2.1. The decreasing mean time to failure class

The notion of a mean time to failure (MTTF) function was first introduced in [Reference Barlow and Proschan7] in the context of an age replacement policy where the replacement of an item takes place at the failure of the item or at a predetermined time t, whichever comes first. It is a standard practice to adopt an age replacement policy in order to control in-service failure of systems. Let $X_{[t]}$ denote the time to the first in-service failure of an item under an age replacement policy with the age replacement time t. Then the MTTF function of F is defined as

\begin{equation*} M_F(t)\;{:}{=}\; \mathbb{E}[X_{[t]}] = \frac{\int_0^t\bar{F}(y)\,{\text{d}} y}{F(t)} \quad \text{for all } t>0.\end{equation*}

This function itself has a special importance in the literature [Reference Kayid, Ahmad, Izadkhah and Abouammoh21, Reference Klefsjö27, Reference Li and Xu31]. We now have the following formal definition of the DMTTF and increasing mean time to failure (IMTTF) classes.

Another ageing family, NBRU $_{\text{rh}}$ (new better than renewal used in the reversed hazard rate order), which was introduced in [Reference Li and Xu31], turns out to be identical to the DMTTF class. The preservation of the DMTTF class under some basic reliability operations was investigated in [Reference Knopik28], while the weak convergence issues as well as closure under convolution for this class were studied in [Reference Knopik29]. An extensive study on the monotonic as well as non-monotonic classes based on the MTTF function can be found in [Reference Izadi, Sharafi and Khaledi18, Reference Khan, Bhattacharyya and Mitra23]. However, characterization results for this class are so far absent in the literature. The next theorem addresses this. We first prove a lemma that is essential for the next theorem.

Proof. We need to show that, for all $0\leq n_1 < n_2$ and $s>0$ ,

$$ \frac{1}{s}\sum_{i=1}^{n_1+1}\alpha_i(s)/(1-\alpha_{n_1+1}(s)) \geq \frac{1}{s}\sum_{i=1}^{n_2+1}\alpha_i(s)/(1-\alpha_{n_2+1}(s)). $$

Define $g(k)= ({1}/{s})\sum_{i=1}^{k+1}\alpha_i(s)$ and

\begin{equation*} c_j = \frac{g(n_j)}{1-\alpha_{n_j+1}(s)}, \quad j=1,2. \end{equation*}

Now, $({1}/{s})\sum_{i=1}^{k+1}\alpha_i(s)-c(1-\alpha_{k+1}(s))$ has at most one change of sign from $+$ to $-$ for each $c>0$ . So this happens for $c_1$ and $c_2$ in particular. Thus, $g(k)-c_j(1-\alpha_{k+1}(s))$ has at most one change of sign from $+$ to $-$ , $j=1,2$ . Consequently, by choice of $c_j$ , $j=1,2$ , we have

\begin{alignat*}{2} g(k) & \geq c_j(1-\alpha_{k+1}(s)) \qquad & & (k \leq n_j) \\ & \leq c_j(1-\alpha_{k+1}(s)) & & (k > n_j). \end{alignat*}

Now, if possible, let

(2.3) \begin{equation} c_1 = \frac{g(n_1)}{1-\alpha_{n_1+1}(s)} < \frac{g(n_2)}{1-\alpha_{n_2+1}(s)} =c_2. \end{equation}

Then, taking $n_1<k<n_2$ where $k\in \mathbb N$ , we have

\begin{alignat*}{2} g(k) & \leq c_1(1-\alpha_{k+1}(s)), \qquad & & (k > n_1) \\ & < c_2(1-\alpha_{k+1}(s)), & & (\text{by ({2.3})}) \\ & \leq g(k), & & (k < n_2), \end{alignat*}

which is a contradiction. Therefore, we have ${g(n_1)}/({1-\alpha_{n_1+1}(s)}) \geq {g(n_2)}/({1-\alpha_{n_2+1}(s)})$ for all $0\leq n_1 < n_2$ and $s>0$ , i.e. $({1}/{s})\sum_{i=1}^{n+1}\alpha_i(s)/(1-\alpha_{n+1}(s))$ is decreasing in $n\geq 0$ for all $s>0$ .

Proof. Necessity. For any life DF F we have, by Proposition 2.1,

\begin{equation*} s\int_0^\infty\frac{{\text{e}}^{-st}(st)^n}{n!}\bigg(\int_0^t\bar{F}(y)\,{\text{d}} y\bigg)\,{\text{d}} t = \sum_{i=0}^n\int_0^\infty\frac{{\text{e}}^{-sy}(sy)^i}{i!}\bar{F}(y)\,{\text{d}} y = \frac{1}{s}\sum_{i=0}^n\alpha_{i+1}(s). \end{equation*}

Therefore, for $n\geq 0$ , $s >0$ , and for any $c> 0$ ,

(2.4) \begin{equation} s\int_0^\infty\frac{{\text{e}}^{-st}(st)^n}{n!}\bigg[\int_0^t\bar{F}(y)\,{\text{d}} y - cF(t)\bigg]\,{\text{d}} t = \frac{1}{s}\sum_{i=1}^{n+1}\alpha_i(s) - c(1-\alpha_{n+1}(s)). \end{equation}

Observe that the kernel $K(n,t)= {\text{e}}^{-st}(st)^n/n!$ is TP $_2$ in $t\in(0,\infty$ ) for $n\in \{0,1,2,\ldots\}$ . Since F is DMTTF, the variation diminishing property (VDP) of K(n,t) together with (2.4) imply that, for any $c> 0$ , the expression $({1}/{s})\sum_{i=1}^{n+1}\alpha_i(s)-c(1-\alpha_{n+1}(s))$ has at most one sign change from $+$ to $-$ . Therefore, by virtue of Lemma 2.4, $({1}/{s})\sum_{i=1}^{n+1}\alpha_i(s)/(1-\alpha_{n+1}(s))$ is decreasing in $n\geq 0$ for all $s>0$ . This further implies that $\sum_{i=1}^{n}\alpha_i(s)/(1-\alpha_{n}(s))$ is decreasing in $n> 0$ for all $s>0$ . Now adding 1 to this expression and using the condition $\alpha_0(s)=1$ , the necessary part is proved.

