A note on some preservation results on the Laplace transform ordering of residual lives

Abstract

In this paper, we provide several preservation results for Laplace transform ordering of residual lives (Lt-rl) under the reliability operations convolutions, mixtures and series systems. We also show that this order is preserved under completely monotone transformations.

Keywords

Laplace transform convolution mixture series systems completely monotone function

1. Introduction

Stochastic comparisons between probability distributions play a fundamental role in probability, statistics, and some related areas, such as reliability theory, survival analysis, economics and actuarial science (see Shaked and Shanthikumar [10] for an exhaustive monograph on this topic). Several concepts of the stochastic comparisons between random variables have been defined and studied in the literature, since they are useful tools in many different areas of the applied probability and statistics (see Shaked and Shanthikumar [11]). Laplace transform as an useful notion in applied mathematics and engineering, is also important in many areas of the probability and statistics. Rolski and Stoyan [9] defined the Laplace transform order ( $⩽_{Lt}$ ) based on the Laplace transform. Then Klefsjo [7] obtained an useful ageing property of it and Alzaid et al. [2] presented general preservation properties and applications of this order. Shaked and Wong [12] introduced stochastic orders based on the ratios of the Laplace transforms. Belzunce et al. [4] introduced Laplace transform ordering of residual lives (Lt-rl). They also considered some properties and relationships of this new definition to some well known orders. Ahmed and Kayid [1] studied some preservation results of this order. Gandotra et al. [5] also developed some preservation properties of the Laplace transform ordering of residual lives and three other orders. In particular, they established for any nonnegative random variable Y independent from $X_{1}$ and $X_{2}$ and under suitable conditions, $X_{1} ⩽_{Lt - rl} X_{2}$ implies $X_{1} - Y ⩽_{Lt - rl} X_{2} - Y$ . In this paper, we provide other preservation properties of this order. Section 2, contains definitions and some basic propositions used through the paper. In Section 3, we present some preservation results under the operations of convolution and mixtures, series systems and completely monotone transformations. Finally, Section 4 is devoted to the conclusion.

Throughout this paper, the terms “increasing” and “decreasing” mean “non-decreasing” and “non-increasing”, respectively. All integrals and expectations are implicitly assumed to exist whenever they are written. Also, all distribution functions under consideration are restricted to be absolutely continuous with support in the positive real line $R^{+}$ .

2. Preliminaries

In this section, we first briefly recall some definitions, two propositions and one lemma which will be needed later.

Let X and Y be two nonnegative random variables, with distribution functins $F_{X}$ and $F_{Y}$ , and denote their survival functions by ${\overline{F}}_{X}$ and ${\overline{F}}_{Y}$ , respectively. Also, denote their Laplace transform functions by $L_{X} (s)$ and $L_{Y} (s)$ , respectively. The Laplace–Stieltjes transform of $F_{X}$ is given by $\begin{matrix} L_{X} (s) = \int_{0}^{\infty} e^{- s u} d F_{X} (u), s > 0 . \end{matrix}$ In the following, the definitions of Laplace transform order and Laplace transform order of residual lives are given.

Definition 2.1.
A random variable X is smaller than a random variable Y in the Laplace transform order (denoted by $X ⩽_{Lt} Y$ ), if $L_{X} (s) ⩾ L_{Y} (s)$ , for all $s > 0$ .

If $\begin{array}{rcl} L_{X}^{} (s) = \int_{0}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u = \frac{1 - L_{X} (s)}{s}, \end{array}$ then $X ⩽_{Lt} Y$ , if and only if $L_{X}^{} (s) ⩽ L_{Y}^{} (s)$ , for all $s > 0$ .

In general, for any nonnegative random lifetime X, the residual lifetime of X is defined by $X_{t} = [X - t | X > t]$ , where $t \in (0, l_{X})$ and $l_{X} = sup {t : F_{X} (t) < 1}$ .

