Some stochastic inequalities for weighted sums

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1 Some stochastic inequalities for weighted sums Yaming Yu Department of Statistics University of California Irvine, CA 92697, USA Abstract We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let Y i be i.i.d. random variables on R +. Assuming that log Y i has a log-concave density, we show that a i Y i is stochastically smaller than b i Y i, if (log a 1,...,log a n ) is majorized by (log b 1,...,log b n ). On the other hand, assuming that Y p i has a log-concave density for some p > 1, we show that a i Y i is stochastically larger than b i Y i, if (a q 1,...,aq n) is majorized by (b q 1,...,bq n), where p 1 + q 1 = 1. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko (1998) on Weibull variables is proved. Some applications in reliability and wireless communications are mentioned. Keywords: gamma distribution, log-concavity, majorization, Rayleigh distribution, tail probability, usual stochastic order, Weibull distribution, weighted sum. 1 Main results and examples This note aims to unify and generalize certain stochastic comparison results concerning weighted sums. Let Y 1,...,Y n be i.i.d. random variables on R +. We are interested in comparing two weighted sums, n i=1 a iy i and n i=1 b iy i, a i, b i R +, with respect to the usual stochastic order. A random variable X is said to be no larger than Y in the usual stochastic order, written as X st Y, if Pr(X > t) Pr(Y > t) for all t R. For an introduction to various stochastic orders, see Shaked and Shanthikumar (27). Ordering in terms of st may be used to bound 1

2 the tail probability of a i Y i, for example, in terms of the tail probability of Y i. For specific distributions, such comparisons have been explored in several contexts, including reliability (Boland et al. 1994; Bon and Paltanea 1999). We shall use the notion of majorization (Marshall and Olkin 1979). A real vector b = (b 1,...,b n ) is said to majorize a = (a 1,...,a n ), written as a b, if (i) n i=1 a i = n i=1 b i, and (ii) n i=k a (i) n i=k b (i), k = 2,...,n, where a (1)... a (n) and b (1)... b (n) are (a 1,...,a n ) and (b 1,...,b n ) arranged in increasing order, respectively. A function φ(a) symmetric in the coordinates of a = (a 1,...,a n ) is said to be Schur concave, if a b implies φ(a) φ(b). A function φ(a) is Schur convex if φ(a) is Schur concave. A nonnegative function f(x), x R n, is log-concave, if supp(f) is convex, and log f(x) is concave on supp(f). Log-concavity plays a critical role in deriving our main results. For other stochastic comparison results involving log-concavity, see, for example, Karlin and Rinott (1981), and Yu (28, 29b, 29c). In this section, after stating our main results (Theorems 1 and 2), we illustrate with several examples, and mention potential applications. The main results are proved in Section 2. Some technical details in the proof of Theorem 2 are collected in the appendix. Theorem 1. Let Y 1,...,Y n be i.i.d. random variables with density f(y) on R + such that f(e x ) is log-concave in x R. Then, for each t >, Pr ( a i Y i t) is a Schur concave function of Equivalently, if a, b R n +, then log a (log a 1,...,log a n ). log a log b = a i Y i st bi Y i. (1) Theorem 2. Let p > 1, and let Y 1,...,Y n be i.i.d. random variables with density f(y) on R + such that the function min{, 2/p 1} log x + log f(x 1/p ) (2) is concave in x R +. Then, for each t >, Pr ( a i Y i t) is a Schur convex function of a q (a q 1,...,aq n) R n +, where p 1 + q 1 = 1. Equivalently, if a, b R n +, then a q b q = b i Y i st ai Y i. (3) 2