Sufficiency. It is given that $\sum_{k=0}^{n-1} \alpha_k(s)/ (1-\alpha_n(s))$ is decreasing in $n> 0$ for all $s>0$ . Therefore, for all $n,k \geq 0$ ,

(2.5) \begin{equation} (1-\alpha_{n+k+1}(s))\sum_{i=1}^{n+1}\alpha_i(s) \geq (1-\alpha_{n+1}(s))\sum_{i=1}^{n+k+1}\alpha_i(s). \end{equation}

Now,

\begin{align*} \sum_{i=1}^{n+1}\alpha_i(s) = s\sum_{i=0}^{n}\int_0^\infty\frac{{\text{e}}^{-su}(su)^{i}}{i!}\bar{F}(u)\,{\text{d}} u & = s\int_0^\infty\Bigg(\sum_{i=0}^n\frac{{\text{e}}^{-su}(su)^{i}}{i!}\Bigg)\bar{F}(u)\,{\text{d}} u \\ & = s\int_0^\infty\bar{G}_n(u)\bar{F}(u)\,{\text{d}} u. \end{align*}

Let $\lim_{n \to \infty}({n}/{s}) = x > 0$ . Then, using Lemma 2.1 and appealing to the dominated convergence theorem (DCT), we get

(2.6) \begin{equation} \lim_{n\to\infty}\frac{1}{s}\sum_{i=1}^{n+1}\alpha_i(s) = \lim_{n\to\infty}\int_0^\infty\bar{G}_n(u)\bar{F}(u)\,{\text{d}} u = \int_0^x\bar{F}(u)\,{\text{d}} u. \end{equation}

Now let x and $x+y$ be two continuity points of F, where $x,y>0$ . Also let $s=s(n,x)={n}/{x}$ and $k= [{ny}/{x}]$ , where [w] indicates the greatest integer less than or equal to w. Then $\lim_{n\to\infty}({n+k})/{s} = x+y$ . Now, taking the limit as $n\to\infty$ on both sides of (2.5) and using Lemma 2.2 and (2.6), we finally obtain $F(x+y)\int_0^x\bar{F}(u)\,{\text{d}} u \geq F(x)\int_0^{x+y}\bar{F}(u)\,{\text{d}} u$ . This inequality holds for all $x,y \geq 0$ since the continuity points of F are dense in $[0,\infty)$ . Therefore, F is DMTTF.

2.2. The increasing mean inactivity time class

For any fixed $t>0$ , the inactivity time (IT) [Reference Ruiz and Navarro38] defined by $X_{(t)}\;{:}{=}\;[t-X\mid X\leq t]$ is the random time that has elapsed until time t after the failure occurs. Then the mean inactivity time (MIT) function is given by

\begin{align*} m_F(t)\;{:}{=}\; \mathbb{E}[X_{(t)}] = \frac{\int_0^tF(y)\,{\text{d}} y}{F(t)}, \quad t>0.\end{align*}

This function has widespread applications in diverse fields and has also been called the mean waiting time (MWT) function [Reference Finkelstein15] or mean past life (MPL) function [Reference Asadi5, Reference Goliforushani and Asadi17]. It was pointed out in [Reference Finkelstein15] that the MIT function can be treated as the dual of the mean residual life (MRL) function. In recent work, preservation of MIT ordering for coherent systems was studied in [Reference Rao and Naqvi37]. The monotonic behaviour of $m_F(t)$ induces the following ageing class.

Note that the concept of a decreasing mean inactivity time class does not hold for non-negative distributions [Reference Chandra and Roy13]. The behaviour of the excess lifetime of a renewal process with IMIT interarrivals was investigated in [Reference Li and Xu30]. Characterization results for the IMIT class using some ordering relations such as right spread order, increasing convex order, etc. can be found in [Reference Ahmad, Kayid and Pellerey4]. Closure under weak convergence of the IMIT class was established in [Reference Khan, Bhattacharyya and Mitra24]. In order to obtain a BS type characterization result, we proceed as follows. We consider a quantity analogue of $\alpha_n(s)$ in this situation. This was first considered in [Reference Kebir22] while characterizing various probabilistic orderings using the Laplace transform. Define

\begin{equation*} \bar{a}_n(s) = \frac{({-}1)^n}{n!}\frac{{\text{d}}^n}{{\text{d}} s^n}\bigg[\frac{\phi_F(s)}{s}\bigg], \quad n\geq0,\,s> 0,\end{equation*}

and $\bar{\alpha}_{n+1}(s)=s^{n+1}\bar{a}_n(s)$ for $n\geq 0$ , $s>0$ . Therefore,

\begin{equation*} \bar{\alpha}_{n+1}(s) = s\int_0^\infty\frac{(sx)^n}{n!}{\text{e}}^{-sx}F(x)\,{\text{d}} x, \quad n\geq0,\,s> 0.\end{equation*}

Recalling (1.2), it is clear that

(2.7) \begin{equation} \begin{cases} \bar{\alpha}_{n+1}(s) = 1-\alpha_{n+1}(s), & n\geq0,\,s>0, \\ \bar{\alpha}_0(s) = 0, & s > 0. \end{cases}\end{equation}

Observe that, under the setup of the cumulative damage shock model with random threshold as described in the introduction, $\bar{\alpha}_{n}(s)$ denotes the discrete probability distribution for that specific model. $\bar{\alpha}_{n}(s)$ can also be viewed as $\bar{\alpha}_{n}(s) = \mathbb{E}[F(X_n(s))]$ , where the random variable $X_n(s)$ has a Gamma(n, s) distribution. Next, we establish the characterization result for the IMIT class.

