DISPERSIVE FUNCTIONS AND STOCHASTIC ORDERS

Transcription

1 APPLICATIONES MATHEMATICAE 24,4(1997), pp J. BARTOSZEWICZ(Wroc law) DISPERSIVE UNCTIONS AND STOCHASTIC ORDERS Abstract. eneralizations of the hazard functions are proposed and general hazard rate orders are introduced. Some stochastic orders are defined as general ones. A unified derivation of relations between the dispersive order and some other orders of distributions is presented. 1. Introduction. Denote by φ, ψ, χ real functions, A, B, C, D intervals on the real line R, X, Y, Z random variables,,, H their respective probability distribution functions and by f, g, h their respective density functions, if they exist. Define = 1 and 1 (u) = inf{x : (x) u}, u (0,1) (and analogously for ). We identify the distribution functions,, H with the respective probability distributions and denote their supports, which are intervals, by S, S, S H respectively. We denote by φψ the superposition of functions φ and ψ: in particular, 1 denotes the superposition of 1 and. We use increasing in place of nondecreasing and decreasing in place of nonincreasing. Alzaid and Proschan (1992) have introduced the following definition. Definition 1. A function φ : A R is dispersive on the interval A R if φ(x) x is increasing on A. It is easy to see that the superposition of two dispersive functions is also dispersive. We give two theorems on dispersive functions which are modifications of two known results Mathematics Subject Classification: 60E05, 62N05. Key words and phrases: dispersive function, preservation theorem, hazard function, mean residual life, partial orders, stationary renewal distribution. Research supported by KBN(State Committee for Scientific Research, Poland), under rant 2 PO3A [429]

2 430 J. Bartoszewicz Bartoszewicz (1987) has proved a theorem on the preservation of the dispersiveness under transformations. Its modified version is the following (see also Shaked and Shanthikumar (1994)). Theorem 1. Let A, B, C, D be intervals on the real line R and B C A. Let φ : A B be a dispersive function on A such that there exists x 0 for which φ(x) x if x < x 0, and φ(x) x if x > x 0. If a function ψ : C D is increasing (decreasing) concave (convex) on the set {x : x = ψ 1 (u) < x 0, u D} and convex (concave) on the set {x : x = ψ 1 (u) > x 0, u D}, then the function ψφψ 1 is dispersive on D. We recall that φ : R R is superadditive (subadditive) on A R if φ(x + y) ( ) φ(x) + φ(y) for x,y,x + y A. It is well known that if φ is convex (concave), then it is superadditive (subadditive) and if φ is increasing superadditive (subadditive), then φ 1 is subadditive (superadditive). It is also well known that the superposition of two increasing superadditive (subadditive) functions is also superadditive (subadditive). Theorem 2.3 of Ahmed et al. (1986) can be modified as follows (see also Bartoszewicz (1987)). Theorem 2. Let A R be an interval and φ : A R be an increasing function such that there exists x 0 for which φ(x) φ(x) lim 1 and lim x x 0 0 x x x 0 +0 x 1. If φ is subadditive for x < x 0 and superadditive for x > x 0, then φ is dispersive on A. Recall the definitions of two stochastic orders. is stochastically smaller than ( st or X st Y ) if (x) (x) for every x. is less dispersive than ( disp or X disp Y ) if (1) 1 (β) 1 (α) 1 (β) 1 (α) whenever 0 < α < β < 1. Shaked (1982) has proved that disp if and only if 1 (x) x is increasing on S, i.e. φ = 1 is dispersive on S. Thus Theorem 1 may be treated as a preservation theorem for the dispersive order. In particular, we have the following statement (Bartoszewicz (1987), Shaked and Shanthikumar (1994)). Corollary 1. Let (0) = (0) = 0, S and S be intervals and ψ be an increasing convex function. If X st Y and X disp Y, then ψ(x) disp ψ(y ).