3 Remark. In Theorem 1, the condition that f(e x ) is log-concave is equivalent to log Y i having a log-concave density. In Theorem 2, a sufficient condition for (2) is that x 1/p 1 f(x 1/p ) is log-concave, or, equivalently, Y p i has a log-concave density (this special case is mentioned in the abstract). Theorems 1 and 2 are quite applicable, since log-concavity is associated with many well-known densities (see Corollaries 1 and 2 below). Theorem 1 is reminiscent of the following result of Proschan (1965). Theorem 3. Let Y i, i = 1,...,n, be i.i.d. random variables on R with a log-concave density that is symmetric about zero. Then for each t >, Pr( a i Y i t) is a Schur-concave function of a R n +. Theorem 2 is closely related to Theorem 4, which is a version (with a stronger assumption) of Theorem 24 of Karlin and Rinott (1983). Theorem 4. Let < p < 1, and let Y 1,...,Y n be i.i.d. random variables on R + such that Y p i has a log-concave density. Then, for each t >, Pr ( a i Y i t) is a Schur concave function of (a q 1,...,aq n) R n +, where p 1 + q 1 = 1. Karlin and Rinott (1983) gave an elegant proof of Theorem 4 using the Prékopa-Leindler inequality. Our proofs of Theorems 1 and 2 (Section 2) borrow ideas from both Proschan (1965) and Karlin and Rinott (1983). See Shaked and Tong (1988) for more related inequalities. Bounds on the distribution function of a i Y i are readily obtained in terms of the distribution function of Y i. In Theorem 1, for example, (1) gives ) ( ) ( ) 1/n Pr( bi Y i t Pr b Yi t, b = bi, t >. (4) In Theorem 2, (3) gives ) ( Pr( bi Y i t Pr b ) ( Y i t, b = n 1/q 1 bi) q, t >. (5) More generally, we obtain inequalities for the expectations of monotone functions, since X st Y implies Eg(X) Eg(Y ) for every increasing function g such that the expectations exist. Let us mention some specific distributions to which Theorems 1 and 2 can be applied. Corollary 1 follows from Theorem 1. The log-concavity condition is easily verified in each case (for more distributions that satisfy this condition, see Hu et al. 24, Example 1). Related 3

4 results on sums of uniform variables can be found in Korwar (22). The gamma case has recently been discussed by Bon and Paltanea (1999), Korwar (22), Khaledi and Kochar (24), and Yu (29a). Corollary 1. For a, b R n +, (1) holds when Y i are i.i.d. having one of the following distributions: 1 uniform on the interval (, s), s > ; 2 gamma(α, β), α, β > ; 3 any log-normal distribution; 4 the Weibull distribution with parameter p >, whose density is f(y) = py p 1 e yp, y > ; 5 the generalized Rayleigh distribution with parameter ν >, whose density is f(y) y ν 1 e y2 /2, y >. The inequality (4) holds for each of these distributions. The gamma case is interesting in that the upper bound in (4) is in terms of a single gamma variable, Y i. The gamma case with α = 1/2 dates back to Okamoto (196). See also Bock et al. (1987) and Székely and Bakirov (23) for related inequalities. Corollary 2. Let p > 1, and define q by p 1 + q 1 = 1. Then, for a, b R n +, (3) holds in the following cases: 1, Y p i 1 Y i are i.i.d. Weibull variables with parameter p; 2 p = q = 2, and Y i are i.i.d. generalized Rayleigh variables with parameter ν 1. Corollary 2 follows from Theorem 2. The condition (2) is easily verified. For example, in Case has a log-concave density, which implies (2). Case 1 confirms a conjecture of Hitczenko (1998). Case 2 recovers some results of Hu and Lin (2, 21). 4

5 The Weibull case and the generalized Rayleigh case are interesting in that Corollary 1 is also applicable, and we obtain a double bound through (4) and (5). For example, if Y i are i.i.d. generalized Rayleigh variables with parameter ν 1, then ( Pr a ) ) ( ) Y i t Pr( ai Y i t Pr a Yi t, a i >, t >, where a = ( n 1 a 2 i ) 1/2 and a = ( a i ) 1/n. Manesh and Khaledi (28) present related inequalities. We briefly mention some applications. Weighted sums of independent χ 2 variables arise naturally in multivariate statistics as quadratic forms in normal variables. Stochastic comparisons between such weighted sums are therefore statistically interesting, and can lead to bounds on the distribution functions. Suppose the component lifetimes of a redundant standby system (without repairing) are modeled by a scale family of distributions. Then the total lifetime is of the form i a iy i. When Y i are i.i.d exponential variables, Bon and Paltanea (1999) obtain comparisons of the total lifetime with respect to several stochastic orders. Our Corollary 1 shows that, for the usual stochastic order, (1) actually holds for a broad class of distributions including the commonly used gamma, Weibull, and log-normal distributions. When Y i are i.i.d. exponential variables and a i R +, the quantity E log(1 + a i Y i ) appears in certain wireless communications problems (Jorswieck and Boche, 27). By the monotonicity of log(1 + x), we have ai Y i st bi Y i = E log (1 + a i Y i ) E log (1 + ) a i Y i. Corollary 1 therefore leads to qualitative comparisons for this expected value. Other weighted sums, e.g., of Rayleigh variables, also appear in the context of communications. It would be interesting to see whether results similar to Theorems 1, 2 and 4 can be obtained for the hazard rate order, or the likelihood ratio order. For sums of independent gamma variables, such results have been obtained by Boland et al. (1994), Bon and Paltanea (1999), Korwar (22), Khaledi and Kochar (24), and Yu (29a). 5