Proof. Necessity. Using Proposition 2.1, it is easy to check that for $n\geq 0$ , $s>0$ , and for any $c\geq0$ ,

\begin{equation*} s\int_0^\infty\frac{{\text{e}}^{-st}(st)^n}{n!}\bigg[\int_0^tF(y)\,{\text{d}} y - cF(t)\bigg]\,{\text{d}} t = \frac{1}{s}\sum_{i=1}^{n+1}\bar{\alpha}_i(s) - c\bar{\alpha}_{n+1}(s). \end{equation*}

Since F is IMIT, the function $\int_0^tF(y)\,{\text{d}} y-cF(t)$ can undergo at most one change of sign from $-$ to $+$ . Thus, using the VDP of the kernel $K(n,t) = {\text{e}}^{-st}(st)^n/n!$ , we obtain that $\sum_{i=1}^{n+1}\bar{\alpha}_i(s)/\bar{\alpha}_{n+1}(s)$ is increasing in $n\geq 0$ and for all $s>0$ . Observing that $\bar{\alpha}_0(s) = 0$ from (2.7), we get that $\sum_{i=0}^{n}\bar{\alpha}_i(s)/\bar{\alpha}_{n}(s)$ is increasing in $n> 0$ for all $s>0$ . We complete the proof by adding $-1$ to this expression.

Sufficiency. From the given condition we have

(2.8) \begin{equation} \bar{\alpha}_{n+k+1}(s)\sum_{i=1}^{n+1}\bar{\alpha}_i(s) \leq \bar{\alpha}_{n+1}(s)\sum_{i=1}^{n+k+1}\bar{\alpha}_i(s) \quad \text{for all } n,k \geq 0,\,s>0. \end{equation}

A calculation analogous to the DMTTF case yields $({1}/{s})\sum_{i=1}^{n+1}\bar{\alpha}_i(s) = \int_0^\infty\bar{G}_n(u)F(u)\,{\text{d}} u$ . Changing the order of integration and substituting $\bar{G}_n(u)$ from (2.2), we have

\begin{equation*} \int_0^\infty\bar{G}_n(u)F(u)\,{\text{d}} u = \int_0^\infty\int_0^y\frac{s^{n+1}}{n!}{\text{e}}^{-sy}y^nF(u)\,{\text{d}} u\,{\text{d}} y = \int_0^\infty h(y)\,{\text{d}} G_n(y), \end{equation*}

where $h(y)=\int_0^yF(u)\,{\text{d}} u$ . Let $\lim_{n\to\infty}({n}/{s}) = x > 0$ ; applying [Reference Billingsley10, Theorem 5.2] for $n\to\infty$ we obtain

(2.9) \begin{equation} \frac{1}{s}\sum_{i=1}^{n+1}\bar{\alpha}_i(s) \to h(x) = \int_0^xF(u)\,{\text{d}} u. \end{equation}

Consider x and $x+y$ to be two continuity points of F, where $x,y>0$ . As before, let $s={n}/{x}$ and $k=[{ny}/{x}]$ . Letting $n\to\infty$ in (2.8) and using Lemma 2.2 in conjunction with (2.9), we obtain $F(x+y)\int_0^xF(u)\,{\text{d}} u \leq F(x)\int_0^{x+y}F(u)\,{\text{d}} u$ . The proof is now completed by noting that the set of continuity points of F is dense in $[0,\infty)$ .

2.3. Some important connections

The techniques employed by Block and Savits serve quite well in establishing characterization results for a number of ageing classes. However, in some cases either the method fails or the calculations become intractable. Searching for an alternative route, we have developed a new approach involving the equilibrium distribution of F. Let the DF of the equilibrium (or first derived) distribution of F(t) be denoted by $F_{(1)}(t)\;{:}{=}\; ({1}/{\mu})\int_0^t\bar{F}(u)\,{\text{d}} u$ for all $t\geq0$ , where $\mu$ is the mean of F. For a life-distribution function F, it is natural to assume that the expected lifetime is positive and finite. We start out by looking at an analogue of $\alpha_n(s)$ for the equilibrium distribution. Let $\phi_{F_{(1)}}(s) = \int_0^\infty{\text{e}}^{-sx}\,{\text{d}} F_{(1)}(x)$ , $s>0$ , so that

$$a'_n(s) = \frac{({-}1)^n}{n!}\frac{{\text{d}}^n}{{\text{d}} s^n}\bigg[\frac{1-\phi_{F_{(1)}}(s)}{s}\bigg], \quad n\geq0,\, s\geq 0,$$

and

(2.10) \begin{equation} \begin{cases} \alpha'_{n+1}(s) =s^{n+1}a^{\prime}_n(s), & n\geq0,\,s\geq 0, \\ \alpha'_0(s) = 1 & \text{for all } s \geq 0. \end{cases}\end{equation}

Note that we use a $'$ to denote the quantities corresponding to (1.1) for the equilibrium distribution. We now have the following important relation, which is the key to our approach.

Proposition 2.2. For $\alpha_{n}(s)$ and $\alpha'_{n}(s)$ as defined in (1.1) and (2.10),

\begin{equation*} \alpha'_{n}(s) = \frac{\sum_{k=n+1}^\infty\alpha_{k}(s)}{\sum_{k=1}^\infty\alpha_{k}(s)} \quad \text{for all } n\geq0,\,s\geq 0. \end{equation*}

Proof. We have, for all $n\geq0$ and $s\geq 0$ ,

\begin{align*} \alpha'_{n+1}(s) & = s\int_0^\infty\frac{(st)^n}{n!}{\text{e}}^{-st}\bar{F}_{(1)}(t)\,{\text{d}} t \\ & = s\int_0^\infty\frac{(st)^n}{n!}{\text{e}}^{-st}\bigg(\frac{1}{\mu}\int_t^\infty\bar{F}(u)\,{\text{d}} u\bigg)\,{\text{d}} t \\ & = \frac{s}{\mu}\int_0^\infty\int_0^u\frac{(st)^n}{n!}{\text{e}}^{-st}\bar{F}(u)\,{\text{d}} t\,{\text{d}} u \\ & = \frac{1}{\mu s}\sum_{k=n+1}^\infty s\int_0^\infty\frac{{\text{e}}^{-su}(su)^k}{k!}\bar{F}(u)\,{\text{d}} u \qquad \text{(using Proposition 2.1)} \\ & = \frac{1}{\mu s}\sum_{k=n+2}^\infty\alpha_{k}(s). \end{align*}

The proof is completed by noting that

\begin{equation*} \mu s = s\int_0^\infty\bar{F}(u)\,{\text{d}} u = \sum_{k=0}^\infty s\int_0^\infty\frac{(sx)^k{\text{e}}^{-sx}}{\Gamma(k+1)}\bar{F}(x)\,{\text{d}} x = \sum_{k=1}^\infty\alpha_{k}(s). \end{equation*}

From Proposition 2.2, it is easy to observe that

(2.11) \begin{equation} \alpha'_{n}(s) - \alpha'_{n+1}(s) = \frac{\alpha_{n+1}(s)}{\sum_{k=1}^\infty\alpha_{k}(s)}.\end{equation}

At this juncture, it would be appropriate to highlight a few comments on $\alpha'_{n}(s)$ . To this end, we have the following lemma.