Belzunce et al.* [4] defined the strong comparison based on Laplace transform functions of X and Y as follows.
Definition 2.2.
A nonnegative random variable X is said to be smaller than a random variable Y in the Laplace transform of residual lives order (denoted by $X ⩽_{Lt - rl} Y$ ) if $\begin{array}{rcl} (1) & X_{t} ⩽_{Lt} Y_{t}, for all t \in (0, l_{X}) \cap (0, l_{Y}) . \end{array}$

They obtained equivalent condition for this order which is given in the following.
Proposition 2.3.
Let X and Y be two continuous nonnegative random variables, then $X ⩽_{Lt - rl} Y$ if and only if $\begin{array}{rcl} (2) & \frac{\int_{t}^{\infty} e^{- s u} {\overline{F}}_{Y} (u) d u}{\int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u}, \end{array}$ is increasing in $t > 0$ .

Belzunce et al. [4] also proposed new ageing classes based on Lt-rl order. One of them is decreasing (increasing) residual live in the Laplace order class.
Definition 2.4.
X is ${DRL}_{Lt}$ ( ${IRL}_{Lt}$ ) (decreasing (increasing) residual live in the Laplace order) if $\begin{array}{rcl} (3) & X_{t} ⩾_{Lt} (⩽_{Lt}) X_{t^{'}}, \forall t < t^{'}, t, t^{'} \in (0, l_{X}) . \end{array}$

Here, we recall the definition of completely monotone (c.m.) functions, which will be used in the next section.
Definition 2.5.
A function $h : R^{+} ⟶ R$ is completely monotone if all its derivatives $h^{(n)}$ exist and for all $x > 0$ and $n = 0, 1, 2, \dots$ , $\begin{array}{rcl} {(- 1)}^{n} h^{(n)} (x) ⩾ 0 . \end{array}$

Two following propositions due to Shaked and Shantikumar [11] gives conditions under which the Laplace transform order is closed under operation of taking minima and transformations with completely monotone derivative.
Proposition 2.6.
Let the independent nonnegative random variables $X_{1}, X_{2}, \dots, X_{n}, Y_{1}, Y_{2}, \dots, Y_{n}$ have the survival functions ${\overline{F}}_{1}, {\overline{F}}_{2}, \dots, {\overline{F}}_{n}, {\overline{G}}_{1}, {\overline{G}}_{2}, \dots, {\overline{G}}_{n}$ , respectively. If for $i = 1, 2, \dots, n$ , $X_{i} ⩽_{Lt} Y_{i}$ and ${\overline{F}}_{i}$ and ${\overline{G}}_{i}$ are completely monotone, then $\begin{array}{l} min {X_{1}, X_{2}, \dots, X_{n}} \\ (4) & ⩽_{Lt} min {Y_{1}, Y_{2}, \dots, Y_{n}} . \end{array}$
Proposition 2.7.
Suppose that X and Y be two nonnegative random variables and $X ⩽_{Lt} Y$ . Then $h (X) ⩽_{Lt} h (Y)$ for all nonnegative functions h with a completely monotone derivative.
Definition 2.8 (Karlin [6]).

Let χ and γ be two subsets of the real line. A non-negative function h defined on $χ \times γ$ is said to be TP₂ if $\begin{matrix} h (x, y) h (x^{'}, y^{'}) ⩾ h (x^{'}, y) h (x, y^{'}), \end{matrix}$ whenever $x ⩽ x^{'}$ , $y ⩽ y^{'}$ and $x, x^{'} \in χ$ , $y, y^{'} \in γ$ .

Here, we give one lemma which is needed in the sequel.

Lemma 2.9 (Marshall and Olkin [8]).

The function $k (x, y) = h (y - x)$ is TP ₂ in $(x, y) \in R^{+} \times R^{+}$ if and only if h is nonnegative and logconcave.

3. Main results

Useful properties of the stochastic orders are their closure with respect to typical reliability operations like convolution or mixture (see Barlow and Proschan [3] and Shaked and Shanthikumar [10]). In this section, we present some preservation properties of the Laplace transform order of residual lives. First, we prove one proposition which will be used later.