3 Dispersive functions and stochastic orders 431 Bartoszewicz (1987) has used Theorems 1 and 2 to prove some relations between the dispersive and the hazard rate orders of distributions. In this paper we remark that many other stochastic orders may be defined by dispersive functions. In Section 2 a general approach is proposed by introducing operators and general hazard functions. Particular cases are considered. Many orders of distributions are defined by dispersiveness of these hazard functions. Properties of the hazard operators are given in Section 3. Some relations between the dispersive and some other orders are established in Section eneralizations of the hazard function. Let be the class of distributions with a common support S which is an interval. It is well known that the function R (x) = log (x), x S, is the hazard function of. If the density f exists, we have where R (x) = r (t)dt, (2) r (t) = f(t)/(t) is the hazard rate function of the distribution. Thus may be represented in the form (3) (x) = 1 e R (x), x S. We generalize (3) in two ways Hazard functions defined on S. Let H be a class of real functions defined on S. Consider a one-to-one mapping (operator) Λ : H with H = Λ(). Denote by Λ 1 : H the inverse operator. Thus for every we have (x) = (Λ 1 Λ())(x), x S, where denotes the superposition of operators. By analogy we introduce the following definition. Definition 2. The function R (x) = Λ()(x), x S, is called the Λ-hazard function of the distribution. Also by analogy with the hazard rate order (see Example 1 below) we propose the following definition. Definition 3. Let,. We say that is smaller than in the Λ-order ( Λ or X Λ Y ) if R (x) R (x) is increasing on S. It is easy to see that under general assumptions about Λ the Λ-order is a preorder. If R 1, the inverse of R, exists and is increasing, then we may

4 432 J. Bartoszewicz define the Λ-order in the following way: Λ R R 1 is dispersive on R (S). Now we give examples of stochastic orders which may be defined as Λ-orders. Properties of these may be found in Shaked and Shanthikumar (1994). The orders defined in Examples 1, 3 and 4 are commonly called uniform stochastic orders and were studied in detail by Keilson and Sumita (1982) and Lynch et al. (1987). Example 1 (The hazard rate order). Let be the class of continuous distributions with common support S = [0, ). Let,. We say that is smaller than in the hazard rate order ( hr or X hr Y ) if (4) / decreases in x S. If the densities f and g exist, then (4) holds if, and only if, r (x) r (x), x S, where r and r are the hazard rate functions of and respectively, defined by (2). We define Λ() = log for, and Λ 1 (φ) = 1 e φ for φ H. Hence Λ hr. Example 2 (The generalized hazard rate order). Let be the class of continuous distributions with a common support S which is an interval. Let H be a distinguished distribution from and Λ() = H 1. We may call this Λ-order a generalized hazard order and denote it by ghr. The function R (x) = H 1 (x), x S, is called a generalized hazard function. If the density h exists, the function d dx R (x) = f(x) hh 1 (x) is called a generalized hazard rate function of (Barlow et al. (1972)). If S = [0, ) and H(x) = 1 e x, x 0, we obtain the usual hazard rate function and the usual hazard rate order. Example 3 (The reversed hazard rate order). Let now be the class of continuous distributions with common support S = (,a], a <. Let,. We say that is smaller than in the reversed hazard rate order ( rh or X rh Y ) if (5) (x)/(x) decreases in x S.

5 Dispersive functions and stochastic orders 433 If the densities f and g exist, we may define the corresponding reversed hazard rate functions r (x) = f(x) (x) and and then (5) holds if, and only if, r (x) = g(x) (x), x S, r (x) r (x), x S. We define Λ() = log for, and Λ 1 (φ) = e φ for φ H. Thus from (5) we have Λ rh. R e m a r k 1. The reversed hazard rate order may also be treated as the generalized hazard rate order defined in Example 2 with H(x) = e x a, x a. Example 4 (The likelihood ratio order). Let be the class of absolutely continuous distributions with respect to the Lebesgue measure with a common support S. Let,. We say that is smaller than in the likelihood ratio order ( lr or X lr Y ) if f(x)/g(x) decreases in x S. We define the operator Λ : H by (6) Λ() = log d,. dx The inverse operator has the form Λ 1 (φ) = e φ, φ H. Thus we have Λ lr. Example 5 (The mean residual life order). Let be the class of continuous distributions on S = [0, ) with finite means. If X is a random variable with distribution, we define the mean residual life of X at x > 0 as Ì x (7) m (x) = E[X x X > x] = (u)du. (x) Since µ = E(X) = Ì 0 (x)dx < we have m (x) < for x <. or a random variable Y with distribution function, m and µ are similarly defined. We say that is smaller than in the mean residual life order ( mrl or X mrl Y ) if m (x) m (x), x > 0.

