1. PRELIMINARIES
Let X and Y be two positive random variables, F and G their respective probability distribution functions, and f and g their respective density functions, if they exist. Denote by F = 1 − F the tail (or survival function) of F, for G analogously. We identify distribution functions F and G with respective probability distributions and denote their supports, which are intervals by SF and SG, respectively. Let lF and lG be leftpoints of SF and SG. We denote by φψ the composition of functions φ and ψ: In particular, G−1F denotes the composition of G−1 and F. A distribution F or X is said to be increasing failure rate (IFR) [or decreasing failure rate (DFR)] if −log F is convex (or concave). A distribution F with SF = [a,b], −∞ ≤ a < b < ∞, or X, is said to be IRFR (increasing reversed failure rate) if log F is convex. A distribution F or X is said to be DRFR (decreasing reversed failure rate) if log F is concave. A distribution F or X is said to be increasing failure rate average (IFRA) [or decreasing failure rate average (DFRA)] if −log F is star-shaped (or antistar-shaped) [i.e., −log F(t)/t is increasing (or decreasing) for t > 0]. A distribution F or X is said to be new better than used (NBU) [or new worse than used (NWU)] if −log F is superadditive (or subadditive) on (0,∞) [i.e., if F(s + t) ≤ (≥)F(s)F(t) for all s,t ≥ 0]. A distribution F or X with finite mean is said to be increasing mean residual life (IMRL) [or decreasing mean residual life (DMRL)] if the mean residual life function
is increasing (or decreasing) for all t such that F(t) > 0. We use increasing in place of nondecreasing and decreasing in place nonincreasing. We assume that a/0 is equal to ∞ whenever a > 0.
We deal with some stochastic orders, which are defined not only for positive random variables. Recall their definitions and some properties for completeness. Similarly to Shaked and Shanthikumar [14], we use notation involving random variables. However, stochastic orders are relations between probability distributions.
We say that X is smaller than Y in the likelihood ratio order (X ≤lr Y) if g/f is increasing on SF ∪ SG.
We say that X is smaller than Y in the hazard rate order (X ≤hr Y) if G/F is increasing on SF ∪ SG.
We say that X is smaller than Y in the reversed hazard rate order (X ≤rh Y) if G/F is increasing on SF ∪ SG.
Let X and Y have finite means. We say that X is smaller than Y in the mean residual life order (X ≤mrl Y) if mF(t) ≤ mG(t) for every t > 0.
We say that X is stochastically smaller than Y (X ≤st Y) if F(x) ≥ G(x) for every x or, equivalently, if F(x) ≤ G(x) for every x.
We say that X is smaller than Y in the dispersive order (X ≤disp Y) if G−1F(x) − x is increasing on SF.
Let X and Y have finite means and F(0) = G(0). We say that X is smaller than Y in the excess wealth order (X ≤ew Y) if
for all u ∈ (0,1).
Let now F(0) = G(0) = 0.
We say that X is smaller than Y in the convex transform order (X ≤c Y) if G−1F is convex on SF.
We say that X is smaller than Y in the star order (X ≤* Y) if G−1F is star-shaped (i.e., G−1F(x)/x is increasing on SF).
Let
denote the Laplace–Stieltjes transforms of the distributions F and G, respectively.
We say that Y is smaller than X in the Laplace transform order (Y ≤Lt X) if f*(s) ≤ g*(s) for every s ≥ 0.
Shaked and Wong [16] defined two stochastic orders based on ratios of Laplace transforms.
We say that Y is smaller than X in the Laplace transform ratio order (Y ≤Lt−r X) if g*(s)/f*(s) is increasing in s > 0.
We say that Y is smaller than X in the reverse Laplace transform ratio order (Y ≤r−Lt−r X) if [1 − g*(s)]/[1 − f*(s)] is increasing in s > 0.
The following relations among the above-defined stochastic orders are well known; see Shaked and Shanthikumar [14,15] and Shaked and Wong [16]:
It is easy to notice that
might be considered as a mixture of exponential distributions with means 1/x, x > 0, and a mixing distribution F. Thus,
is the distribution with the density
and is DFR as is the mixture of DFR (exponential) distributions. Similarly, we define
for the distribution G.
Let
be random variables with distributions
, respectively. It is evident that orders based on the Laplace transform can be studied as stochastic orders in the class of mixtures of exponential distributions:
and
The orders based on the Laplace transform have a variety of interpretations corresponding to models in reliability, inventory, and maintenance (see Klefsjö [13], Alzaid, Kim, and Proschan [1], Shaked and Shanthikumar [14], and Shaked and Wong [16]). Mixtures of exponential distributions were intensively studied by many authors (e.g., Steutel [17], Keilson and Steutel [12], and Jewell [10]).