Proposition 3.1.
Let X be a nonnegative continuous random variable with survival function ${\overline{F}}_{X}$ , and $\begin{array}{rcl} (5) & ψ_{X} (t) = \int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u . \end{array}$ If $X \in {DRL}_{Lt}$ , then $ψ_{X} (t)$ is logconcave in $t \in R^{+}$ , for all $s > 0$ .
Proof.
Suppose that $X \in {DRL}_{Lt}$ . Therefore according to definition 2.4, we have $X_{t} ⩾_{Lt} X_{t^{'}}$ for all $t < t^{'}$ , $t, t^{'} \in (0, l_{X})$ , i.e. $\begin{matrix} \frac{\int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u}{e^{- s t} {\overline{F}}_{X} (t)} ⩾ \frac{\int_{t^{'}}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u}{e^{- s t^{'}} {\overline{F}}_{X} (t^{'})}, \end{matrix}$ which is equivalent to $\begin{array}{l} \frac{\partial}{\partial t} log [\int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u] \\ ⩾ \frac{\partial}{\partial t^{'}} log [\int_{t^{'}}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u] . \end{array}$ The above inequality shows that $\frac{\partial}{\partial t} log [\int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u]$ is decreasing in t. Thus $log \int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u$ is a concave function. So, the desired result is obtained. □
Theorem 3.2.
Suppose that the lifetime random variable X is ${DRL}_{Lt}$ . Then $X ⩽_{Lt - rl} X + Y$ for all nonnegative random variables Y which are independent of X.
Proof.
We must prove that $X ⩽_{Lt - rl} X + Y$ which is equivalent to $\begin{array}{rcl} \frac{\int_{t}^{\infty} e^{- s u} {\overline{F}}_{X + Y} (u) d u}{\int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u) d u} = \frac{ψ_{X + Y} (t)}{ψ_{X} (t)} \end{array}$ is increasing in $t ⩾ 0$ . Since $X \in {DRL}_{Lt}$ and using Proposition 3.1, we have $ψ_{X} (t)$ is logconcave. Therefore using Lemma 2.9, we have $ψ_{X} (t - y)$ is TP₂ in $(t, y)$ i.e. $\frac{ψ_{X} (t - y)}{ψ_{X} (t)}$ is increasing in $t ⩾ 0$ for all $y ⩾ 0$ . But $\begin{array}{rcl} ψ_{X + Y} (t) & = & \int_{t}^{\infty} e^{- s u} {\overline{F}}_{X + Y} (u) d u \\ = & \int_{0}^{\infty} \int_{t}^{\infty} e^{- s u} {\overline{F}}_{X} (u - y) d u d F_{Y} (y) \\ = & \int_{0}^{\infty} e^{- s y} ψ_{X} (t - y) d F_{Y} (y) . \end{array}$ Thus $\begin{array}{rcl} (6) & \frac{ψ_{X + Y} (t)}{ψ_{X} (t)} = \int_{0}^{\infty} \frac{ψ_{X} (t - y)}{ψ_{X} (t)} e^{- s y} d F_{Y} (y) . \end{array}$ Since $\frac{ψ_{X} (t - y)}{ψ_{X} (t)}$ is increasing in $t ⩾ 0$ and from relation (6), we have $\frac{ψ_{X + Y} (t)}{ψ_{X} (t)}$ is increasing in $t ⩾ 0$ . So, the proof is complete. □