Proof. Using Proposition 2.2, the proof follows directly from the facts that

$$\frac{1}{s}\sum_{k=n+1}^\infty\alpha_{k}(s)\to\int_x^\infty\bar{F}(u)\,{\text{d}} u$$

as $n\to\infty$ and $\mu s = \sum_{k=1}^\infty\alpha_{k}(s)$ .

Again, $\alpha'_{n}(s)$ can be written as

(2.12) \begin{equation} \alpha'_{n}(s) = \mathbb{E}[\bar{F}_{(1)}(X_n(s))],\end{equation}

where $X_n(s)$ is a random variable having a gamma distribution with parameters n and s. Thus, by virtue of Lemma 2.5 and (2.12) it is easy to observe that properties of $\alpha'_{n}(s)$ can be translated into properties of $\bar{F}_{(1)}$ , and vice versa. The same was shown to be true in [Reference Block and Savits11] for the functions $\alpha_{n}(s)$ and $\bar F$ . Now consider a cumulative damage shock model with random threshold $X_e$ having DF $F_{(1)}$ . Let the damages $Y_1, Y_2, \ldots$ caused by the successive shocks be mutually independent, each having exponential distribution with parameter $s>0$ . Also, $Y_1, Y_2, \ldots$ are independent of the threshold $X_{e}$ . Then $\alpha'_{n}(s)$ can be considered as the discrete survival probabilities for that particular model.

Let us now consider a homogeneous Poisson process $\{N_s(t)\colon t\geq 0\}$ with intensity $s>0$ . Then

\begin{equation*} \alpha_{n+1}(s) = s\int_0^\infty\frac{(su)^n}{n!}{\text{e}}^{-su}\bar{F}(u)\,{\text{d}} u = \int_0^\infty\mathbb{P}(N_s(u)>n)\,{\text{d}} F(u) = \mathbb{P}(N_s(X)>n).\end{equation*}

As in [Reference Block and Savits11], we define

(2.13) \begin{equation} \begin{cases} \beta_{n}(s) = \mathbb{P}(N_s(X)=n), & n\geq0,\,s>0, \\ \beta_n(0) = 0, & n > 0, \\ \beta_0(0) = 1. & \end{cases}\end{equation}

Then, for $n\geq0$ , $s\geq 0$ , we obtain

\begin{equation*} \beta_{n}(s) = \int_0^\infty\frac{(su)^n}{n!}{\text{e}}^{-su}\,{\text{d}}{F}(u) = \frac{({-}s)^n}{n!}\frac{{\text{d}}^n}{{\text{d}} s^n}\phi_F(s).\end{equation*}

Therefore, $\beta_{n}(s) = \alpha_n(s) - \alpha_{n+1}(s)$ . Considering the above quantities for the equilibrium distribution $F_{(1)}$ , this leads to

(2.14) \begin{equation} \beta'_{n}(s) = \alpha'_n(s) - \alpha'_{n+1}(s),\end{equation}

where

\begin{equation*} \beta'_{n}(s) = \int_0^\infty\frac{(su)^n}{n!}{\text{e}}^{-su}\,{\text{d}}{F_{(1)}}(u) = \frac{({-}s)^n}{n!}\frac{{\text{d}}^n}{{\text{d}} s^n}\phi_{F_{(1)}}(s).\end{equation*}

From (2.11) and (2.14) we now note that

(2.15) \begin{equation} \beta'_{n}(s) = \frac{1}{\mu s}\alpha_{n+1}(s).\end{equation}

Observe that (2.15) is another connection between the DF and its equilibrium distribution by means of the Laplace transform. Let $f_{(1)}(x) = {\bar{F}(x)}/{\mu}$ be the density of $F_{(1)}$ . Then, in the presence of the relation in (2.15), we can restate [Reference Block and Savits11, Corollary 3.7] for the first derived distribution. Before going to the lemma, we first recall the definition of PF $_r$ .

Note that ‘ $f_{(1)}$ is PF $_{r}$ in $x>0$ ’ is equivalent to ‘ ${\bar{F}(x)}/{\mu}$ is PF $_{r}$ in $x>0$ ’, i.e. $\bar{F}(x)$ is PF $_{r}$ in $x>0$ , as $\mu >0$ . Again using (2.15), ‘ $\beta'_{n}(s)$ is PF $_{r}$ in $n \geq 0$ for any $s>0$ ’ is equivalent to saying ‘ $({1}/{\mu s})\alpha_{n+1}(s)$ is PF $_{r}$ in $n \geq 0$ for any $s>0$ ’, i.e. if and only if, for any $s>0$ , $\alpha_{n+1}(s)$ is PF $_{r}$ in $n \geq 0$ . Thus we have the following proposition.

The particular case $r=2$ yields the following corollary, which is nothing but Vinogradov’s result in the continuous case.

Therefore, exploiting the theory of total positivity in conjunction with our results involving the equilibrium distribution, we arrive at the idea of characterizing the IFR class in the special case when F is continuous.

Recall the theorem in [Reference Vinogradov40] that states that a necessary and sufficient condition for a DF F to be an IFR class is $a^{2}_n(s) \geq a^{}_{n-1}(s)a^{}_{n+1}(s)$ for all $n \geq 0$ , i.e., using (1.1),

\begin{equation*} \frac{\alpha^{}_{n+1}(s)}{\alpha^{}_{n}(s)}\geq \frac{\alpha^{}_{n+2}(s)}{\alpha^{}_{n+1}(s)}.\end{equation*}

This is equivalent to ${\alpha^{}_{n+1}(s)}/{\alpha^{}_{n}(s)}$ being decreasing in $n=0,1,2,\ldots$ for all $s>0$ . Now it is easy to prove that F has decreasing mean residual life if and only if $F_{(1)}$ is IFR. Using this together with Proposition 2.2, we can provide an alternative proof of Block and Savits’ characterization for the DMRL class. For completeness, we first state the theorem as given in [Reference Block and Savits11] and then present our proof.