We are interested in finding out whether the exponential mixtures are ordered, or under what conditions they are, in the same order as the mixing distributions. Directly from relations (1)–(4) we have the following implications:
Note that the exponential mixtures are ordered in the reversed way. In the following sections we will prove the above results for the likelihood ratio as well as new statements for hazard rate and reversed hazard rate orders.
Bartoszewicz [6] has proved preservation results for the star order (without changing the way of ordering and without any assumptions) and for the dispersive order (with a reversed relation between mixtures and under some additional assumptions concerning classes of distributions). Using known relations between stochastic orders, involving assumptions of classes of life distributions also, we will obtain preservation results (with reversed inequalities) for exponential mixtures in cases when mixing distributions are ordered with respect to the mean residual life, excess wealth, and dispersive orders. In Section 2.7, by the preservation theorem for the star order and under some assumptions of distributions, a similar result will be proven for the convex transform order.
We will apply the following results concerning relations between stochastic orders, involving assumptions about classes of life distributions.
Lemma 1 (Bartoszewicz [3], Bagai and Kochar [2]): Let X and Y be two nonnegative random variables.
(a) If X ≤hr Y and X or Y is DFR, then X ≤disp Y.
(b) If X ≤disp Y and X or Y is IFR, then X ≤hr Y.
Lemma 2 (Bartoszewicz [4]): Let X and Y be two nonnegative random variables. If X is NBU and Y is NWU, then
Lemma 3 (Shaked and Shanthikumar [15]): Let X and Y be two random variables with finite means and with lF = lG = 0.
(a) If X ≤ew Y and X or Y is DMRL, then X ≤mrl Y.
(b)If X ≤mrl Y and X or Y is IMRL, then X ≤ew Y.
Lemma 4 (Shaked and Shanthikumar [14]): Let X and Y be random variables with finite means. If X ≤mrl Y and mF /mG is increasing, then X ≤hr Y.
Lemma 5: Let X and Y be random variables with finite means and lF = lG = 0. If X ≤ew Y, X is DFR and Y is DMRL, then X ≤disp Y.
Proof: Let X ≤ew Y. Y is DMRL and so from Lemma 3(a) we obtain X ≤mrl Y. Because the ratio of mean residual life functions mF /mG is increasing (since X is IMRL and Y is DMRL), it follows from Lemma 4 that X ≤hr Y, and from Lemma 1(a), we have X ≤disp Y. █
Lemmas 4 and 5 are similar: Under monotonicity assumptions of a ratio of some functions, a weaker order implies the stronger one. The following lemma gives a similar result for the convex transform and star orders.
Lemma 6: If X ≤* Y and, for every a ∈ (0,1), {[G−1(u + a)/G−1(u)] − 1}/{[F−1(u + a)/F−1(u)] − 1} is increasing in u ∈ (0,1 − a), then X ≤c Y.
Proof: G−1F is convex if and only if
is increasing in x for every h > 0. First, putting a = F(x + h) − F(x) in (5), we have
and next putting u = F(x), we obtain that G−1F is convex if and only if, for every a ∈ (0,1),
is increasing in u ∈ (0,1 − a). We can write (6) in the following form:
From assumptions, it follows that (6) is increasing and, hence, G−1F is convex (i.e., X ≤c Y). █
In particular, we easily obtain the following result.
Corollary 1:
(a) If X ≤* Y, log G−1 is convex, and log F−1 is concave, then X ≤c Y.
(b) Let F and G be absolutely continuous distributions. If X ≤* Y, xf (x) is increasing, and xg(x) is decreasing in x, then X ≤c Y.
Proof:
(a) Since log G−1 is convex, for every h ∈ (0,1) we have that log[G−1(x + h)/G−1(x)] is increasing in x ∈ (0,1 − h). Then G−1(x + h)/G−1(x) is increasing in x. Similarly, if log F−1 is concave, then F−1(x + h)/F−1(x) is decreasing in x. So, [G−1(x + h)/G−1(x) − 1]/[F−1(x + h)/F−1(x) − 1] is increasing and the proof follows from Lemma 6.
(b) Notice that if G is an absolutely continuous distribution with a density function g, then the monotonicity of xg(x) in x means that log G−1 is convex (or concave). █
2. RESULTS
2.1. The Likelihood Ratio Order
Consider a family of random variables
where U(x) has density hx. Let X and Y be random variables with support S and distributions F and G, respectively. Denote by U(X) and U(Y) the random variables with density functions hF(t) = ∫S hx(t) dF(x) and hG(t) = ∫S hx(t) dG(x), respectively.
