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The observed total time on test and the observed excess wealth
Statistics & Probability Letters 68 (2004) 247 – 258
The observed total time on test and the observed excess wealth Xiaohu Lia , Moshe Shakedb;∗ a
b
Department of Mathematics, Lanzhou University, Gansu, Lanzhou 730000, China Department of Mathematics, University of Arizona, 617 N Santa Rita, P.O. Box 210089, Tucson, AZ 85721-0089, USA Received November 2003; received in revised form January 2004; accepted March 2004
Abstract Let X be a nonnegative random variable with mean 6 ∞, and let TX be the total time on test transform of X . It has been observed in the literature that the inverse of TX is a distribution function with support (0; ). In this paper, we identify the random variable that has this distribution, and we study some of its properties. We also study an analogous random variable that is based on what is called the excess wealth transform. c 2004 Elsevier B.V. All rights reserved. MSC: 60E15; 60K10 Keywords: Stochastic orders; Pareto distribution; NBUE; IFR; IFRA; HNBUE
1. Introduction Let X be a nonnegative random variable with a continuous distribution function F and mean 6 ∞. Denote the corresponding survival function by F< = 1 − F. The total time on test (TTT) transform TX , of X , is de>ned by F −1 (p) < F(x) d x; p ∈ (0; 1): (1.1) TX (p) = 0
Research supported by the National Natural Science Foundations of China (10201010) and Action Programming of Lanzhou University. ∗ Corresponding author. Tel.: +1-520-621-6858; fax: +1-520-621-8322. E-mail address: [email protected] (M. Shaked). c 2004 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2004.03.003
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The TTT transform is a theoretical version of the empirical TTT transform that is often used in statistical reliability theory. Roughly speaking, TX (p) gives the average time that an item spends on test if the test is terminated when a fraction p of all the items on the test fail. It has been observed in the literature (Barlow and Doksum, 1972; Bartoszewicz, 1986, 1995) that the inverse of TX is a distribution function with support (0; ). In this paper, we identify the random variable that has this distribution, and we study some of its properties. We also study an analogous random variable that is based on what Kochar et al. (2002) called the excess wealth transform. In this paper ‘increasing’ and ‘decreasing’ stand for ‘non-decreasing’ and ‘non-increasing’, respectively. 2. Denitions and basic properties Let X be a nonnegative random variable with a continuous distribution function F and mean 6 ∞. Let Xttt denote the observed TTT when X is observed; that is, let Xttt = TX (F(X )): Remark 2.1. The random variable Xttt arises in applications in which an item, in a pilot study, with lifetime X , determines the p (that is, p = F(X )) in (1.1), according to which a subsequent lifetime test, of similar items, is terminated. The random variable Xttt , then, is the theoretical version of the empirical TTT transform that is observed in that subsequent lifetime test. Since F(X ) has a uniform(0; 1) distribution, we see that Xttt =st TX (U ); where U is a uniform(0; 1) random variable, and =st denotes equality in law. Note that X < Xttt = F(x) d x: 0
(2.1)
Proposition 2.2. The distribution function of Xttt is the inverse of the TTT transform. Proof. We compute the distribution function of Xttt as follows: P{Xttt 6 y} = P{TX (U ) 6 y} = P{U 6 TX−1 (y)} = TX−1 (y);
y ∈ (0; );
(2.2)
and this gives the stated result. The inverse of the TTT transform is denoted in Barlow and Doksum (1972) and in Bartoszewicz (1986, 1995) by HF . Let us recall now, from Kochar et al. (2002), another transform. This is the excess wealth transform WX , of X , de>ned by ∞ < F(x) d x; p ∈ (0; 1); (2.3) WX (p) = F −1 (p)
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whenever ¡ ∞. In such a case (that is when ¡ ∞) let Xew denote the observed excess wealth when X is observed; that is, let Xew = WX (F(X )) =st WX (U ); where, again, U is a uniform(0; 1) random variable. Note that ∞ < Xew = F(x) d x:
(2.4)
X
Proposition 2.3. The survival function of Xew is the inverse of the excess wealth transform. Proof. We compute the survival function of Xew as follows: P{Xew ¿ y} = P{WX (U ) ¿ y} = P{U 6 WX−1 (y)} = WX−1 (y);
y ∈ (0; );
(2.5)
and this gives the stated result. ∞ < The function X , de>ned as X (t) = t F(x) d x, t ¿ 0, is often called the stop-loss transform; see, for example, MLuller (1996). By (2.4), Xew = X (X ):
(2.6)
We see that relation (2.6) is of the same spirit as the probability integral transform U = F(X ) < ). However, whereas U and V always have the uniform(0; 1) distribution (when F is or V = F(X continuous), the distribution of Xew varies with the distribution of X ; see, for instance, Examples 2.6 and 2.7. Remark 2.4. A referee pointed out that the TTT transform, and the excess wealth transform, have important interpretations in actuarial science. Suppose, an insurer accepts a random risk X . Suppose further that the insurer arranges a reinsurance for this risk determined by a retention level ‘; that is, the reinsurer will pay any excess of a claim above the level ‘ (in other words, ‘ may be thought of as a deductible); see, for instance, Waters (1983). If ‘ is determined as a particular percentile of the claim distribution, then the TTT transform, given in (1.1), is the expected claim below the retention level; this is an important quantity for the insurer. Also, the excess wealth transform, given in (2.3), is the expected claim above the retention level; this is an important quantity for the reinsurer. Comparisons of these transforms, and of the observed TTT’s and excess wealths (see Section 3), are therefore of interest in the area of reinsurance, and may have potential valuable applications in this area. Note that from (2.1) and (2.4) it follows, when ¡ ∞, that: Xew = − Xttt : Thus, many results about Xew can be derived from results about Xttt , and vice versa.