Let now $X (θ)$ be a random variable having survival function ${\overline{F}}_{θ}$ and let $Θ_{i}$ be a random variable having distribution function $G_{i}$ , for $i = 1, 2$ , and support $R^{+}$ . The following theorem is a closure of Lt-rl order under mixture.
Theorem 3.3.
Let ${X (θ), θ \in R^{+}}$ be a family of random variables independent of $Θ_{1}$ and $Θ_{2}$ . If $Θ_{1} ⩽_{lr} Θ_{2}$ and $X (θ_{1}) ⩽_{Lt - rl} X (θ_{2})$ , for all $θ_{1} ⩽ θ_{2}$ , then $\begin{matrix} X (Θ_{1}) ⩽_{Lt - rl} X (Θ_{2}) . \end{matrix}$
Proof.
Let ${\overline{F}}_{i}$ be the survival function of $X (Θ_{i})$ , for $i = 1, 2$ . We known that $\begin{matrix} {\overline{F}}_{i} (x) = \int_{0}^{\infty} {\overline{F}}_{θ} (x) d G_{i} (θ) . \end{matrix}$ We should prove that $Φ (i, t) = \int_{t}^{\infty} e^{- s u} {\overline{F}}_{i} (u) d u$ is TP₂ in $i \in {1, 2}$ and $t \in R^{+}$ . But $\begin{array}{rcl} Φ (i, t) & = & \int_{t}^{\infty} e^{- s u} {\overline{F}}_{i} (u) d u \\ = & \int_{t}^{\infty} e^{- s u} \int_{0}^{\infty} {\overline{F}}_{θ} (u) d G_{i} (θ) d u \\ (7) & = & \int_{0}^{\infty} g_{i} (θ) ϕ (θ, t) d θ, \end{array}$ where $\begin{matrix} ϕ (θ, t) = \int_{t}^{\infty} e^{- s u} {\overline{F}}_{θ} (u) d u . \end{matrix}$ Note that $X (θ_{1}) ⩽_{Lt - rl} X (θ_{2})$ implies that $ϕ (θ, t)$ is ${TP}_{2} (θ, t)$ in $θ \in R^{+}$ and $t \in R^{+}$ . Assumption $Θ_{1} ⩽_{lr} Θ_{2}$ means that $g_{i} (θ)$ , as a function of θ and of $i \in {1, 2}$ is TP₂. Now in view of the general composition theorem of Karlin [6], $Φ (i, t)$ is TP₂ in $i \in {1, 2}$ and $t \in ℜ^{+}$ , which completes the proof. □

It is a well-known fact that some stochastic orders are preserved under the formation of a series/parallel systems. In the following result, we show that the Lt-rl order is preserved by a series system.
Theorem 3.4.
Let $X_{1}, X_{2}, \dots, X_{n}$ and $Y_{1}, X_{2}, \dots, Y_{n}$ have the survival functions ${\overline{F}}_{1}, {\overline{F}}_{2}, \dots, {\overline{F}}_{n}$ and ${\overline{G}}_{1}, {\overline{G}}_{2}, \dots, {\overline{G}}_{n}$ , respectively. If for $i = 1, \dots, n$ , $X_{i} ⩽_{Lt - rl} Y_{i}$ and ${\overline{F}}_{i}$ and ${\overline{G}}_{i}$ are completely monotone functions, then $\begin{matrix} min {X_{1}, \dots, X_{n}} ⩽_{Lt - rl} min {Y_{1}, Y_{2}, \dots, Y_{n}} . \end{matrix}$
Proof.
Let $T_{n} = min {X_{1}, X_{2}, \dots, X_{n}}$ and $W_{n} = min {Y_{1}, Y_{2}, \dots, Y_{n}}$ . According to definition of residual lifetime function, we have $\begin{array}{rcl} {(T_{n})}_{t} & = & (T_{n} - t | T_{n} > t) \\ = & min {[X_{i} - t | X_{i} > t], i = 1, \dots, n} \\ = & min_{1 ⩽ i ⩽ n} {(X_{i})}_{t} . \end{array}$ Similarly, $\begin{array}{rcl} {(W_{n})}_{t} & = & (W_{n} - t | W_{n} > t) \\ = & min_{1 ⩽ i ⩽ n} {(Y_{i})}_{t} . \end{array}$ Assumption $X_{i} ⩽_{Lt - rl} Y_{i}$ is equivalent to ${(X_{i})}_{t} ⩽_{Lt} {(Y_{i})}_{t}$ . For $i = 1, \dots, n$ , we denote the survival functions of ${(X_{i})}_{t}$ and ${(Y_{i})}_{t}$ by ${\overline{F}}_{i | t}$ and ${\overline{G}}_{i | t}$ . Thus $\begin{array}{l} {\overline{F}}_{i | t} (x) = \frac{{\overline{F}}_{i} (x + t)}{{\overline{F}}_{i} (t)}, \\ {\overline{G}}_{i | t} (x) = \frac{{\overline{G}}_{i} (x + t)}{{\overline{G}}_{i} (t)} . \end{array}$ The functions ${\overline{F}}_{i | t}$ and ${\overline{G}}_{i | t}$ are both completely monotone, because $\begin{array}{l} {(- 1)}^{n} {\overline{F}}_{i | t}^{(n)} (x) \\ (8) & = \frac{1}{{\overline{F}}_{i} (t)} {(- 1)}^{n} {{\overline{F}}_{i} (x + t)}^{(n)} ⩾ 0 . \end{array}$ Since ${\overline{F}}_{i}$ is completely monotone, (8) is satisfied. Similarly, the function ${\overline{G}}_{i | t}$ is completely monotone. From Proposition 2.6, we conclude that $\begin{array}{rcl} min_{1 ⩽ i ⩽ n} {(X_{i})}_{t} ⩽_{Lt} min_{1 ⩽ i ⩽ n} {(Y_{i})}_{t}, \end{array}$ which is equivalent to ${(T_{n})}_{t} ⩽_{Lt} {(W_{n})}_{T}$ , i.e. $T_{n} ⩽_{Lt - rl} W_{n}$ . Therefore the desired result is obtained. □