Theorem 2.3 ([Reference Block and Savits11].) For non-negative distributions F, F is DMRL if and only if

$$\frac{\sum_{k=n}^\infty\alpha_{k}(s)}{\alpha_{n}(s)}$$

is decreasing in $n = 1,2,\ldots$ for all $s > 0$ .

Proof. $F_{(1)}$ is IFR if and only if ${\alpha'_{n+1}(s)}/{\alpha'_{n}(s)}$ is decreasing in $n=0,1,2,\ldots$ and $s>0$ . Replacing $\alpha'_{n}(s)$ by using Proposition 2.2, we have that F is DMRL if and only if

\begin{alignat*}{2} & \sum_{k=n+1}^\infty\alpha_{k}(s)/\sum_{k=n}^\infty\alpha_{k}(s) & & \text{is decreasing in } n>0 \text{ and } s>0 \\ \Leftrightarrow\ & \Bigg[\alpha_{n}(s)+\sum_{k=n+1}^\infty\alpha_{k}(s)\Bigg]/\sum_{k=n+1}^\infty\alpha_{k}(s) \quad & & \text{is increasing in } n>0 \text{ and } s>0 \\ \Leftrightarrow\ & \sum_{k=n+1}^\infty\alpha_{k}(s)/\alpha_{n}(s) & & \text{is decreasing in } n>0 \text{ and } s>0 \\ \Leftrightarrow\ & \sum_{k=n}^\infty\alpha_{k}(s)/\alpha_{n}(s) & & \text{is decreasing in } n>0 \text{ and } s>0. \end{alignat*}

This illustrates our new approach of obtaining BS type characterizations using the machinery developed here. We shall now apply this technique to prove further interesting results.

2.4. The decreasing mean residual life in harmonic average class

In [Reference Deshpande, Kochar and Singh14], the authors illustrated various new concepts of ageing criteria that arise as a consequence of positive ageing of equipment, and the notion of this particular ageing class is one such. Let $e_F(t)=({1}/{\bar{F}(t)})\int_t^\infty\bar{F}(u)\,{\text{d}} u$ be the mean residual life (MRL) function of F. We have the following definition.

The interrelationships of the DMRLHA class with the other ageing classes are also discussed in [Reference Deshpande, Kochar and Singh14]. There are well-established BS type characterization results for various ageing classes such as IFRA, NBUE, HNBUE, etc. in the existing literature [Reference Block and Savits11, Reference Klefsjö26, Reference Klefsjö27, Reference Vinogradov40]. We have already demonstrated the possibility of obtaining such characterizations using the appropriate ageing properties of the corresponding equilibrium distribution in the context of the DMRL class. Motivated by this, we now try to exploit the same technique to obtain new BS type characterizations. In particular, our new approach yields a simple proof of the characterization result for the DMRLHA class, whereas the standard techniques, outlined in [Reference Block and Savits11], require an involved calculation. We thus adopt this alternative method of proof.

As discussed earlier, an interesting observation in the context of BS type characterization results is that we can often anticipate the exact forms that would appear in the characterization results of the classes. However, for the DMRLHA class this is not at all obvious. Next we present an intuitive justification behind the specific form that arises in Theorem 2.4.

Utilizing the inversion formula of the MRL function [Reference Meilijson35],

\begin{equation*} \bar{F}(t) = \begin{cases} \dfrac{e_F(0)}{e_F(t)}\exp\bigg[{-}\displaystyle\int_0^t\dfrac{{\text{d}} u}{e_F(u)}\bigg], & 0\leq t < {F}^{-1}(1), \\ 0, & t\geq {F}^{-1}(1), \end{cases}\end{equation*}

where ${F}^{-1}(1)\;{:}{=}\; \sup\{t\mid F(t)<1\}$ , we have

\begin{equation*} \frac{1}{t}\int_0^t\frac{{\text{d}} u}{e_F(u)} = \ln\bigg(\frac{\mu}{e_F(t)\bar{F}(t)}\bigg)^{1/t},\end{equation*}

where $\mu = e_F(0)$ is the mean of the DF F. Therefore, F is DMRLHA if and only if

\begin{align*} \bigg[\frac{1}{t}\int_0^t\frac{1}{e_F(x)}\,{\text{d}} x\bigg]^{-1} \text{ is decreasing in } t & \Leftrightarrow \ln\bigg(\frac{\mu}{e_F(t)\bar{F}(t)}\bigg)^{1/t} \text{ is increasing in } t \\ & \Leftrightarrow \bigg(\frac{\mu}{e_F(t)\bar{F}(t)}\bigg)^{1/t} \text{ is increasing in } t.\end{align*}

Observing that $e_F(t)\bar{F}(t)$ corresponds to $({1}/{s})\sum_{k=n+1}^\infty\alpha_{k}(s)$ and $\mu s = \sum_{k=1}^\infty \alpha_{k} (s)$ , the expression for the DMRLHA class no longer comes as a surprise.

2.5. The new better than used in failure rate class

The new better than used in failure rate (NBUFR) ageing class was introduced in [Reference Abouammoh and Ahmed1]. The definition given there for this class is reproduced here.

Definition 2.6. An absolutely continuous DF F is said to be NBUFR (NWUFR) with failure rate $r_F(t)$ if $r_F(0) \leq (\geq)\, r_F(t)$ for all $t>0$ , where

(2.16) \begin{equation} r_F(0)\;{:}{=}\; \lim_{t\to 0^+} t^{-1}\int_0^t r_F(x)\,{\text{d}} x. \end{equation}

The following equivalent form of this definition was also suggested in [Reference Abouammoh and Ahmed1].