(2.7)
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Using (2.1), the moments of Xttt can be obtained as follows. For any integer k ¿ 1 we have k ∞ x k < E[Xttt ] = F(y) dy dF(x) 0
0
∞
=
=
···
0
0
x
∞
0
···
E[Xttt ] =
∞
0
∞
0
0
F<
< i ) dy1 · · · dyk dF(x) F(y
i=1
max yi
16i6k
= k! In particular,
x k
···
06y1 6···6yk
k
< i ) dy1 · · · dyk F(y
i=1
< 1 ) · · · F(y < k −1 )[F(y < k )]2 dy1 · · · dyk : F(y
F< 2 (x) d x:
Similarly (when ¡ ∞), k E[Xew ] = k! · · ·
06y1 6···6yk
< 1 ) · · · F(y < k ) dy1 · · · dyk F(y1 )F(y
and, in particular, ∞ < F(x)F(x) d x: E[Xew ] = 0
A closure property of the observed TTT and the observed excess wealth random variables, under scale transformation, is given below. Proposition 2.5. Let a ¿ 0. Then (aX )ttt =st aXttt . If ¡ ∞ then also (aX )ew =st aXew . Proof. It is easy to verify (either directly, or by looking at Fig. 1 of Kochar et al. (2002)) that −1 TaX (p) = aTX (p), p ∈ (0; 1). Therefore, TaX (y) = TX−1 (y=a), y ∈ (0; ). It follows that −1 (y) = TX−1 ( ya ) = P{aXttt 6 y}; P{(aX )ttt 6 y} = TaX
y ∈ (0; a)
and the >rst stated result follows. The second result follows from the >rst result, combined with (2.7). Example 2.6. Let X be an exponential random variable with mean 1. Then, from (2.1) it is easy to see that Xttt = 1 − e−X : From (2.7) we thus get Xew = e−X :
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It follows that both Xttt and Xew have the uniform(0; 1) distribution. If X has an exponential distribution with rate (that is, mean −1 ), then, by Proposition 2.5, both Xttt and Xew have the uniform(0; −1 ) distribution. The next example shows an instance in which = ∞. Example 2.7. Let X have the Pareto distribution with survival function < F(x) =
1 ; 1+x
x ¿ 0:
From (2.1) it is seen that X 1 Xttt = d x = log(1 + X ): 1+x 0 The distribution function of Xttt can be computed as follows: P{Xttt 6 y} = P{log(1 + X ) 6 y} = P{X 6 ey − 1} = 1 −
1 = 1 − e− y ; 1 + ey − 1
y ¿ 0;
that is, Xttt has an exponential distribution with mean 1. 3. Some stochastic order relations In Remark 2.1 we indicated how the random observed TTT, say Xttt , arises in the planning of lifetime experiments. Similarly, Xttt and Xew have useful interpretations in actuarial science (see Remark 2.4). Therefore, it is of interest to see how such random variables can be compared and bounded. This is the subject matter of this section. Let the nonnegative random variable X have the TTT transform TX , as de>ned in (1.1). Let Y be another nonnegative random variable with TTT transform TY . If TX (p) 6 TY (p) for all p ∈ (0; 1), then Kochar et al. (2002) denoted this by X 6ttt Y . Note that if X 6ttt Y then EX 6 EY (even if EY or EX are in>nite). Recall that, by de>nition, X 6st Y (that is, X is stochastically smaller than Y ) if P{X ¿ x} 6 P{Y ¿ x} for all x. Proposition 3.1. Let X and Y be two nonnegative random variables. Then Xttt 6st Yttt ⇐⇒ X 6ttt Y: Proof. Note that any of the above two inequalities implies EX 6 EY . By (2.2) we have that Xttt 6st Yttt if, and only if, TX−1 (y) ¿ TY−1 (y) for all y. That is, if and only if, TX (p) 6 TY (p) for all p ∈ (0; 1). That is, if, and only if, X 6ttt Y . As a corollary, we see that if X and Y are ordered with respect to 6st or 6ttt , then so are Xttt and Yttt .
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Corollary 3.2. Let X and Y be two nonnegative random variables. Then X 6st Y ⇒ Xttt 6st Yttt and X 6ttt Y ⇒ Xttt 6ttt Yttt : Proof. Recall from Kochar et al. (2002) that the order 6st is stronger than the order 6ttt . Therefore, using Proposition 3.1 we see that X 6st Y ⇒ X 6ttt Y ⇐⇒ Xttt 6st Yttt ⇒ Xttt 6ttt Yttt ; and the two stated results follow. Now let the >nite mean nonnegative random variable X have the excess wealth transform WX , as de>ned in (2.3). Let Y be another >nite mean nonnegative random variable with excess wealth transform WY . If WX (p) 6 WY (p) for all p ∈ (0; 1), then Shaked and Shanthikumar (1998) denoted RS
this by X 6ew Y . The inequality X 6ew Y is denoted as X Y in Fernandez-Ponce et al. (1998). Note that, for nonnegative random variables, if X 6ew Y then EX 6 EY . Proposition 3.3. Let X and Y be two nonnegative random variables with ?nite means. Then Xew 6st Yew ⇐⇒ X 6ew Y: The proof of Proposition 3.3 is similar to the proof of Proposition 3.1, and is therefore omitted. From Proposition 3.3, and from (2.3) of Fernandez-Ponce et al. (1998), it follows that: Xew 6st Yew ⇒ Var(X ) 6 Var(Y ); provided the variances exist; and from (2.4) in that paper it follows that: Xew 6st Yew ⇒ E|X1 − X2 | 6 E|Y1 − Y2 |;
(3.1)
where X1 and X2 [respectively, Y1 and Y2 ] are independent copies of X [respectively, Y ]. The following result is an interesting generalization of (3.1). Its proof can be found in the appendix. Proposition 3.4. Let X and Y be two nonnegative random variables with ?nite means, and let X1 ; X2 ; : : : ; Xn and Y1 ; Y2 ; : : : ; Yn be independent copies of X and of Y , respectively. If [min{X1 ; X2 ; : : : ; Xn−r }]ew 6st [min{Y1 ; Y2 ; : : : ; Yn−r }]ew
(3.2)
for some 1 6 r 6 n − 1, then E(X(r+1) − X(r) ) 6 E(Y(r+1) − Y(r) ) for that r, where X(r) , X(r+1) , Y(r) , and Y(r+1) denote the corresponding rth and (r + 1)st order statistics.