The Lt-rl order is closed under transformations with completely monotone derivative as the following theorem shows. Theorem 3.5.
If $X ⩽_{Lt - rl} Y$ and h is any positive function with a completely monotone derivative, then $h (X) ⩽_{Lt - rl} h (Y)$ .
Proof.
We must prove that ${(h (X))}_{t} ⩽_{Lt} {(h (Y))}_{t}$ . From definition of residual life function, this condition is equivalent to $\begin{array}{l} [h (X) | h (X) > t] \\ (9) & ⩽_{Lt} [h (Y) | h (Y) > t], \forall t \in R^{+} . \end{array}$ From assumption $X ⩽_{Lt - rl} Y$ , we have $X_{t} ⩽_{Lt} Y_{t}$ . Therefore $\begin{matrix} X_{h^{(- 1)} (t)} ⩽_{Lt} Y_{h^{(- 1)} (t)}, \forall t \in R^{+}, \end{matrix}$ or equivalently $\begin{array}{l} [X | X > h^{(- 1)} (t)] \\ (10) & ⩽_{Lt} [Y | Y > h^{(- 1)} (t)], \forall t \in R^{+}, \end{array}$ where $h^{(- 1)} (u) = sup {x : h (x) ⩽ u}, \forall u \in R$ . Since derivative of h is completely monotone, we conclude that $h^{' (0)} (x) ⩾ 0$ , i.e. $h (x)$ is increasing in x. Because the continuity of h and the definition of $h^{- 1}$ , we have $\begin{matrix} x > h^{(- 1)} (t) \Leftrightarrow h (x) > t . \end{matrix}$ Therefore (10) can be rewritten as $\begin{array}{l} [X | h (X) > t] \\ (11) & ⩽_{Lt} [Y | h (Y) > t], \forall t \in R^{+} . \end{array}$ Now, using Proposition 2.7, we obtain $\begin{matrix} h ([X | h (X) > t]) ⩽_{Lt} h ([Y | h (Y) > t]) \end{matrix}$ which is equivalent to (9). This completes the proof. □

4. Conclusions

In this paper, we studied some preservation properties of the Laplace transform ordering of residual lives (Lt-rl) under the reliability operations and completely monotone transformations.

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