The parameters of a distribution are typically considered to be real or perhaps vector values. In the literature, families of distributions are considered that are characterized by having a parameter that is itself a distribution function. These families are called semiparametric because they also contain a real parameter. Choosing the parameter that is first a distribution function is one possible way to use a semiparametric model. The underlying distribution is the formal name for this distribution function. In practice, the selection of an underlying distribution leads to the selection of a parametric model, but the selection is limited to families with the structure of the semiparametric model.

The underlying distribution F might already have one or more parameters, in which case a semiparametric family might provide a way to include a new parameter, extending the family from which F originates. One can imagine that the standard families of the Gamma and Weibull distributions are derived from the exponential distribution via semiparametric families that include a second parameter. The Weibull and Gamma families can both be found as special cases of a three-parameter family using the same technique. The study of semiparametric families is therefore advantageous for two reasons: it provides a new understanding of traditional distribution families, and it offers strategies for extending families to make data fitting more flexible (see, e.g., Marshall and Olkin [18]).

Stochastic orderings of random variables have long been a useful tool for making comparisons between probability distributions (see Müller and Stoyan [19], Shaked and Shanthikumar [23], Belzunce et al. [3], and Li and Li [16]). Some researchers used stochastic orders comparing distributions in terms of the magnitude of random variables to perform stochastic comparisons between semiparametric models, including those presented in equations (1), (5), and (9) in Section 2. To this end, we quantify the effect of varying the parameters of the model on the variation of the response variables and, furthermore, the effect of changing the underlying distribution on changing the distribution of the response variables using several known stochastic orders. For example, in the context of the proportional hazard rate (PHR) model for the case where $\lambda$ is a random variable (frailty), Gupta and Kirmani [10] and subsequently Xu and Li [24] identified some stochastic ordering properties of the model. Considering the proportional reversed hazard rates (PRHR) model, Di Crescenzo [6] made some stochastic comparisons between two candidate distributions of the model that differ in their parameters. Kirmani and Gupta [13] derived some stochastic ordering results for the model proportional odds rates (POR) model.

Recently, however, many researchers have focused on stochastic orders that compare lifetime distributions according to aging behavior, namely the faster aging stochastic orders. One of the key ideas in reliability theory and survival analysis is stochastic aging. It broadly outlines the pattern of aging/degradation of a system over time. Three different notions of aging are presented in the literature: positive aging, negative aging, and no aging. Positive aging implies a stochastically decreasing remaining lifetime of the system, while negative aging implies just the opposite. The system does not mature with time if there is no aging. To study different characteristics of system aging, various aging classes (including increasing failure rate (IFR), decreasing failure rate (DFR), increasing failure rate on average (IFRA), decreasing failure rate on average (DFRA), increasing likelihood ratio (ILR), and decreasing likelihood ratio (DLR), to name a few) have been presented in the literature based on these three aging principles. The reader can consult Barlow and Proschan [2] and Lai and Xie [15] for further discussion on this topic. In addition to these ideas about aging, relative aging is a useful concept to use when studying system reliability. Relative aging is used to measure how a system changes over time relative to another system.

In real life, there are many situations where we deal with multiple systems of the same type (e.g., TVs from different manufacturers, CPUs from different brands, etc.). In these circumstances, we often encounter the following problem: how to determine whether one system is aging faster than others over time? The idea of relative aging provides a compelling answer to this problem. When dealing with the crossover hazards/medium remaining life phenomena, another component of relative aging proves helpful. Many real-life situations involve this type of circumstance. For example, when Pocock et al. [20] examined survival data on the effects of two different treatments on breast cancer patients and became aware of the phenomenon of crossover hazards. In addition, Champlin et al. [4] described several cases in which the superiority of one treatment over another lasted only for a short period of time. The above considerations suggest that increasing/decreasing hazard ratio models are a viable option in a variety of real-world scenarios. In fact, Kalashnikov and Rachev [12] have developed a concept of relative aging based on the monotonicity of the ratio of two hazard rate functions called relative hazard rate order. This concept is known as faster hazard rate aging. Sengupta and Deshpande [22] presented another idea in a similar way based on the monotonicity of the ratio of two cumulative hazard rate functions. Rezaei et al. [21] proposed a relative order based on the ratio of the reversed hazard rates of two random lifetimes and called it relative reversed hazard rate order.

The aim of this paper is to perform stochastic comparisons between two newly defined semiparametric models, the modified proportional hazard rate model and the modified proportional reversed hazard rate model, corresponding to the relative hazard rate and the relative reversed hazard rate order.

The rest of the paper is organized as follows. In Section 2 we give some advanced preliminary considerations and auxiliary results. In Section 3, we consider the modified proportional hazard rate model for comparison in terms of relative hazard rate order. In Section 4, we consider the modified proportional reversed hazard rate model to give some ordering properties according to the relative reversed hazard rate order. In Section 5, we conclude the paper with a more detailed summary and provide an outlook on possible future studies.