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Note that (3.1) is a special case of the above result when n = 2 and r = 1. Remark 3.5 (NBUE characterization): A nonnegative random variable X is said to be NBUE (new better than used in expectation) if EX ¿ E[X − t|X ¿ t] for all t ¿ 0. Belzunce (1999) showed that X is NBUE if, and only if, X 6ew E(EX ), where E(EX ) denotes an exponential random variable with mean EX . Since X and E(EX ) have the same mean, the above condition is equivalent to X ¿ttt E(EX ) (see, Kochar et al., 2002). From Proposition 3.1 and Example 2.6 we thus see that X is NBUE if, and only if, Xttt ¿st U(0; EX ); where U(0; EX ) is a uniform(0; EX ) random variable. Let X and Y be two nonnegative random variables with continuous distribution functions F and G, respectively, and with supports (0; a) and (0; b), respectively, for some >nite or in>nite constants a and b. Recall that X is said to be smaller than Y in the convex transform order (denoted as X 6c Y ) IFR
if G −1 F is convex. Note that X 6c Y is denoted in Fernandez-Ponce et al. (1998) as X Y . Proposition 3.6. Let X and Y be two nonnegative random variables with continuous distribution functions F and G, respectively, and supports (0; ∞), and ?nite or in?nite means X and Y , respectively. Then Xttt 6c Yttt ⇐⇒ X 6c Y: Proof. We note that Xttt 6c Yttt ⇐⇒ TY TX−1 (x) is convex in x ∈ (0; X )
d T (p)
−1 Y dp
p=TX (x) is increasing in x ∈ (0; X ) ⇐⇒
d d p TX (p)
−1 p=TX (x)
⇐⇒
d dp d dp
⇐⇒
(1 − p)=g(G −1 (p)) is increasing in p ∈ (0; 1) (1 − p)=f(F −1 (p))
⇐⇒
f(F −1 (p)) is decreasing in p ∈ (0; 1) g(G −1 (p))
TY (p) TX (p)
is increasing in p ∈ (0; 1)
⇐⇒ G −1 F is convex on (0; ∞) and the stated result follows.
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Remark 3.7. Recall that a nonnegative random variable X has the IFR (increasing failure rate) property if, and only if, X 6c E(1), where E(1) denotes a mean 1 exponential random variable. It follows, from Proposition 3.6 and Example 2.7, that Xttt is IFR if, and only if, X 6c Y , where Y has the Pareto distribution given in Example 2.7. The proof of the following lemma is straightforward, and is therefore omitted. Lemma 3.8. Let V and W be two random variables with continuous distribution functions, and supports of the form [0; X ] and [0; Y ], respectively, for some ?nite constants X and Y . Then V 6c W ⇐⇒ X − V ¿c Y − W: Proposition 3.9. Let X and Y be two nonnegative random variables with continuous distribution functions, and ?nite means X and Y , respectively. Then Xew ¿c Yew ⇐⇒ X 6c Y: Proof. We note that Xew ¿c Yew ⇐⇒ X − Xttt ¿c Y − Yttt
(by (2:7))
⇐⇒ Xttt 6c Yttt
(by Lemma 3:8)
⇐⇒ X 6c Y
(by Proposition 3:6)
and the stated result follows. Proposition 3.9 is mathematically equivalent to Theorem 3.1(a) of Fernandez-Ponce et al. (1998), although the notation and the terminology vary signi>cantly. Let X and Y be two random variables with continuous distribution functions F and G, respectively, >nite means X and Y , respectively, and with the common support (0; ∞). Denote the corresponding survival functions by F< ≡ 1 − F and G< ≡ 1 − G. According to Kochar and Wiens (1987) X is smaller than Y in the DMRL (decreasing mean residual life) order (denoted as X 6dmrl Y ) if ∞ < dx (1=Y ) G−1 (p) G(x) ∞ is increasing in p ∈ (0; 1): (3.3) < (1=X ) −1 F(x) dx F
(p)
Furthermore, the random variable X above is said to be smaller than the random variable Y above in the star order (denoted as X 6∗ Y ) if G −1 F is starshaped on (0; ∞) (that is, if G −1 F(x)=x increases in x ∈ (0; ∞)); see, for example, Shaked and Shanthikumar (1994). Proposition 3.10. Let X and Y be two random variables with continuous distribution functions, ?nite means X and Y , respectively, and common support (0; ∞). Then Xew ¿∗ Yew ⇐⇒ X 6dmrl Y:
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255
Proof. Let us, in this proof, denote the distribution and survival function of Xew by FXew and F< Xew , respectively. Similarly, we use the notation FYew and F< Yew . Recall from (2.5) that F< Xew (y) = WX−1 (y), y ∈ [0; X ], and that F< Yew (y) = WY−1 (y), y ∈ [0; Y ]. Now, we note that Xew ¿∗ Yew ⇐⇒ FX−ew1 FYew is starshaped on (0; ∞) 1 < ⇐⇒ F< − Xew F Yew is starshaped on (0; ∞)
⇐⇒
1 < (because FX−ew1 FYew = F< − Xew F Yew )
1 F< − Yew (p) is increasing in p ∈ (0; 1) 1 F< − X (p)
(by diPerentiation and substitution)
WY (p) is increasing in p ∈ (0; 1) WX (p)
(by (2:5)):
ew
⇐⇒
The latter condition is equivalent to (3.3), and the stated result follows. Proposition 3.10 is mathematically equivalent to Theorem 3.1(b) of Fernandez-Ponce et al. (1998), although the notation and the terminology vary signi>cantly. It is of interest to mention here that Bartoszewicz (1995, 1998) proved that X 6∗ Y ⇒ Xttt 6∗ Yttt ;
(3.4)
where X and Y are random variables with 0 being the left endpoint of their support. Recall that a nonnegative random variable X has the IFRA (increasing failure rate average) property if, and only if, X 6∗ E(1), where E(1) denotes a mean 1 exponential random variable. As in Remark 3.7, it follows, from (3.4) and Example 2.7, that if X 6∗ Y , where Y has the Pareto distribution given in Example 2.7, then Xttt is IFRA. Recall that two nonnegative random variables X and Y , with respective survival functions F< < are ordered with respect to the increasing concave order (denoted as X 6icv Y ) if x F(t) < dt 6 and G, 0 x < dt for all x ¿ 0, or, equivalently, if Eg(X ) 6 Eg(Y ) for all increasing concave functions g G(t) 0 for which the expectations exist. In the next result it is shown that if X and Y are ordered with respect to 6icv , then so are Xttt and Yttt . Proposition 3.11. Let X and Y be two nonnegative random variables. Then X 6icv Y ⇒ Xttt 6icv Yttt : Proof. Let F< and G< be the survival functions of X and Y , respectively. It is easy to see that the x < order 6icv is closed under increasing concave transformations. The function 0 F(t) dt is increasing and concave in x. Thus, if X 6icv Y then Y X < dt 6icv < dt: Xttt = F(t) F(t) 0
0
Also, from X 6icv Y we obtain Y Y < dt = Yttt : < G(t) F(t) dt 6st 0
0
The stated result follows from the two inequalities above.