We will use the following lemma to obtain the preservation result for the likelihood ratio order.
Lemma 7 (Shaked and Shanthikumar [14]): If U(x) ≤lr U(x′) for all x ≤ x′ and X ≤lr Y, then U(X) ≤lr U(Y).
We can see that if {U(x); x > 0} is a family of random variables with exponential distributions (i.e., hx(t) = xe−xt, x > 0, t > 0), then
are such that
. It is evident that U(x) ≤lr U(x′) (and so ≤rh, ≤hr, ≤mrl) whenever x′ ≤ x. Hence, from Lemma 7 modified according to the basic composition theorem and Lemma 1.1 of Karlin [11, pp.17 and 99] we obtain the following result.
Theorem 1: Let F and G be absolutely continuous distributions. If X ≤lr Y, then
.
2.2. The Hazard Rate Order
The result of Lemma 7 is also true for the hazard rate and the mean residual life orders.
Lemma 8 (Hu, Kundu, and Nanda [8], Hu, Nanda, Xie, and Zhu [9]): Let X ≤hr Y.
(a) If U(x) ≤hr U(x′) for all x ≤ x′, then U(X) ≤hr U(Y).
(b) If U(x) ≤mrl U(x′) for all x ≤ x′, then U(X) ≤mrl U(Y).
From Lemma 8(a) we obtain the following property in the same way as in the case of the preservation result for the likelihood ratio order.
Theorem 2: If X ≤hr Y, then
.
The theorem means that the hazard rate order between mixing distributions implies the reversed hazard rate and the hazard rate orders between exponential mixtures.
2.3. The Reversed Hazard Rate Order
It is not hard to see that Lemmas 7 and 8 can be modified for the reversed hazard rate order in the following. The proof is similar to those presented by Shaked and Shanthikumar [14] and Hu et al. [8]; therefore, we omit it.
Lemma 9: If U(x) ≤rh U(x′) for all x ≤ x′ and X ≤rh Y, then U(X) ≤rh U(Y).
Hence, since U(x) ≤rh U(x′) whenever x′ ≤ x and using again known results of Karlin [11], we can obtain the following preservation result for reversed hazard rate order.
Theorem 3: If X ≤rh Y, then
.
Similar to the hazard rate order, the reversed hazard rate order between mixing distributions implies both mentioned orderings of exponential mixtures. Note that considered orders (the usual stochastic, the likelihood ratio, the hazard rate, and the reversed hazard rate) are preserved under exponential mixtures without any additional assumptions about distributions.
2.4. The Mean Residual Life Order
Unfortunately, there is no result for the mean residual life order like the one for the previous orders, as it is shown in the following example.
Example 1: Let Fα,β be the distribution function of the gamma distribution; that is, the density function is fα,β(t) = αβtβ−1e−αt/Γ(β), t > 0. Then the mean residual life function and the Laplace transform are of the form
If β > 1, then the mean residual life function of the distribution
exists and
.
Let random variables X and Y have gamma distributions F = F1,3 and G = F1/3,2, respectively. Then
and mG(t) − mF(t) > 0 for all t > 0. So X ≤mrl Y. Since
, we see that
are not ordered in the mean residual life order.
To obtain a preservation result for the mean residual life order we need some additional assumptions.
Theorem 4: If X ≤mrl Y and mF /mG is increasing, then
.
Proof: If X ≤mrl Y and mF /mG is increasing, then X ≤hr Y, from Lemma 4. The theorem follows from Lemma 8(b). █
Immediately we obtain the following corollary.
Corollary 2: If X ≤mrl Y, X is IMRL and Y is DMRL, then
.
2.5. The Dispersive Order
Bartoszewicz [6] discussed the problem of the preservation of the dispersive ordering and, among others, he proved the following result: If 0 ≤ lG < lF, Y ≤st X, X ≤disp Y, and X or Y is DRFR, then
. Now we can state the following theorems.
Theorem 5: If X ≤disp Y and X or Y is IFR, then
.
Proof: It follows from Lemma 1(b) that if X or Y is IFR and X ≤disp Y, then X ≤hr Y. By Theorem 2 we have
. Since
are DFR, Lemma 1(a) implies that
. █
Theorem 6: If X ≤disp Y and X is NBU and Y is NWU, then
.
Proof: The proof runs in the same way as the previous one, using mainly Lemma 2. █
2.6. The Excess Wealth Order
It is shown in the following example that a preservation result for the excess wealth order cannot be obtained without some additional assumptions.