Preservation properties under various reliability operations as well as shock model theory for both NBUFR and NWUFR have been discussed in [Reference Abouammoh and Ahmed1, Reference Gohout and Kuhnert16]. The non-monotonic analogue of the NBUFR class, namely new worse then better than used in failure rate (NWBUFR), was introduced in [Reference Pandey and Mitra36]. For the NWBUFR class, weak convergence issues, reliability bounds, and interrelationships with some other non-monotonic ageing families are well-studied in [Reference Majumder and Mitra33]. However, the literature so far does not contain any definition of the NBUFR class without the assumption of the existence of the density. We try to redress this. Our definition for this class is motivated by the definition of the IFR class; the notion is broadened by dropping the requirement for absolute continuity. Taking our cue from this, we adopt the definition of $r_F(0)$ as in [Reference Loh32]:

(2.17) \begin{equation} r_F(0)\;{:}{=}\; -\lim_{t\to 0^+} \frac{1}{t}\log \bar{F}(t).\end{equation}

Henceforth, $r_F(0)$ is defined to be the quantity given in (2.17). We thus propose the following definition.

In fact, we shall show later that the condition $\mathbb{E}(X^{r+s+2}) < \infty$ can be dropped (see Remark 2.3). Next we state the main result.

Proof. Necessity. Fix $s>0$ . From the definition of $a_{n}(s)$ , we have

\begin{equation*} a_{n}(s) = \int_0^\infty\frac{u^n}{n!}{\text{e}}^{-su}\bar{F}(u)\,{\text{d}} u = \int_0^\infty\frac{u^n}{n!}\bar{G}(u)\,{\text{d}} u, \end{equation*}

where $\bar{G}(u)= {\text{e}}^{-su}\bar{F}(u)$ . Note that $\bar{G}(u)$ is NBUFR with $r_G(0)= (r_F(0)+s)$ if and only if $\bar{F}(u)$ is NBUFR with $r_F(0)$ as defined by (2.17). Note that

(2.18) \begin{align} a_n(s) = \frac{\mu_{n+1}}{(n+1)!} = \lambda_{n+1}, \end{align}

where $\mu_{n+1}$ is the $(n+1)$ th moment of G. Now, using Lemma 2.7 for the DF G, we have $\lambda_{n+m+2}\leq \lambda_{m+1}/[r_G(0)]^{n+1}$ for all $n, m \geq 0$ . Utilizing (2.18) gives $a_{n+m+2}\leq a_{m+1}/[r_G(0)]^{n+1}$ for all $n, m \geq0$ , $s>0$ . Now, considering (1.1), we further obtain $\alpha_{n+m+2}(s) \leq s^{n+1}/[r_G(0)]^{n+1} \alpha_{m+1}(s)$ . Finally, substituting the value of $r_G(0)$ , the necessary part is proved.

Sufficiency. We have

(2.19) \begin{equation} \alpha_{n+m}(s) \leq \frac{1}{[ 1+r_F(0)/s]^n}\alpha_m(s) \quad \text{for all } n, m >0,\, s>0. \end{equation}

Now let x and $x+y$ be two continuity points of F, where $x,y>0$ . For $n=1,2,\ldots$ , define $s={n}/{x}$ and $m=[{ny}/{x}]$ . Now letting $n\to\infty$ and using Lemma 2.2, we have, from (2.19), $\bar{F}(x+y) \leq {\text{e}}^{-r_F(0)x}\bar{F}(y)$ . The continuity points of F are dense in $[0,\infty)$ , hence the above inequality holds for all $x,y \geq 0$ . Therefore, F is NBUFR.

Remark 2.2. Note that the condition of Theorem 2.5 is equivalent to

(2.20) \begin{equation} \alpha_{m+1}(s) \leq \frac{1}{[1+r_F(0)/s]}\alpha_m(s) \quad \text{for all } m >0,\,s>0. \end{equation}

Thus, (2.20) can also be treated as the necessary and sufficient condition for characterizing the NBUFR class.

At this juncture, it is appropriate to highlight an interesting application of our results. Theorem 2.5 in conjunction with Proposition 2.2 can be used to provide a short and elegant alternative proof of Block and Savits’ type characterization result for the NBUE class. The simplicity of this proof makes it rather attractive.

Theorem 2.6 ([Reference Block and Savits11].) For non-negative distributions F, F is NBUE if and only if

$$\alpha_{n}(s)\sum_{k=0}^\infty\alpha_{k}(s) \geq \sum_{k=n}^\infty\alpha_{k}(s)$$

for all $n>0$ and all $s>0$ .

Proof. From [Reference Deshpande, Kochar and Singh14, Proposition 6.3], $F_{(1)}$ is NBUFR $\Leftrightarrow$ F is NBUE. Also, from (2.20) we have that $F_{(1)}$ is NBUFR if and only if

(2.21) \begin{equation} \alpha'_{n+1}(s) \leq \frac{1}{[1+r_{F_{(1)}}(0)/s]}\alpha'_{n}(s) \quad \text{for all } n>0, \, s>0. \end{equation}

Now it is easy to observe that $r_{F_{(1)}}(0)={1}/{\mu}$ , where $\mu$ is the mean of F. Proposition 2.2 together with (2.21) shows that, for all $n>0$ , $s>0$ ,

\begin{equation*} \sum_{k=n+2}^\infty\alpha_{k}(s) \leq \frac{1}{[1+{1}/{\mu s}]}\sum_{k=n+1}^\infty\alpha_{k}(s), \end{equation*}

i.e. $[1+\mu s]\sum_{k=n+2}^\infty\alpha_{k}(s) \leq \mu s\sum_{k=n+1}^\infty\alpha_{k}(s)$ , i.e. $\sum_{k=n+1}^\infty\alpha_{k}(s) \leq \alpha_{n}(s)\sum_{k=1}^\infty\alpha_{k}(s)$ . As $\alpha_{0}(s)=1$ , we finally obtain that F is NBUE if and only if $\sum_{k=n}^\infty\alpha_{k}(s) \leq \alpha_{n}(s)\sum_{k=0}^\infty\alpha_{k}(s)$ for all $n>0$ and $s>0$ .

2.6. The new better than used average in failure rate class

The new better than used average in failure rate (NBAFR) class was introduced in [Reference Loh32] as a generalization of the NBU class. A noticeable property of the NBAFR class is that it neither contains nor is contained in the NBUE family, which is an immediate generalization of the NBU class. First we look into the definition of the NBAFR class as proposed in [Reference Loh32].