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Remark 3.12 (HNBUE random variables): Recall that a nonnegative random variable X has the HNBUE (harmonic new better than used in expectation) property if X ¿icv E(EX ), where E(EX ) denotes an exponential random variable with mean EX ; see, for example, Shaked and Shanthikumar (1994, p. 68). From Proposition 3.11 and Example 2.6 we thus see that if X is HNBUE then Xttt ¿icv U(0; EX ); where U(0; EX ) is a uniform(0; EX ) random variable. We end this section with some comparisons with respect to the dispersive order. Recall that two random variables X and Y , with continuous distribution functions F and G, respectively, are ordered in the dispersive order (denoted by X 6disp Y ) if F −1 (!) − F −1 (") 6 G −1 (!) − G −1 (") for all 0 ¡ " ¡ ! ¡ 1. The proof of the next proposition can be found in the Appendix. Proposition 3.13. Let X and Y be two nonnegative random variables with continuous distribution functions. Then X 6disp Y ⇐⇒ Xttt 6disp Yttt ⇐⇒ Xew 6disp Yew : Acknowledgements We thank the referee for his useful and encouraging comments. Appendix A. Proofs of Propositions 3.4 and 3.13 Proof of Proposition 3.4. Let F and G denote the distribution functions of X and Y , respectively. Let Fmin and Gmin denote the distribution functions of min{X1 ; X2 ; : : : ; Xn−r } and min{Y1 ; Y2 ; : : : ; Yn−r }, respectively. Then F< min (x) = F< n−r (x), and therefore, −1 Fmin (p) = F −1 (1 − (1 − p)1=(n−r) );
p ∈ (0; 1):
Similarly, −1 Gmin (p) = G −1 (1 − (1 − p)1=(n−r) );
p ∈ (0; 1):
Now, by Proposition 3.3, condition (3.2) is equivalent to min{X1 ; X2 ; : : : ; Xn−r } 6ew min{Y1 ; Y2 ; : : : ; Yn−r }; that is,
∞
−1 Fmin (p)
< n− r
F
(x) d x 6
∞
−1 Gmin (p)
G< n−r (x) d x;
p ∈ (0; 1):
In view of the arbitrariness of p, the latter inequality is the same as ∞ ∞ n− r < G< n−r (x) d x; p ∈ (0; 1): F (x) d x 6 F −1 (p)
G −1 (p)
X. Li, M. Shaked / Statistics & Probability Letters 68 (2004) 247 – 258
The above inequality can be written as 1 (1 − u)n−r d[G −1 (u) − F −1 (u)] ¿ 0; p
257
p ∈ (0; 1):
From Lemma 7.1(a) of Barlow and Proschan (1975, p. 120) it follows that 1 ur (1 − u)n−r d[G −1 (u) − F −1 (u)] ¿ 0; p ∈ (0; 1); p
that is,
∞
F −1 (p)
F (x)F< n−r (x) d x 6 r
In particular, n r
0
∞
∞
G −1 (p)
F r (x)F< n−r (x) d x 6
G r (x)G< n−r (x) d x;
n r
∞
0
p ∈ (0; 1):
G r (x)G< n−r (x) d x:
By Kendall and Stuart (1977, p. 368), ∞ n E(X(r+1) − X(r) ) = F r (x)F< n−r (x) d x; r 0 and a similar expression holds for E(Y(r+1) − Y(r) ). These facts yield the stated inequality. Proof of Proposition 3.13. Let F and G be the distribution functions of X and Y , respectively. Consider the inequality F −1 (!) G−1 (!) < < F(x) dx 6 G(x) d x; for all 0 ¡ " ¡ ! ¡ 1: (A.1) F −1 (")
G −1 (")
From (1.1) we see that (A.1) is the same as TX (!) − TX (") 6 TY (!) − TY (");
for all 0 ¡ " ¡ ! ¡ 1;
and, by Proposition 2.2, (A.1) is the same as Xttt 6disp Yttt . Furthermore, from (2.3) it is seen that (A.1) is the same as WX (") − WX (!) 6 WY (") − WY (!);
for all 0 ¡ " ¡ ! ¡ 1;
and, by Proposition 2.3, (A.1) is the same as Xew 6disp Yew . Therefore Xttt 6disp Yttt ⇐⇒ Xew 6disp Yew . Assume now that X 6disp Y , that is, ! d[G −1 (u) − F −1 (u)] ¿ 0; for all 0 ¡ " ¡ ! ¡ 1: (A.2) "
From Lemma 7.1(b) of Barlow and Proschan (1975, p. 120) it follows that ! (1 − u) d[G −1 (u) − F −1 (u)] ¿ 0; for all 0 ¡ " ¡ ! ¡ 1: "
(A.3)
A straightforward computation shows that (A.3) is equivalent to (A.1). Thus, X 6disp Y ⇒ Xttt 6disp Yttt . Conversely, assume that (A.3) holds (that is, Xttt 6disp Yttt ). Since 1=(1 − u) is
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X. Li, M. Shaked / Statistics & Probability Letters 68 (2004) 247 – 258
increasing in u ∈ (0; 1), it follows from Lemma 7.1(a) of Barlow and Proschan (1975, p. 120) that (A.2) holds. That is, Xttt 6disp Yttt ⇒ X 6disp Y . References Barlow, R.E., Doksum, K.A., 1972. Isotonic tests for convex orderings. In: Le Cam, L.M., Neyman, J., Scott, E.L. (Eds)., Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability I, 293–323. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Reliability Models. Holt, Rinehart and Winston, New York. Bartoszewicz, J., 1986. Dispersive ordering and the total time on test transformation. Statist. Probab. Lett. 4, 285–288. Bartoszewicz, J., 1995. Stochastic order relation and the total time on test transform. Statist. Probab. Lett. 22, 103–110. Bartoszewicz, J., 1998. Application of a general composition theorem to the star order of distributions. Statist. Probab. Lett. 38, 1–9. Belzunce, F., 1999. On a characterization of right spread order by the increasing convex order. Statist. Probab. Lett. 45, 103–110. Fernandez-Ponce, J.M., Kochar, S.C., Mu˜noz-Perez, J., 1998. Partial orderings of distributions based on right-spread functions. J. Appl. Probab. 35, 221–228. Kendall, M., Stuart, A., 1977. The advanced Theory of Statistics, Vol. 1, Distribution Theory, 4th Edition. Charles GriUn and Company, London. Kochar, S.C., Wiens, D.P., 1987. Partial orderings of life distributions with respect to their aging properties. Naval Res. Logist. 34, 823–829. Kochar, S.C., Li, X., Shaked, M., 2002. The total time on test transform and the excess wealth stochastic orders of distributions. Adv. Appl. Probab. 34, 826–845. MLuller, A., 1996. Orderings of risks: A comparative study via stop-loss transforms. Insurance Math. Econom. 17, 215–222. Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and their Applications. Academic Press, San Diego. Shaked, M., Shanthikumar, J.G., 1998. Two variability orders. Probab. Eng. Inform. Sci. 12, 1–23. Waters, H.R., 1983. Some mathematical aspects of reinsurance. Insurance Math. Econom. 2, 17–26.
