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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.

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 [

R. C. Gupta and R. D. Gupta [

Recently, Nanda and Das [

Aranda-Ordaz [

Assume that

Sometimes the hazard rate functions of

In order to guarantee that

Assume that

if

In Section

Throughout the paper, assume that all random variables under consideration have

At first we introduce some concepts of aging notions that will be useful in the section. Recall that a random variable

In the following we give some aging closure properties between the random variables

If the random variable

In the following, we give two examples related to this proposition. Example

Let

Let

If the random variable

For

Example

Let

If the random variable

We only give the proof for the case of NBU. In order to prove that

Note that

From the fact that

Example

Assume that

Plot of the

If the random variable

If the random variable

We only give the proof for the case of NBAFR. It is noted that

Example

Firstly let us recall the concepts of some stochastic orders that are closely related to the main results in this section. A random variable

In the following we give some sufficient (and necessary) conditions of stochastic ordering between random variables

Suppose

Note that

The following example indicates that the condition of the monotone property of the

Assume that

Plot of the

Suppose

The following corollary follows immediately from the proposition above.

If

Suppose

Suppose that

Note that

Suppose that

Its proof is similar to that of Proposition

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research was supported by the National Natural Science Foundation of China (71361020).