The preservation properties of the NBAFR class under well-known reliability operations were also investigated in [Reference Loh32]. A non-monotonic generalization of this class was introduced in [Reference Majumder and Mitra34], which also investigated a moment bound, weak convergence, and related issues for this class. Motivated by [Reference Shokry, Ahmed, Rakha and Hewedi39, Theorem 2.1], we now give the following proposition that gives a moment inequality for the NBAFR class.

Now it would be appropriate to propose the characterization result for the NBAFR class.

Proof. Necessity. Fix $s>0$ . Consider the DF G defined by $\bar{G}(u) = {\text{e}}^{-su}\bar{F}(u)$ . Then G is NBAFR with $r_G(0) = (r_F(0)+s)$ if and only if F is NBAFR with $r_F(0)$ as defined in (2.17). As before, we have

\begin{equation*} a_{n}(s) = \int_0^\infty\frac{u^n}{n!}\bar{G}(u)\,{\text{d}} u = \frac{\mu_{n+1}}{(n+1)!} = \lambda_{n+1}. \end{equation*}

Using Proposition 2.4 for the DF G we have $\lambda_{n+1} \leq {1}/{[r_G(0)]^{n+1}}$ for all $n\geq 0$ , $s>0$ . Replacing $r_G(0)$ , we get that if F is NBAFR then $a_{n}(s) \leq {1}/{[r_F(0)+s]^{n+1}}$ for all $n\geq0$ , $s>0$ . Finally, using (1.1), the proof of the necessary part is completed.

Sufficiency. Now we have

(2.22) \begin{equation} \alpha_{n}(s) \leq \frac{1}{[1+r_F(0)/s]^n} \quad \text{for all } n>0,\,s>0. \end{equation}

Let $x>0$ be a continuity point of F and define $s=s(n,x)$ such that $s={n}/{x}$ . Then letting $n\to\infty$ in (2.22) and using Lemma 2.2 we obtain $\bar{F}(x) \leq {\text{e}}^{-r_F(0)x}$ . Since the continuity points of F are dense, the above inequality holds for all $x>0$ . Hence F is NBAFR.

Using Theorem 2.7, we now furnish an alternative proof for the BS type characterization for the HNBUE class as given in [Reference Klefsjö26].

For the NBAFR class, another significant insight from considering the equilibrium distribution is that it enables us to obtain an upper bound for $r_F(0)$ .

Proof. We have

\begin{align*} \bar{F}_{(1)} (t) = \frac{1}{\mu}\int_t^\infty\bar{F}(u)\,{\text{d}} u & \leq \frac{1}{\mu}\int_t^\infty{\text{e}}^{-r_F(0)u}\,{\text{d}} u \qquad \text{(using the definition of NBAFR)} \\ & = \frac{1}{\mu}\frac{{\text{e}}^{-r_F(0)t}}{r_F(0)}. \end{align*}

Taking the $\log$ of both sides, we have $\log\bar{F}_{(1)}(t) \leq -r_F(0)t - \log\mu r_F(0)$ , i.e. $-{\log\bar{F}_{(1)}(t)}/{t} \geq r_F(0) + {\log\mu r_F(0)}/{t}$ . Taking the limit $t\to 0^+$ on both sides, we obtain

(2.23) \begin{equation} r_{F_{(1)}}(0) \geq r_F(0) + \lim_{t \to 0^+}\frac{1}{t} \log \mu r_F(0). \end{equation}

Suppose $\mu r_F(0) > 1$ . In this case the right-hand side of (2.23) tends to $+\infty$ as $t\to 0^+$ , which implies that $r_{F_{(1)}}(0) = +\infty$ ; this is a contradiction since $r_{F_{(1)}}(0)={1}/{\mu}$ . Therefore, $\mu r_F(0) \not> 1$ . This observation leads to an upper bound of $r_F(0)$ , which is $ r_F(0)\leq{1}/{\mu}$ when F is NBAFR.

Note that our conclusion agrees with [Reference Majumder and Mitra34, Theorem 3.2(iii)] for $k=1$ as we can look upon the NBAFR class as a special case of the NWBAFR class with change point $t_0 = 0$ .

3.1. The NBURFR and NBARFR classes

Proposed in [Reference Abouammoh and Ahmed2], the notions of new better (worse) than used renewal failure rate, NBURFR (NWURFR), and new better (worse) than average used renewal in failure rate, NBARFR (NWARFR), classes arise from the comparison between the failure rate function of a new unit and that of the asymptotic remaining life of a used unit. Let $r_{F_{(1)}}(t)$ be the failure rate functions of $F_{(1)}$ and $r_F(0)$ as defined in (2.17). Then the formal definitions of these classes are as follows.

Proof. For the necessary part, we have, for $n\geq0$ , $s\geq 0$ ,

\begin{align*} \sum_{i=n+1}^\infty\alpha_i(s) & = \sum_{i=n}^\infty s\int_0^\infty\frac{(sy)^i}{i!}{\text{e}}^{-sy}\bar{F}(y)\,{\text{d}} y \\ & = s^2\int_0^\infty\frac{{\text{e}}^{-st}(st)^n}{n!}\bigg(\int_t^\infty\bar{F}(y)\,{\text{d}} y\bigg)\,{\text{d}} t \qquad \text{(by virtue of Proposition 2.1)} \\ & \leq s^2\int_0^\infty\frac{{\text{e}}^{-st}(st)^{n-1}}{\Gamma(n)}\frac{\bar{F}(t)}{r_F(0)}\,{\text{d}} t = \frac{1}{r_F(0)/s}\alpha_n(s). \end{align*}

For the other part, given that $n\geq0$ and $s > 0$ ,

(3.1) \begin{equation} \sum_{i=n+1}^\infty\alpha_i(s) \leq \frac{1}{r_F(0)/s}\alpha_n(s). \end{equation}

Now let $x>0$ be a continuity point of F and $\lim_{n\to \infty}({n}/{s})=x$ . Then, appealing to the DCT and arguing as in the derivation of (2.6), we get

\begin{equation*} \lim_{n\to\infty}\frac{1}{s}\sum_{i=n+1}^{\infty}\alpha_i(s) = \int_x^\infty\bar{F}(u)\,{\text{d}} u. \end{equation*}

Therefore, letting $n\to\infty$ in condition (3.1), we obtain $\int_x^\infty\bar{F}(u)\,{\text{d}} u \leq \bar{F}(x)/r_F(0)$ . Note that this inequality holds for all $x>0$ . Thus, F is NBURFR.