The observed total time on test and the observed excess wealth Xiaohu Lia , Moshe Shakedb;∗ a
b
Department of Mathematics, Lanzhou University, Gansu, Lanzhou 730000, China Department of Mathematics, University of Arizona, 617 N Santa Rita, P.O. Box 210089, Tucson, AZ 85721-0089, USA Received November 2003; received in revised form January 2004; accepted March 2004
Abstract Let X be a nonnegative random variable with mean 6 ∞, and let TX be the total time on test transform of X . It has been observed in the literature that the inverse of TX is a distribution function with support (0; ). In this paper, we identify the random variable that has this distribution, and we study some of its properties. We also study an analogous random variable that is based on what is called the excess wealth transform. c 2004 Elsevier B.V. All rights reserved. MSC: 60E15; 60K10 Keywords: Stochastic orders; Pareto distribution; NBUE; IFR; IFRA; HNBUE
1. Introduction Let X be a nonnegative random variable with a continuous distribution function F and mean 6 ∞. Denote the corresponding survival function by F< = 1 − F. The total time on test (TTT) transform TX , of X , is de>ned by F −1 (p) < F(x) d x; p ∈ (0; 1): (1.1) TX (p) = 0
Research supported by the National Natural Science Foundations of China (10201010) and Action Programming of Lanzhou University. ∗ Corresponding author. Tel.: +1-520-621-6858; fax: +1-520-621-8322. E-mail address: [email protected] (M. Shaked). c 2004 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2004.03.003
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The TTT transform is a theoretical version of the empirical TTT transform that is often used in statistical reliability theory. Roughly speaking, TX (p) gives the average time that an item spends on test if the test is terminated when a fraction p of all the items on the test fail. It has been observed in the literature (Barlow and Doksum, 1972; Bartoszewicz, 1986, 1995) that the inverse of TX is a distribution function with support (0; ). In this paper, we identify the random variable that has this distribution, and we study some of its properties. We also study an analogous random variable that is based on what Kochar et al. (2002) called the excess wealth transform. In this paper ‘increasing’ and ‘decreasing’ stand for ‘non-decreasing’ and ‘non-increasing’, respectively. 2. Denitions and basic properties Let X be a nonnegative random variable with a continuous distribution function F and mean 6 ∞. Let Xttt denote the observed TTT when X is observed; that is, let Xttt = TX (F(X )): Remark 2.1. The random variable Xttt arises in applications in which an item, in a pilot study, with lifetime X , determines the p (that is, p = F(X )) in (1.1), according to which a subsequent lifetime test, of similar items, is terminated. The random variable Xttt , then, is the theoretical version of the empirical TTT transform that is observed in that subsequent lifetime test. Since F(X ) has a uniform(0; 1) distribution, we see that Xttt =st TX (U ); where U is a uniform(0; 1) random variable, and =st denotes equality in law. Note that X < Xttt = F(x) d x: 0
(2.1)
Proposition 2.2. The distribution function of Xttt is the inverse of the TTT transform. Proof. We compute the distribution function of Xttt as follows: P{Xttt 6 y} = P{TX (U ) 6 y} = P{U 6 TX−1 (y)} = TX−1 (y);
y ∈ (0; );
(2.2)
and this gives the stated result. The inverse of the TTT transform is denoted in Barlow and Doksum (1972) and in Bartoszewicz (1986, 1995) by HF . Let us recall now, from Kochar et al. (2002), another transform. This is the excess wealth transform WX , of X , de>ned by ∞ < F(x) d x; p ∈ (0; 1); (2.3) WX (p) = F −1 (p)
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whenever ¡ ∞. In such a case (that is when ¡ ∞) let Xew denote the observed excess wealth when X is observed; that is, let Xew = WX (F(X )) =st WX (U ); where, again, U is a uniform(0; 1) random variable. Note that ∞ < Xew = F(x) d x:
(2.4)
X
Proposition 2.3. The survival function of Xew is the inverse of the excess wealth transform. Proof. We compute the survival function of Xew as follows: P{Xew ¿ y} = P{WX (U ) ¿ y} = P{U 6 WX−1 (y)} = WX−1 (y);
y ∈ (0; );
(2.5)
and this gives the stated result. ∞ < The function X , de>ned as X (t) = t F(x) d x, t ¿ 0, is often called the stop-loss transform; see, for example, MLuller (1996). By (2.4), Xew = X (X ):
(2.6)
We see that relation (2.6) is of the same spirit as the probability integral transform U = F(X ) < ). However, whereas U and V always have the uniform(0; 1) distribution (when F is or V = F(X continuous), the distribution of Xew varies with the distribution of X ; see, for instance, Examples 2.6 and 2.7. Remark 2.4. A referee pointed out that the TTT transform, and the excess wealth transform, have important interpretations in actuarial science. Suppose, an insurer accepts a random risk X . Suppose further that the insurer arranges a reinsurance for this risk determined by a retention level ‘; that is, the reinsurer will pay any excess of a claim above the level ‘ (in other words, ‘ may be thought of as a deductible); see, for instance, Waters (1983). If ‘ is determined as a particular percentile of the claim distribution, then the TTT transform, given in (1.1), is the expected claim below the retention level; this is an important quantity for the insurer. Also, the excess wealth transform, given in (2.3), is the expected claim above the retention level; this is an important quantity for the reinsurer. Comparisons of these transforms, and of the observed TTT’s and excess wealths (see Section 3), are therefore of interest in the area of reinsurance, and may have potential valuable applications in this area. Note that from (2.1) and (2.4) it follows, when ¡ ∞, that: Xew = − Xttt : Thus, many results about Xew can be derived from results about Xttt , and vice versa.