6 434 J. Bartoszewicz One can easily prove (Shaked and Shanthikumar (1994)) that mrl We define the operator Λ : H by (8) Λ()(x) = R (x) = log The inverse operator Λ 1 is of the form Thus we have Ì x Ì (u)du decreases in x > 0. (u)du x x (u)du, x > 0. Λ 1 (φ) = 1 + d du e φ, φ H. Λ mrl. Since R is increasing, mrl R R 1 is dispersive on ( log µ, ). Example 6 (The memory order). Let be the class of continuous distributions on S = [0, ) with finite means and (0) = 0 for all. Let m and m be the respective mean residual lifes of distributions and from, defined by (7), and µ and µ their respective means. ollowing Ebrahimi and Zahedi (1992) we say that is smaller than in the memory order ( m or X m Y ) if µ µ and m (x) m (x) increases in x > 0. Ebrahimi and Zahedi (1992) have proved that m implies mrl. We define the operator Λ : H by Ì x Λ()(x) = R (x) = m (x) = (u)du. (x) Thus for, with µ µ, we have Λ m. Example 7 (Monotone convex and concave orders). Let X and Y be two random variables and and be their respective distribution functions. We say that is smaller than in the increasing convex [concave] order ( icx or X icx Y [ icv or X icv Y ]) if E[φ(X)] E[φ(Y )] for all increasing convex [concave] functions φ : R R. One can prove (Shaked and Shanthikumar (1994)) that icx x (u)du x (u)du for all x

7 and Dispersive functions and stochastic orders 435 icv (u)du (u)du for all x. It is easy to notice that we may define these orders as Λ-orders. or the increasing convex order we put Λ()(x) = and for the increasing concave order we put Λ()(x) = t t (u)dudt (u)dudt Hazard functions defined on (0,1). Let Λ : H be a one-to-one operator, where H = Λ() is the class of real functions defined on (0,1). Denote by Λ 1 : H the inverse operator. We introduce the following definition. Definition 5. The function R (u) = Λ()(u), u (0,1), is called the Λ-hazard function of the distribution. We also propose the following definition. Definition 6. Let,. We say that is smaller than in the Λ-order ( Λ or X Λ Y ) if R (u) R (u) is increasing on (0,1). Similarly to the Λ-order, it is easy to see that under general assumptions about Λ the Λ-order 1 is a preorder. If R, the inverse of R, exists and is increasing, then we may define the Λ-order as follows: (9) Λ R R 1 is dispersive on R ((0,1)). Example 8 (The dispersive order). This order defined by (1) may also be defined as the Λ-order for Λ() = 1, i.e. R (u) = 1 (u), u (0,1). The inverse operator is Λ 1 (φ) = φ 1, φ H. Thus (1) holds iff R (u) R (u) is increasing on (0,1), i.e. Λ disp. Example 9 (The star order). Let be the class of continuous distributions on (0, ) and,. We say that is smaller than in the star order ( or X Y ) if 1 is starshaped, i.e. 1 (x)/x is increasing in x > 0, i.e. 1 (u)/ 1 (u) is increasing in u (0,1). Putting Λ() = log 1 and Λ 1 (φ) = ψ 1, where ψ = e φ, φ H, we have Λ.

8 436 J. Bartoszewicz Example 10 (The convex transform order). We say that is smaller than in the convex transform order ( c or X c Y ) if (10) 1 is convex on S. If,, the class of absolutely continuous distributions on an interval S, then (10) is equivalent to f 1 (u) g 1 (u) is increasing on (0,1). Thus putting Λ() = log d du 1 and Λ 1 (φ) = ψ 1, where ψ = Ì e φ, φ H, we have Λ c. Example 11 (The s-order). Let and be continuous symmetric distributions on R, i.e. (x) = 1 ( x), x R (and analogously for ). We say that is smaller than in the s-order ( s or X s Y ) (van Zwet (1964)) if (11) 1 is concave on (,0) and convex on (0, ). If,, the class of absolutely continuous symmetric distributions on R, then (11) is equivalent to f 1 (u) g 1 (u) is decreasing on (0,1/2) and increasing on (1/2,1). Thus putting { Λ()(u) = R log d (u) = du 1 (u), u (0,1/2), log d du 1 (u), u (1/2,1), and Λ 1 (φ) = ψ 1, where ψ = sign(ψ) Ì e φ, φ H, we have for,, Λ s. Example 12 (The Lorenz order). Let be the class of distributions on [0, ) with a common mean µ. Let,. We say that is smaller than in the Lorenz order ( L or X L Y ) if Putting 0 1 (t)dt Λ()(u) = 0 u (t)dt, x [0,1]. 1 (t)dt dx, u (0,1), and Λ 1 (φ) = ψ 1, where ψ = d2 du 2 φ, φ H, we have Λ L.