Example 2: Let F and G be distribution functions of discrete random variables X and Y, such that
. Then we have the excess wealth functions of X and Y:
Since WX(t) ≤ WY(t) for every t ∈ (0,1), we see that X ≤ew Y. The respective distribution and quantile functions of exponential mixtures are
We can calculate excess wealth functions:
Since
, we see that
are not ordered in the excess wealth order.
We formulate the following theorems.
Theorem 7: If X ≤ew Y, X is DFR, and Y is IFR, then
.
Proof: Immediately from Lemma 5 it follows that X ≤disp Y. From Theorem 5 we obtain that
and, hence,
. █
Theorem 8: If X ≤ew Y, X or Y is DMRL and mF /mG is increasing, then
.
Proof: If X ≤ew Y and X or Y is DMRL, then from Lemma 3(a), X ≤mrl Y. Now, using Theorem 4, we obtain that
. Since the exponential mixtures are DFR and hence IMRL, we have from Lemma 3(b) that
. █
2.7. The Convex Transform and Star Orders
Bartoszewicz [6] studied the preservation of stochastic orders under mixtures of exponential distributions for other orders: star and convex (dilation) ones. Remember for completeness the following result for the star order.
Theorem 9: If X ≤* Y, then
.
If we take G(x) = 1 − e−x, x > 0, then we obtain a necessary condition for X to be IFRA, expressed in terms of the Laplace transform of F.
Corollary 3: If X is IFRA (DFRA), then
is increasing (decreasing) in s > 0.
Proof: If X is IFRA, then X ≤* Y, where Y has exponential distribution G(x) = 1 − e−x. Theorem 9 implies that
. The definition
means that
is star-shaped; that is,
is increasing in s > 0. It is easy to see that
. So, for
we obtain the result. The proof is similar for X being DFRA. █
However, the condition of Corollary 3 is not sufficient. Block and Savits [7] have given the necessary and sufficient condition for X to be IFRA, which involves derivatives of f* of all orders: Verifying monotonicity of (7) requires only the first derivative.
It is easily seen that [1 − f*(s)]/[sE(X)], s ≥ 0, is the Laplace–Stieltjes transform of the renewal equilibrium distribution F0 with a density function f0(x) = [1 − F(x)]/E(X), x ≥ 0. Thus, Corollary 3 can be formulated in another way.
Corollary 3′: Let X0 be a random variable with the distribution F0. If X is IFRA (DFRA), then X0 ≤Lt−r (≥Lt−r)X.
The convex transform order, stronger than the star one, is not preserved under exponential mixtures. It is implied from the following example.
Example 3: Consider the distribution F with
with a density
The failure rate on SF is of the form
It is easy to see that X is IFR. The Laplace transform of F is
and
.
Let G(x) = 1 − e−x, x ≥ 0. Since X is IFR, we have X ≤c Y (G−1F is convex). The ordering is preserved under exponential mixture if
, or simply 1/f*(s), is convex. Calculate
Unfortunately, the function V(s) is not monotone since V(2) = 1.878, V(4) = 2.058, and V(8) = 2.004.
However, under some additional assumptions, we obtain a preservation result for the convex order, too. First we will prove the following lemma.
Lemma 10: If log F−1 is convex (concave), then
is concave (convex).
Proof: Assume that log F−1 is convex (the proof under concavity assumption is similar). Then
is increasing in u ∈ (0,1) for every h > 0 or, equivalently,
is decreasing in x for every h > 0. An easy calculation shows that
For simplicity of notation, let h(s) = se−s. Then
where 0 = s0 < s1 < ··· < sn−1 < sn = M, λ = max Δsi, Δsi = si+1 − si, ξi ∈ [si,si+1], and ΔF(si /x) = F(si+1 /x) − F(si /x) for i = 0,1,…, n − 1. Let yi = si /x. Then from (8),
is decreasing in yi and thus in 1/x. It means that ΔF(si /x) is increasing in x > 0 and, hence, so is
. █
Theorem 10: If X ≤c Y, log F−1 is convex, and log G−1 is concave, then
.
Proof: It is well known that X ≤c Y ⇒ X ≤* Y and, hence, from Theorem 9 we have
. Now the theorem follows from Lemma 10 and Corollary 1(a). █
From Corollary 1(b) and Lemma 10 we obtain also a simple result.
Corollary 4: Let F and G be absolutely continuous distributions. If X ≤c Y, xf (x) is increasing, and xg(x) is decreasing in x, then
.
Acknowledgment
The authors are very grateful to the referee for valuable comments and suggestions that led to an improvement of the article.