(2.7)
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Using (2.1), the moments of Xttt can be obtained as follows. For any integer k ¿ 1 we have k ∞ x k < E[Xttt ] = F(y) dy dF(x) 0
0
∞
=
=
···
0
0
x
∞
0
···
E[Xttt ] =
∞
0
∞
0
0
F<
< i ) dy1 · · · dyk dF(x) F(y
i=1
max yi
16i6k
= k! In particular,
x k
···
06y1 6···6yk
k
< i ) dy1 · · · dyk F(y
i=1
< 1 ) · · · F(y < k −1 )[F(y < k )]2 dy1 · · · dyk : F(y
F< 2 (x) d x:
Similarly (when ¡ ∞), k E[Xew ] = k! · · ·
06y1 6···6yk
< 1 ) · · · F(y < k ) dy1 · · · dyk F(y1 )F(y
and, in particular, ∞ < F(x)F(x) d x: E[Xew ] = 0
A closure property of the observed TTT and the observed excess wealth random variables, under scale transformation, is given below. Proposition 2.5. Let a ¿ 0. Then (aX )ttt =st aXttt . If ¡ ∞ then also (aX )ew =st aXew . Proof. It is easy to verify (either directly, or by looking at Fig. 1 of Kochar et al. (2002)) that −1 TaX (p) = aTX (p), p ∈ (0; 1). Therefore, TaX (y) = TX−1 (y=a), y ∈ (0; ). It follows that −1 (y) = TX−1 ( ya ) = P{aXttt 6 y}; P{(aX )ttt 6 y} = TaX
y ∈ (0; a)
and the >rst stated result follows. The second result follows from the >rst result, combined with (2.7). Example 2.6. Let X be an exponential random variable with mean 1. Then, from (2.1) it is easy to see that Xttt = 1 − e−X : From (2.7) we thus get Xew = e−X :
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It follows that both Xttt and Xew have the uniform(0; 1) distribution. If X has an exponential distribution with rate (that is, mean −1 ), then, by Proposition 2.5, both Xttt and Xew have the uniform(0; −1 ) distribution. The next example shows an instance in which = ∞. Example 2.7. Let X have the Pareto distribution with survival function < F(x) =
1 ; 1+x
x ¿ 0:
From (2.1) it is seen that X 1 Xttt = d x = log(1 + X ): 1+x 0 The distribution function of Xttt can be computed as follows: P{Xttt 6 y} = P{log(1 + X ) 6 y} = P{X 6 ey − 1} = 1 −
1 = 1 − e− y ; 1 + ey − 1
y ¿ 0;
that is, Xttt has an exponential distribution with mean 1. 3. Some stochastic order relations In Remark 2.1 we indicated how the random observed TTT, say Xttt , arises in the planning of lifetime experiments. Similarly, Xttt and Xew have useful interpretations in actuarial science (see Remark 2.4). Therefore, it is of interest to see how such random variables can be compared and bounded. This is the subject matter of this section. Let the nonnegative random variable X have the TTT transform TX , as de>ned in (1.1). Let Y be another nonnegative random variable with TTT transform TY . If TX (p) 6 TY (p) for all p ∈ (0; 1), then Kochar et al. (2002) denoted this by X 6ttt Y . Note that if X 6ttt Y then EX 6 EY (even if EY or EX are in>nite). Recall that, by de>nition, X 6st Y (that is, X is stochastically smaller than Y ) if P{X ¿ x} 6 P{Y ¿ x} for all x. Proposition 3.1. Let X and Y be two nonnegative random variables. Then Xttt 6st Yttt ⇐⇒ X 6ttt Y: Proof. Note that any of the above two inequalities implies EX 6 EY . By (2.2) we have that Xttt 6st Yttt if, and only if, TX−1 (y) ¿ TY−1 (y) for all y. That is, if and only if, TX (p) 6 TY (p) for all p ∈ (0; 1). That is, if, and only if, X 6ttt Y . As a corollary, we see that if X and Y are ordered with respect to 6st or 6ttt , then so are Xttt and Yttt .
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Corollary 3.2. Let X and Y be two nonnegative random variables. Then X 6st Y ⇒ Xttt 6st Yttt and X 6ttt Y ⇒ Xttt 6ttt Yttt : Proof. Recall from Kochar et al. (2002) that the order 6st is stronger than the order 6ttt . Therefore, using Proposition 3.1 we see that X 6st Y ⇒ X 6ttt Y ⇐⇒ Xttt 6st Yttt ⇒ Xttt 6ttt Yttt ; and the two stated results follow. Now let the >nite mean nonnegative random variable X have the excess wealth transform WX , as de>ned in (2.3). Let Y be another >nite mean nonnegative random variable with excess wealth transform WY . If WX (p) 6 WY (p) for all p ∈ (0; 1), then Shaked and Shanthikumar (1998) denoted RS
this by X 6ew Y . The inequality X 6ew Y is denoted as X Y in Fernandez-Ponce et al. (1998). Note that, for nonnegative random variables, if X 6ew Y then EX 6 EY . Proposition 3.3. Let X and Y be two nonnegative random variables with ?nite means. Then Xew 6st Yew ⇐⇒ X 6ew Y: The proof of Proposition 3.3 is similar to the proof of Proposition 3.1, and is therefore omitted. From Proposition 3.3, and from (2.3) of Fernandez-Ponce et al. (1998), it follows that: Xew 6st Yew ⇒ Var(X ) 6 Var(Y ); provided the variances exist; and from (2.4) in that paper it follows that: Xew 6st Yew ⇒ E|X1 − X2 | 6 E|Y1 − Y2 |;
(3.1)
where X1 and X2 [respectively, Y1 and Y2 ] are independent copies of X [respectively, Y ]. The following result is an interesting generalization of (3.1). Its proof can be found in the appendix. Proposition 3.4. Let X and Y be two nonnegative random variables with ?nite means, and let X1 ; X2 ; : : : ; Xn and Y1 ; Y2 ; : : : ; Yn be independent copies of X and of Y , respectively. If [min{X1 ; X2 ; : : : ; Xn−r }]ew 6st [min{Y1 ; Y2 ; : : : ; Yn−r }]ew
(3.2)
for some 1 6 r 6 n − 1, then E(X(r+1) − X(r) ) 6 E(Y(r+1) − Y(r) ) for that r, where X(r) , X(r+1) , Y(r) , and Y(r+1) denote the corresponding rth and (r + 1)st order statistics.