9 Dispersive functions and stochastic orders 437 Example 13 (The usual stochastic order). The order st may be defined as a Λ-order and a Λ-order as well. Let be the class of distribution functions with the common support S = [a, ), a >. We define the operator Λ : H by Λ()(x) = R (x) = a (u)du, x S. The inverse operator is Λ 1 (φ) = d duφ, φ H. Thus for,, (x) (x) for all x S R (x) R (x) increases in x S, i.e. Λ st. It is clear that st if and only if 1 (u) 1 (u) for all u (0,1). Define the operator Λ : H by Λ()(u) = R (u) = u 0 1 (t)dt, u (0,1). The inverse operator is Λ 1 () = ψ 1, where ψ = d du φ, φ H. Thus for,, i.e. (x) (x) for all x S R (u) R (u) increases in u (0,1), Λ st. 3. Preservation theorems. Let be a class of distributions on which the operators Λ and Λ are defined. It is easy to see that if R = Λ() = ψ,, where ψ is a function for which ψ 1 exists, then for, we have (12) R 1 R = 1. Similarly, if R = Λ() = ψ 1,, and ψ 1 exists, then R 1 R = 1. The following lemma gives conditions under which (12) holds. R 1 Lemma 1. Let,, Λ and Λ be defined on such that R 1 1, R exist. Let φ : R R be a function for which φ 1. Then: (a) R 1 R = 1 if, and only if, (13) Λ(φ 1 ) = Λ()φ 1 ;, R 1,

10 438 J. Bartoszewicz (b) R R 1 = 1 if, and only if, (14) Λ(φ 1 ) = φ Λ(). P r o o f. We prove the statement (a) only. The proof of (b) is similar. Sufficiency. Assume (13) holds. Then for φ = R 1 R we have Λ(R 1 R ) = Λ()R 1 R = R R 1 R = R = Λ(). Therefore we have R 1 R =, i.e. R 1 R = 1. Necessity. Assume (12) holds for every,. rom (12) we have (15) Λ() = Λ() 1. Putting := φ 1 in (15) we have Λ(φ 1 ) = Λ() 1 φ 1 = Λ()φ 1. rom Lemma 1 and Definition 1 we obtain the following corollaries. Corollary 2. Let,, Λ satisfy (13) and H = Λ() be a class of strictly increasing functions. Then disp R 1 R is dispersive on S. Corollary 3. Let, and Λ satisfy (13). If R 1 and are increasing, then and R 1 X Λ Y R (X) disp R (Y ) R (X) disp R (Y ). exist P r o o f. Since 0 = R 1 and 0 = R 1 are the distribution functions of R (X) and R (Y ) respectively, we have R R 1 = R R 1 R R 1 = R 1 R 1 = Similarly we prove the second equivalence. Immediately from Lemma 1 and Theorems 1 and 2 we obtain the following results. Corollary 4. Let,, Λ satisfy (13) and H = Λ() be a class of strictly increasing functions. (a) If φ = R R 1 and ψ = R 1 (or ψ = R 1 ) satisfy the assumptions of Theorem 1 with A = C = R (S), B = R (S) and D = S, then Λ disp. (b) If φ = R 1 R and ψ = R (or ψ = R ) satisfy the assumptions of Theorem 1 with A = B = C = S and D = R (S) (or D = R (S)), then disp Λ.