3.2. The NBRUE and HNBRUE classes

Exploiting the concept of renewal theory, two more classes, new better (worse) than renewal used in expectation, NBRUE (NWRUE), and harmonic new better (worse) than renewal used in expectation, HNBRUE (HNWRUE), were introduced in [Reference Abouammoh, Ahmed and Barry3], where the classes are analogues of NBUE and HNBUE in the sense that here the MRL function of a new unit is compared with that of the asymptotic remaining life of a used unit under a repeated renewal policy. Considering $e_{F_{(1)}}(t)$ to be the MRL function of $F_{(1)}$ , we have the following definitions for these classes.

Definition 3.4. A life DF F is said to be HNBRUE (HNWRUE) if

$$\bigg[{\frac{1}{t}\int_0^t\{e_{F_{(1)}}(x)\}^{-1}\,{\text{d}} x}\bigg]^{-1} \leq (\geq)\,\mu$$

for $t \geq 0$ , which is equivalent to $\int_t^\infty\int_x^\infty\bar{F}(u)\,{\text{d}} u\,{\text{d}} x \leq (\geq)\,{\text{e}}^{-t/\mu}\int_0^\infty\int_x^\infty \bar{F}(u)\,{\text{d}} u\,{\text{d}} x$ for all $t \geq 0$ .

Several important issues such as the preservation properties of these classes under various reliability operations, interrelationships with the other classes, etc. are discussed in [Reference Bhattacharjee, Abouammoh, Ahmed and Barry9].

However, establishing BS type characterizations for these classes seems impossible using the techniques of [Reference Block and Savits11]. Even if some tedious manipulation yields the necessary part, the proof for the sufficiency part requires the boundedness of the function $h(y)= \int_y^\infty\int_v^\infty\bar{F}(u)\,{\text{d}} u\,{\text{d}} v$ in order to apply the DCT or even [Reference Billingsley10, Theorem 5.2], which, in turn, results in imposing a strong assumption on the underlying DF F. We face a similar problem when dealing with the HNBRUE class.

Under the circumstances, we are forced to seek an alternative route for obtaining BS type characterization results in the context of these two classes. We noticed that these classes have connections with some other renewal classes in terms of the equilibrium distribution. The findings are recorded in the following lemma.

Lemma 3.1 makes a couple of useful additions to the list compiled in [Reference Deshpande, Kochar and Singh14, p. 756]; furthermore, it also enables us to adopt our new approach in proving the characterization results. We now have the following two theorems.

Proof. From Theorem 3.1 we have that $F_{(1)}$ is NBURFR if and only if

\begin{equation*} \sum_{i=n+1}^\infty\alpha'_k(s) \leq \frac{1}{r_{F_{(1)}}(0)/s}\alpha'_n(s) \quad \text{for all } n\geq0,\,s>0. \end{equation*}

Using this with Proposition 2.2 we obtain

\begin{equation*} \frac{1}{\mu s}\sum_{i=n+1}^\infty\sum_{j=i+1}^\infty\alpha_j(s) \leq \frac{1}{r_{F_{(1)}}(0)/s}\frac{1}{\mu s}\sum_{i=n+1}^\infty\alpha_i(s) \quad \text{for all } n\geq0,\,s>0. \end{equation*}

Recalling that $r_{F_{(1)}}(0)={1}/{\mu}$ and using Lemma 3.1(i) we finally obtain that F is NBRUE if and only if

\begin{equation*} \sum_{i=n+1}^\infty\sum_{j=i+1}^\infty\alpha_j(s) \leq \sum_{k=1}^\infty\alpha_k(s)\sum_{i=n+1}^\infty\alpha_i(s) \quad \text{for all } n \geq 0, \,s>0.\end{equation*}

Theorem 3.4. F is HNBRUE if and only if

$$ \sum_{i=n+1}^\infty\sum_{j=i+1}^\infty\alpha_j(s) \leq \bigg(1-\frac{1}{\alpha(s)}\bigg)^n\sum_{i=0}^\infty\sum_{j=i+2}^\infty\alpha_j(s) $$

for all $n \geq 0$ and $s>0$ , where $\alpha(s) = \sum_{k=0}^\infty \alpha_k(s)=1+\mu s$ .

As the HNBRUE family is analogous to HNBUE, it is natural to ask whether this class also possesses finite moments of all order. To answer this, we first look into the following proposition.

Proof. Let $\mu_{r}$ be the rth moment of F. Then we have

\begin{align*} \mu_{r+1} & = (r+1)\int_0^\infty x^r\bar{F}(x)\,{\text{d}} x \\ & = (r+1)\int_0^\infty\int_0^x ry^{r-1}\bar{F}(x)\,{\text{d}} y\,{\text{d}} x \\ & = (r+1)\int_0^\infty ry^{r-1}\int_y^\infty\bar{F}(x)\,{\text{d}} x\,{\text{d}} y \\ & = \mu(r+1)\int_0^\infty ry^{r-1}\bar{F}_{(1)}(y)\,{\text{d}} y = \mu(r+1)\mu^{(1)}_{r}, \end{align*}

where $\mu^{(1)}_{r}$ is the rth moment of $F_{(1)}$ . The result follows.

Proof. We first demonstrate a proof for the HNBRUE class. From the definition, it is clear that F is HNBRUE if and only if

\begin{equation*} \bar{F}_{(2)}(t) = \exp\bigg({-}\int_0^t\{e_{F_{(1)}}(x)\}^{-1}\,{\text{d}} x\bigg) \leq \exp({-}t/\mu), \end{equation*}

where $\bar{F}_{(2)}(t)$ is the survival function of the second derived distribution of $F(t)$ . Now, this inequality shows that the second derived distribution of F possesses finite moments of all orders. Therefore, successive applications of Proposition 3.1 yield the desired result.

The proof for the NBARFR class is evident from Proposition 3.1.

Also, since NBRUE $\subset$ HNBRUE and NBURFR $\subset$ NBARFR, the results for the NBRUE and NBURFR classes are immediate.