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Note that (3.1) is a special case of the above result when n = 2 and r = 1. Remark 3.5 (NBUE characterization): A nonnegative random variable X is said to be NBUE (new better than used in expectation) if EX ¿ E[X − t|X ¿ t] for all t ¿ 0. Belzunce (1999) showed that X is NBUE if, and only if, X 6ew E(EX ), where E(EX ) denotes an exponential random variable with mean EX . Since X and E(EX ) have the same mean, the above condition is equivalent to X ¿ttt E(EX ) (see, Kochar et al., 2002). From Proposition 3.1 and Example 2.6 we thus see that X is NBUE if, and only if, Xttt ¿st U(0; EX ); where U(0; EX ) is a uniform(0; EX ) random variable. Let X and Y be two nonnegative random variables with continuous distribution functions F and G, respectively, and with supports (0; a) and (0; b), respectively, for some >nite or in>nite constants a and b. Recall that X is said to be smaller than Y in the convex transform order (denoted as X 6c Y ) IFR
if G −1 F is convex. Note that X 6c Y is denoted in Fernandez-Ponce et al. (1998) as X Y . Proposition 3.6. Let X and Y be two nonnegative random variables with continuous distribution functions F and G, respectively, and supports (0; ∞), and ?nite or in?nite means X and Y , respectively. Then Xttt 6c Yttt ⇐⇒ X 6c Y: Proof. We note that Xttt 6c Yttt ⇐⇒ TY TX−1 (x) is convex in x ∈ (0; X )
d T (p)
−1 Y dp
p=TX (x) is increasing in x ∈ (0; X ) ⇐⇒
d d p TX (p)
−1 p=TX (x)
⇐⇒
d dp d dp
⇐⇒
(1 − p)=g(G −1 (p)) is increasing in p ∈ (0; 1) (1 − p)=f(F −1 (p))
⇐⇒
f(F −1 (p)) is decreasing in p ∈ (0; 1) g(G −1 (p))
TY (p) TX (p)
is increasing in p ∈ (0; 1)
⇐⇒ G −1 F is convex on (0; ∞) and the stated result follows.
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Remark 3.7. Recall that a nonnegative random variable X has the IFR (increasing failure rate) property if, and only if, X 6c E(1), where E(1) denotes a mean 1 exponential random variable. It follows, from Proposition 3.6 and Example 2.7, that Xttt is IFR if, and only if, X 6c Y , where Y has the Pareto distribution given in Example 2.7. The proof of the following lemma is straightforward, and is therefore omitted. Lemma 3.8. Let V and W be two random variables with continuous distribution functions, and supports of the form [0; X ] and [0; Y ], respectively, for some ?nite constants X and Y . Then V 6c W ⇐⇒ X − V ¿c Y − W: Proposition 3.9. Let X and Y be two nonnegative random variables with continuous distribution functions, and ?nite means X and Y , respectively. Then Xew ¿c Yew ⇐⇒ X 6c Y: Proof. We note that Xew ¿c Yew ⇐⇒ X − Xttt ¿c Y − Yttt
(by (2:7))
⇐⇒ Xttt 6c Yttt
(by Lemma 3:8)
⇐⇒ X 6c Y
(by Proposition 3:6)
and the stated result follows. Proposition 3.9 is mathematically equivalent to Theorem 3.1(a) of Fernandez-Ponce et al. (1998), although the notation and the terminology vary signi>cantly. Let X and Y be two random variables with continuous distribution functions F and G, respectively, >nite means X and Y , respectively, and with the common support (0; ∞). Denote the corresponding survival functions by F< ≡ 1 − F and G< ≡ 1 − G. According to Kochar and Wiens (1987) X is smaller than Y in the DMRL (decreasing mean residual life) order (denoted as X 6dmrl Y ) if ∞ < dx (1=Y ) G−1 (p) G(x) ∞ is increasing in p ∈ (0; 1): (3.3) < (1=X ) −1 F(x) dx F
(p)
Furthermore, the random variable X above is said to be smaller than the random variable Y above in the star order (denoted as X 6∗ Y ) if G −1 F is starshaped on (0; ∞) (that is, if G −1 F(x)=x increases in x ∈ (0; ∞)); see, for example, Shaked and Shanthikumar (1994). Proposition 3.10. Let X and Y be two random variables with continuous distribution functions, ?nite means X and Y , respectively, and common support (0; ∞). Then Xew ¿∗ Yew ⇐⇒ X 6dmrl Y:
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Proof. Let us, in this proof, denote the distribution and survival function of Xew by FXew and F< Xew , respectively. Similarly, we use the notation FYew and F< Yew . Recall from (2.5) that F< Xew (y) = WX−1 (y), y ∈ [0; X ], and that F< Yew (y) = WY−1 (y), y ∈ [0; Y ]. Now, we note that Xew ¿∗ Yew ⇐⇒ FX−ew1 FYew is starshaped on (0; ∞) 1 < ⇐⇒ F< − Xew F Yew is starshaped on (0; ∞)
⇐⇒
1 < (because FX−ew1 FYew = F< − Xew F Yew )
1 F< − Yew (p) is increasing in p ∈ (0; 1) 1 F< − X (p)
(by diPerentiation and substitution)
WY (p) is increasing in p ∈ (0; 1) WX (p)
(by (2:5)):
ew
⇐⇒
The latter condition is equivalent to (3.3), and the stated result follows. Proposition 3.10 is mathematically equivalent to Theorem 3.1(b) of Fernandez-Ponce et al. (1998), although the notation and the terminology vary signi>cantly. It is of interest to mention here that Bartoszewicz (1995, 1998) proved that X 6∗ Y ⇒ Xttt 6∗ Yttt ;
(3.4)
where X and Y are random variables with 0 being the left endpoint of their support. Recall that a nonnegative random variable X has the IFRA (increasing failure rate average) property if, and only if, X 6∗ E(1), where E(1) denotes a mean 1 exponential random variable. As in Remark 3.7, it follows, from (3.4) and Example 2.7, that if X 6∗ Y , where Y has the Pareto distribution given in Example 2.7, then Xttt is IFRA. Recall that two nonnegative random variables X and Y , with respective survival functions F< < are ordered with respect to the increasing concave order (denoted as X 6icv Y ) if x F(t) < dt 6 and G, 0 x < dt for all x ¿ 0, or, equivalently, if Eg(X ) 6 Eg(Y ) for all increasing concave functions g G(t) 0 for which the expectations exist. In the next result it is shown that if X and Y are ordered with respect to 6icv , then so are Xttt and Yttt . Proposition 3.11. Let X and Y be two nonnegative random variables. Then X 6icv Y ⇒ Xttt 6icv Yttt : Proof. Let F< and G< be the survival functions of X and Y , respectively. It is easy to see that the x < order 6icv is closed under increasing concave transformations. The function 0 F(t) dt is increasing and concave in x. Thus, if X 6icv Y then Y X < dt 6icv < dt: Xttt = F(t) F(t) 0
0
Also, from X 6icv Y we obtain Y Y < dt = Yttt : < G(t) F(t) dt 6st 0
0
The stated result follows from the two inequalities above.