11 Dispersive functions and stochastic orders 439 Corollary 5. Let,, Λ satisfy (13) and H = Λ() be a class of strictly increasing functions. Let x 0 S be a point such that 1 (x) 1 (x) lim 1 and lim 1. x x 0 0 x x x 0 +0 x If R is subadditive for x < x 0 and superadditive for x > x 0 and R is superadditive for x < x 0 and subadditive for x > x 0, then disp Λ. Similar corollaries may be formulated for the Λ-order, putting in Corollaries 4 and 5: R := R 1 and R 1 := R and assuming that Λ satisfies (14). 4. The relation between the dispersive and some other stochastic orders. Some Λ- and Λ-hazard functions satisfy the condition (13) or (14) and hence, applying Lemma 1 and Corollaries 4 and 5, we may obtain particular relations between the dispersive order and some other stochastic orders The generalized hazard rate order. Consider the class of continuous distributions with the common support S = [0, ) and let H be a distinguished distribution. It is easy to see that Λ() = H 1 satisfies (13). Expressing the convexity of R in terms of the convex transform order, i.e. c H (see Example 10) we may reformulate Corollary 4 as follows. Corollary 4.1. Let,. (a) If ghr and H c or H c, then disp. (b) If disp and c H or c H, then ghr. Recall now the following definition (Barlow and Proschan (1975)). We say that is smaller than H in the superadditive order ( su or X su Y ) if H 1 is superadditive on S. Thus Corollary 5 may be stated in the following form. Corollary 5.1. Let,. If su H and H su, then disp ghr. Let 0 be the class of continuous distributions on R symmetric with respect to the origin and let H 0 be a distinguished distribution. It is easy to see that for, 0, ghr implies (x) (x) for x 0 and (x) (x) for x 0. Expressing the concavity and convexity of R in terms of the s-order (see Example 11) we may now formulate Corollary 4 in the following form.

12 440 J. Bartoszewicz Corollary 4.2. Let, 0. (a) If ghr and H s or H s, then disp. (b) If disp and s H or s H, then ghr The hazard rate order. Let be the class of continuous distributions on [0, ). Putting H(x) = 1 e x, x 0, in Section 4.1 we obtain results for the usual hazard rate order. It is well known (Barlow and Proschan (1975)) that if log is convex (H c ), then is a DR (decreasing failure rate) distribution and if log is concave ( c H), then is an IR (increasing failure rate) distribution. If log is superadditive (H su ), then is an NWU (new worse than used) distribution and if log is subadditive ( su H), then is an NBU (new better than used) distribution. Thus Corollary 4 may be formulated for this particular case as follows. Corollary 4.3 (Bartoszewicz (1987)). Let,. (a) If hr and or is DR, then disp. (b) If disp and or is IR, then hr. Similarly Corollary 5 may be stated in the following form. Corollary 5.2 (Bartoszewicz (1987)). Let,. If is NBU and is NWU, then disp hr. R e m a r k 2. It seems interesting to compare Corollary 4.3(a) with the following result. Theorem 3 (Bartoszewicz (1985)). If S = [0,a 1 ], S = [0,a 2 ], 0 < a 1 a 2 and c or S = [0, ), S = [b, ), (b) = 0, b 0 and c, then st disp The reversed hazard rate order. Consider the class of distributions and the reversed hazard rate order defined in Example 3. We say that has increasing reversed failure rate ( is IRR) if log is convex, and has decreasing reversed failure rate ( is DRR) if log is concave. Since Λ() satisfies (13), after some modifications (Λ() = log is decreasing), we obtain from Corollary 4 the following result. Corollary 4.4. Let,. (a) If rh and or is IRR, then disp. (b) If disp and or is DRR, then rh The mean residual life order. Consider now the class of distributions defined in Example 5 and the mean residual life order defined as the Λ-order with Λ() given by (8). We say that is a DMRL (decreasing mean residual life) distribution if m is decreasing, i.e. R is convex. We

