that, for all , . Note that is NBAFR if and only if . Hence the desired result follows from the condition .
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 is a sufficient but not a necessary one in Propositions 9 and 10.
3. 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 for all ; (b) usual stochastic order (denoted by ) if , for all ; (c) hazard rate order (denoted by ) if , for all ; (d) up hazard rate order (denoted by ) if , for all ; (e) down hazard rate order (denoted by ) if , for all . For more details about stochastic orders, please refer to Shaked and Shanthikumar [21].
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.
Proposition 12. Suppose and are two nonnegative random variables satisfying (2). Then, if is increasing (decreasing ) in .
Proof. Note that if and only if, for all , It is equivalent to that . It holds if is increasing in . The proof of the parenthetical statement is similar.
The following example indicates that the condition of the monotone property of the is sufficient but not a necessary one for the aging intensity ordering between and .
Example 13. Assume that is a random variable having exponential distribution with mean 1/2. Let for . By some computations, we have It can be verified that for , and hence is increasing in (see also Figure 2). Note that . Thus , for all , and hence . However, it is easily obtained that is decreasing in but increasing in .
Proposition 14. Suppose and are two nonnegative random variables satisfying (2). Then, if and only if , for all .
The following corollary follows immediately from the proposition above.
Corollary 15. If , for all , then .
Proposition 16. Suppose and are two nonnegative random variables satisfying (2). Then, if and only if , for all .
Proposition 17. Suppose that and are two nonnegative random variables satisfying (2). Then, if and only if , for all and .
Proof. Note that if and only if is increasing in , for all . It is equivalent to the fact that is increasing in , which is equivalent to the fact that its derivative is nonnegative; that is, , for all and . It follows from the condition. The proof of the parenthetical statement is similar.
Proposition 18. Suppose that and are two nonnegative continuous random variables satisfying (2). Then, if and only if , for all and .
Its proof is similar to that of Proposition 17 and hence is omitted.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The research was supported by the National Natural Science Foundation of China (71361020).