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Remark 3.12 (HNBUE random variables): Recall that a nonnegative random variable X has the HNBUE (harmonic new better than used in expectation) property if X ¿icv E(EX ), where E(EX ) denotes an exponential random variable with mean EX ; see, for example, Shaked and Shanthikumar (1994, p. 68). From Proposition 3.11 and Example 2.6 we thus see that if X is HNBUE then Xttt ¿icv U(0; EX ); where U(0; EX ) is a uniform(0; EX ) random variable. We end this section with some comparisons with respect to the dispersive order. Recall that two random variables X and Y , with continuous distribution functions F and G, respectively, are ordered in the dispersive order (denoted by X 6disp Y ) if F −1 (!) − F −1 (") 6 G −1 (!) − G −1 (") for all 0 ¡ " ¡ ! ¡ 1. The proof of the next proposition can be found in the Appendix. Proposition 3.13. Let X and Y be two nonnegative random variables with continuous distribution functions. Then X 6disp Y ⇐⇒ Xttt 6disp Yttt ⇐⇒ Xew 6disp Yew : Acknowledgements We thank the referee for his useful and encouraging comments. Appendix A. Proofs of Propositions 3.4 and 3.13 Proof of Proposition 3.4. Let F and G denote the distribution functions of X and Y , respectively. Let Fmin and Gmin denote the distribution functions of min{X1 ; X2 ; : : : ; Xn−r } and min{Y1 ; Y2 ; : : : ; Yn−r }, respectively. Then F< min (x) = F< n−r (x), and therefore, −1 Fmin (p) = F −1 (1 − (1 − p)1=(n−r) );
p ∈ (0; 1):
Similarly, −1 Gmin (p) = G −1 (1 − (1 − p)1=(n−r) );
p ∈ (0; 1):
Now, by Proposition 3.3, condition (3.2) is equivalent to min{X1 ; X2 ; : : : ; Xn−r } 6ew min{Y1 ; Y2 ; : : : ; Yn−r }; that is,
∞
−1 Fmin (p)
< n− r
F
(x) d x 6
∞
−1 Gmin (p)
G< n−r (x) d x;
p ∈ (0; 1):
In view of the arbitrariness of p, the latter inequality is the same as ∞ ∞ n− r < G< n−r (x) d x; p ∈ (0; 1): F (x) d x 6 F −1 (p)
G −1 (p)
X. Li, M. Shaked / Statistics & Probability Letters 68 (2004) 247 – 258
The above inequality can be written as 1 (1 − u)n−r d[G −1 (u) − F −1 (u)] ¿ 0; p
257
p ∈ (0; 1):
From Lemma 7.1(a) of Barlow and Proschan (1975, p. 120) it follows that 1 ur (1 − u)n−r d[G −1 (u) − F −1 (u)] ¿ 0; p ∈ (0; 1); p
that is,
∞
F −1 (p)
F (x)F< n−r (x) d x 6 r
In particular, n r
0
∞
∞
G −1 (p)
F r (x)F< n−r (x) d x 6
G r (x)G< n−r (x) d x;
n r
∞
0
p ∈ (0; 1):
G r (x)G< n−r (x) d x:
By Kendall and Stuart (1977, p. 368), ∞ n E(X(r+1) − X(r) ) = F r (x)F< n−r (x) d x; r 0 and a similar expression holds for E(Y(r+1) − Y(r) ). These facts yield the stated inequality. Proof of Proposition 3.13. Let F and G be the distribution functions of X and Y , respectively. Consider the inequality F −1 (!) G−1 (!) < < F(x) dx 6 G(x) d x; for all 0 ¡ " ¡ ! ¡ 1: (A.1) F −1 (")
G −1 (")
From (1.1) we see that (A.1) is the same as TX (!) − TX (") 6 TY (!) − TY (");
for all 0 ¡ " ¡ ! ¡ 1;
and, by Proposition 2.2, (A.1) is the same as Xttt 6disp Yttt . Furthermore, from (2.3) it is seen that (A.1) is the same as WX (") − WX (!) 6 WY (") − WY (!);
for all 0 ¡ " ¡ ! ¡ 1;
and, by Proposition 2.3, (A.1) is the same as Xew 6disp Yew . Therefore Xttt 6disp Yttt ⇐⇒ Xew 6disp Yew . Assume now that X 6disp Y , that is, ! d[G −1 (u) − F −1 (u)] ¿ 0; for all 0 ¡ " ¡ ! ¡ 1: (A.2) "
From Lemma 7.1(b) of Barlow and Proschan (1975, p. 120) it follows that ! (1 − u) d[G −1 (u) − F −1 (u)] ¿ 0; for all 0 ¡ " ¡ ! ¡ 1: "
(A.3)
A straightforward computation shows that (A.3) is equivalent to (A.1). Thus, X 6disp Y ⇒ Xttt 6disp Yttt . Conversely, assume that (A.3) holds (that is, Xttt 6disp Yttt ). Since 1=(1 − u) is
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increasing in u ∈ (0; 1), it follows from Lemma 7.1(a) of Barlow and Proschan (1975, p. 120) that (A.2) holds. That is, Xttt 6disp Yttt ⇒ X 6disp Y . References Barlow, R.E., Doksum, K.A., 1972. Isotonic tests for convex orderings. In: Le Cam, L.M., Neyman, J., Scott, E.L. (Eds)., Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability I, 293–323. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Reliability Models. Holt, Rinehart and Winston, New York. Bartoszewicz, J., 1986. Dispersive ordering and the total time on test transformation. Statist. Probab. Lett. 4, 285–288. Bartoszewicz, J., 1995. Stochastic order relation and the total time on test transform. Statist. Probab. Lett. 22, 103–110. Bartoszewicz, J., 1998. Application of a general composition theorem to the star order of distributions. Statist. Probab. Lett. 38, 1–9. Belzunce, F., 1999. On a characterization of right spread order by the increasing convex order. Statist. Probab. Lett. 45, 103–110. Fernandez-Ponce, J.M., Kochar, S.C., Mu˜noz-Perez, J., 1998. Partial orderings of distributions based on right-spread functions. J. Appl. Probab. 35, 221–228. Kendall, M., Stuart, A., 1977. The advanced Theory of Statistics, Vol. 1, Distribution Theory, 4th Edition. Charles GriUn and Company, London. Kochar, S.C., Wiens, D.P., 1987. Partial orderings of life distributions with respect to their aging properties. Naval Res. Logist. 34, 823–829. Kochar, S.C., Li, X., Shaked, M., 2002. The total time on test transform and the excess wealth stochastic orders of distributions. Adv. Appl. Probab. 34, 826–845. MLuller, A., 1996. Orderings of risks: A comparative study via stop-loss transforms. Insurance Math. Econom. 17, 215–222. Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and their Applications. Academic Press, San Diego. Shaked, M., Shanthikumar, J.G., 1998. Two variability orders. Probab. Eng. Inform. Sci. 12, 1–23. Waters, H.R., 1983. Some mathematical aspects of reinsurance. Insurance Math. Econom. 2, 17–26.