13 Dispersive functions and stochastic orders 441 say that is an IMRL (increasing mean residual life) distribution if m is increasing, i.e. R is concave. It is well known (see e.g. Hollander and Proschan (1984)) that if is IR, then is DMRL, and if is DR, then is IMRL. Unfortunately, Λ() does not satisfy (13) and we cannot obtain relations between the mean residual life and the dispersive orders directly from Corollary 4. Notice, however, that [ ] R (x) = log (u)du = log µ (u)du where x = log[1 (x)] log µ = log (x) log µ, (x) = 1 µ 0 0 (u)du is the stationary renewal distribution corresponding to (Barlow and Proschan (1975)). is defined similarly. It is obvious that and thus m (x) = µ (x) = 1 r (x), x S, (16) mrl hr, i.e. Λ hr. rom (16) and Corollary 4.3 we have the following implications. Lemma 2. (a) is IMRL is DR is IMRL; (b) is DMRL is IR is DMRL; (c) is IMRL hr ; (d) is DMRL hr ; (e) is IMRL disp. We also have the following lemma. Lemma 3. Let,. If is DMRL and is IMRL, then and also st disp (17) disp disp. P r o o f. If is DMRL and is IMRL, then from Lemma 2(d) we have st and st. Thus st. rom Lemma 2(b) we have c K, where K is the exponential distribution and also K c.

14 442 J. Bartoszewicz Thus c. Therefore by Theorem 3 we have disp. Since disp st, (17) also holds. Directly from (16), Corollary 4.3 and Lemma 3 we obtain the relation between the mean residual life and the dispersive orders. Corollary 4.5. Let,. (a) If mrl and or is IMRL, then disp. (b) If disp and or is DMRL, then mrl. (c) If disp and is DMRL and is IMRL, then mrl The likelihood ratio order. We consider the particular case presented in Example 4. Let 1 be the class of absolutely continuous distributions with respect to the Lebesgue measure with common support S = [0, ) and with decreasing densities. Let Λ : 1 H be defined by (6). Thus for, 1 the functions R 1 and R 1 exist. As in Section 3.3, Λ() does not satisfy (13) and also we cannot obtain relations between the likelihood ratio and the dispersive orders in the class 1 directly from Corollary 4. However, notice that R (x) = log f(x) = log f(x) log f(0) f(0) where = log (x) log f(0), (x) = 1 f(x) f(0), x S, is a distribution function. It is easy to notice that µ = 1/f(0) and (x) = 1 µ 0 (u)du, so is the stationary renewal distribution corresponding to ( is similarly defined). Since f(x) g(x) = f(0) g(0) (x) (x), we have (18) lr hr, i.e. Λ hr. One can easily verify the following implications. Lemma 4. (a) log f is convex is DR is DR; (b) log f is concave is IR is IR; (c) is IMRL is DR;

15 Dispersive functions and stochastic orders 443 (d) is DMRL is IR; (e) is DR hr ; (f) is IR hr ; (g) log f is convex disp. Putting in Lemma 3: :=, := and :=, := we obtain the following statement. Lemma 5. Let, 1. If is IR and is DR, then and also st disp disp disp. The relation between the likelihood ratio and the dispersive orders in the class 1 follows from (18), Corollary 4.3 and Lemma 5. Corollary 4.6. Let, 1. (a) If lr and log f or log g is convex, then disp. (b) If disp and log f or log g is concave, then lr. (c) If lr, log f is concave and log g is convex, then disp. R e m a r k 3. It is well known that if f is logarithmically concave (convex), then is IR (DR) (Barlow and Proschan (1975)). If is IR (DR), then is DMRL (IMRL) and also NBU (NWU) (Hollander and Proschan (1984)). On the other hand, for distributions, with the common support S = [0, ), we have the implications (Shaked and Shanthikumar (1994)) lr hr mrl disp st Comparing Corollaries 4.3(b), 4.5(b) (c) and 4.6(b) we see that, as usual, stronger assumptions imply stronger results The star order. Consider the star order defined in Example 9 and the class of continuous distributions on (0, ). or, we have (19) R (u) R (u) increases in u (0,1). Since R (u) = log 1 is increasing, (19) is equivalent to R R 1 (x) is dispersive. Assume that st and. Applying Theorem 1 to the functions φ(x) = R R 1 (x) = log 1 (e x ) and ψ(x) = e x we find that ψφψ 1 = 1 is dispersive, i.e. disp. Thus we obtain the following known result (Sathe (1984), Shaked and Shanthikumar (1994)). Corollary 6. Let,. If st and, then disp.

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