In this section we give some mathematical definitions of the notions that will be utilized in this paper. In the literature, many semiparametric families of distributions have been introduced and studied. Among these models some of them find their applicability in the context of lifetime events. The Cox’s PHR model is of the important and frequently used such semiparametric family of distributions (see, Cox [5]). For a review on the PHR model we refer the reader to Kumar and Klefsjö [14]. Let us consider the parameter $\lambda >0$, called the frailty parameter, then the PHR model is defined as where $\overline{F}(·;\lambda )$ is the survival function (sf) of the response random variable and $\overline{F}(·)$ is the baseline sf. Let $X_0$ have an absolutely continuous distribution function (cdf) $F(·)$, with probability density function (pdf) $f(·).$ Then, the hazard rate (hr) of $X_0$, as important reliability quantity in survival analysis, measures the instantaneous risk for failure of a device with lifetime $X_0$ at a certain age (t, say). The hr of $X_0$ for all $t\ge 0$ which fulfills $\overline{F}\left(t\right)>0$ is defined as follows:

It is well-known that h characterizes the underlying sf, $\overline{F}$, as follows:

Suppose that $h(·;\lambda )$ is the hr function associated with the sf (1), then, it is plainly seen that for every $t\ge 0$ for which $min(\overline{F\left(t\right)},\overline{F}(t;\lambda ))>0,$

In contrast to the PHR model, the PRHR model was introduced by Gupta et al. [9]. We refer the reader to Gupta and Gupta [11] for further descriptions of the PRHR model. In the PRHR model, a positive parameter, $\beta$, called the resilience parameter, is considered. The PRHR model is then defined as in which $F(·;\beta )$ is the cdf of the response random variable and $F(·)$ is the baseline cdf or the underlying distribution function in the model. The reversed hazard rate (rhr) of $X_0$, as another reliability quantity, measures the risk for failure of a device (with original lifetime $X_0$) in the past at a certain time point t at which the device is found to be inactive. The rhr of $X_0$ for all $t\ge 0$ which $F\left(t\right)>0$ is derived via the following relation:

It has been verified that $\tilde{h}$ characterizes the underlying cdf, F, as below:

Let us now assume that $\tilde{h}(·;\beta )$ is the rhr function of the distribution with the cdf (5). Then, it is readily realized for all $t\ge 0$ for which $min\left\{F\right(t),F(t;\beta \left)\right\}>0$ that

Another reputable semiparametric family of distributions is the POR model (see, e.g., Marshall and Olkin [17]). This model is defined with cdf

In some situations the following model is alternatively utilized:

The odds rate function of $X_0$, measures the relative odd of the event $\{X_0>t\}$ in terms of the event $\{X_0\le t\}$ where t is some point of time. The odds rate function of $X_0$ for all $t\ge 0$ which $F\left(t\right)>0$ defined as follows:

We assume that $OR(t;\alpha )=\frac{\overline{F}(t;\alpha )}{F(t;\alpha )}$ is the odds rate function of the distribution with the cdf (9). Then, it is easily verified for all $t\ge 0$ for which $min\left\{F\right(t),F(t;\alpha \left)\right\}>0$ that

Balakrishnan et al. [1], utilized the PHR (resp. PRHR) model as baseline model in (9) (resp. (10)) to propose two new models, referred to as modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models.

Suppose that $X_0$ is a baseline random variable with survival function $\overline{F}$. Let $X_{11},\cdots ,X_{1n}$ are independent and identically distributed (i.i.d.) lifetimes of n components of a system with a common distribution function $F(·;\alpha ,\lambda )$. Then, $X_{11},\cdots ,X_{1n}$ are said to follow the MPHR model with tilt parameter $\alpha$, modified proportional hazard rate $\lambda$ and baseline survival function $\overline{F}$ (denoted as $MPHR(\alpha ;\lambda ;\overline{F})$) if, and only if,

For the case $\alpha =1$, (13) simply reduces to the PHR model. The MPHR model in (13) includes some well-known distributions such as extended exponential and extended Weibull distributions (Marshall and Olkin [18]), extended Pareto distribution (Ghitany [7]) and extended Lomax distribution (Ghitany et al. [8]).

On the other hand, suppose $X_1,\cdots ,X_n$ are i.i.d. lifetimes of n components of a system with a common distribution functions F. Then, $X_1,\cdots ,X_n$ are said to follow the MPRHR model with tilt parameter $\alpha$, modified proportional reversed hazard rate $\beta$ and baseline distribution function F (denoted as $MPRHR(\alpha ;\beta ;F)$) if and only if

Note that the PRHR model is a sub-model of (14) when $\alpha =1$.

We assume that the random variables X and Y have distribution functions F and G, survival functions $\overline{F}=1-F$ and $\overline{G}=1-G$, density functions f and g, hazard rate functions $h_X=f/\overline{F}$ and $h_Y=g/\overline{G}$ and reversed hazard rate functions ${\tilde{h}}_X=f/F$ and ${\tilde{h}}_Y=g/G$, respectively. To compare the magnitude of random variables some notions of stochastic orders are introduced below.

Some stochastic orders in Definition 1 are connected to each other. In this regard, $X{⪯}_{lr}Y$ implies $X{⪯}_{hr}Y$ and also $X{⪯}_{lr}Y$ implies $X{⪯}_{rh}Y$. Furthermore, $X{⪯}_{hr}Y$ gives $X{⪯}_{st}Y$ and also $X{⪯}_{rh}Y$ yields $X{⪯}_{st}Y.$ For further relations and properties of the stochastic orders ${⪯}_{lr},{⪯}_{hr},{⪯}_{rh}$ and ${⪯}_{st}$ we refer the reader to Shaked and Shanthikumar [23]. For more descriptions of the relative order ${⪯}_c$ we refer the reader to Kalashnikov and Rachev [12] and also Sengupta and Deshpande [22]. For further properties of the relative order ${⪯}_b$ the reader can see Rezaei et al. [21].

In this section, we obtain a relative ordering property in the MPHR model according to the relative hazard rate order. We will consider the MPHR model in two settings where two sets of parameters $\mathbf{\alpha }=({\alpha }_1,{\alpha }_2)$ and $\mathbf{\lambda }=({\lambda }_1,{\lambda }_2)$ which are possibly different are assigned and also possibly different baseline sfs $\overline{F}$ and $\overline{G}$ are taken into account. Finding conditions on $\mathbf{\alpha }$ and $\mathbf{\beta }$ and also conditions on $\overline{F}$ and $\overline{G}$ to establish the preservation of the relative hazard rate ordering property in the MPHR model is the main objective of this section.

Before stating next result we introduce some notation. Let $X_0$ and $Y_0$ have pdfs f and g, and sfs $\overline{F}$ and $\overline{G}$, respectively, and, further, $X_1\sim MPHR({\alpha }_1;{\lambda }_1;\overline{F})$ and $Y_1\sim MPHR({\alpha }_2;{\lambda }_2;\overline{G})$. Then, using (13), the sfs of $X_1$ and $Y_1$ which are denoted by $\overline{F}(x;{\alpha }_1,{\lambda }_1)$ and $\overline{G}(x;{\alpha }_2,{\lambda }_2)$, respectively, can be written as follows:

Now, let us denote by $f(·;{\alpha }_1,{\lambda }_1)$ and $g(·;{\alpha }_2,{\lambda }_2)$ the pdfs of $X_1$ and $Y_1$, which can be obtained by taking derivatives of cdfs in (15) as follows:

Appealing to (15) together with (16) the hazard rate function of $X_1$ and the hazard rate function of $Y_1$ are acquired as: where $h(·)$ and $s(·)$ are the hazard rate functions of $X_0$ and $Y_0,$ respectively, and the function $\mathsf{\Phi }(u;\alpha ,\lambda )$ is given by

We define here two measures of relative hazard rates of $Y_0$ and $X_0$ with hazard rate functions $s(·)$ and $h(·),$ respectively. Let us denote two limiting points of hazard rates ratio $\frac{s\left(t\right)}{h\left(t\right)}$ as follows:

In the following example, we show that the result of Theorem 1 is applicable.

The following theorem states another setting for the parameters of two MPHR distributions so that the result of Theorem 1 is obtained under a different condition.

The following example provides a situation where the result of Theorem 2 is applicable.

In this section, we investigate the relative reversed hazard rate ordering property in two MPRHR models with possibly different sets of parameters $(\mathbf{\alpha },\mathbf{\beta })$, where $\mathbf{\alpha }=({\alpha }_1,{\alpha }_2)$ and $\mathbf{\beta }=({\beta }_1,{\beta }_2)$, and also under possibly different baseline distributions F and G.

We start with introducing some notation. Let $X_0$ and $Y_0$ have pdfs f and g, with underlying cdfs F and G, respectively. Let us assume that $X_1^{★}\sim MPRHR({\alpha }_1;{\beta }_1;F)$ and $Y_1^{★}\sim MPRHR({\alpha }_2;{\beta }_2;G)$ and denote by $F^{★}(·;{\alpha }_1,{\beta }_1)$ and $G^{★}(·;{\alpha }_2,{\beta }_2)$. Then, using (14), we have

Using (26), the pdfs of $X_1^{★}$ and $Y_1^{★}$ (signified by $f^{★}(·;{\alpha }_1,{\beta }_1)$ and $g^{★}(·;{\alpha }_2,{\beta }_2)$) are acquired as below:

By dividing the pdfs in (27) into the cdfs given in (26), the reversed hazard rate function of $X_1$ and the reversed hazard rate function of $Y_1$ are derived as follows: where $\tilde{h}(·)$ and $\tilde{s}(·)$ are the reversed hazard rate functions of $X_0$ and $Y_0,$ respectively, and further the function $\Psi (u;\alpha ,\beta )$ is defined as