6 2 Proofs Two proofs are presented for Theorem 1. The first one uses the Prékopa-Leindler inequality (Lemma 1) and is inspired by Karlin and Rinott (1983). Lemma 1. If g(x, y) is log-concave in (x, y) R m R n, then R g(x, y)dx is log-concave in m y R n. We also use a basic criterion for Schur-concavity. Proposition 1. If h(α), α R n, is log-concave and permutation invariant in α, then it is Schur-concave. First proof of Theorem 1. For t >, define n { g(x, α) 1 E e x i f(e x i ), E (x, α) R 2n : i=1 } n e x i+α i t, i=1 where x = (x 1,...,x n ), α = (α 1,...,α n ) R n. Note that E is a convex set (1 E denotes the indicator function). Since f(e x i ) is log-concave, we know that g(x, α) is log-concave in (x, α). By Lemma 1, ) h(α) Pr( e α i Y i t = g(x, α)dx R n is log-concave in α R n. Since h(α) is permutation invariant, it is Schur-concave in α by Proposition 1, and the claim is proved. The second proof is inspired by Proschan (1965), and serves as an introduction to the proof of Theorem 2. Properties of majorization imply that it suffices to prove (1) for a b such that a and b differ only in two components. Since st is closed under convolution (Shaked and Shanthikumar (27)), we only need to prove (1) for n = 2. We shall use log-concavity in the following form. If g(x), x R, is log-concave, and (x 1, x 2 ) (y 1, y 2 ), then g(x 1 )g(x 2 ) g(y 1 )g(y 2 ). Second proof of Theorem 1. Fix t >, and let F denote the distribution function of Y 1. It suffices to show that h(β) Pr ( β 1 Y 1 + βy 2 t ) = 6 F ( tβ β 2 y ) f(y)dy

7 increases in β (, 1]. We may assume that supp(f) [ǫ, ) for some ǫ >. The general case follows by a standard limiting argument. We can then justify differentiation under the integral sign and obtain h (β) = = t/(2β) (t 2βy)f(tβ β 2 y)f(y)dy t/β (t 2βy)f(tβ β 2 y)f(y)dy + (t 2βy) f(tβ β 2 y)f(y)dy. (6) t/(2β) By a change of variables y t/β y in the second integral in (6), we get h (β) = t/(2β) (t 2βy) [ f(tβ β 2 y)f(y) f(β 2 y)f(t/β y) ] dy. If < y < t/(2β) and < β 1, then β 2 y min{y, tβ β 2 y}. That is, Since f(e x ) is log-concave, we have (log(tβ β 2 y), log y) (log(β 2 y), log(t/β y)). f(tβ β 2 y)f(y) f(β 2 y)f(t/β y), < y < t/(2β), which leads to h (β), as required. Our proof of Theorem 2 is similar to (but more involved than) the second proof of Theorem 1. Under the stronger assumption that Y p i has a log-concave density, we actually obtain a simpler proof of Theorem 2 following the first proof of Theorem 1 (see Karlin and Rinott, 1983). It seems difficult, however, to extend this argument assuming only that (2) is concave. Proof of Theorem 2. We may assume n = 2 as in the second proof of Theorem 1. Fix t >. Effectively we need to show that ( ) h(β) Pr β 1/q Y 1 + (1 β) 1/q Y 2 t = F ( ) tβ 1/q (β 1 1) 1/q y f(y)dy increases in β [1/2, 1) (F denotes the distribution function of Y 1 ). We have where q(1 β) 1/p β 1/q+1 h (β) = = y g(y) dy g(y)dy + y1 y g(y)dy, y = t(1 β) 1/p, y 1 = t(1 β) 1/q, (7) 7

8 and g(y) = (y y )f(x(y))f(y), x(y) = (β 1 1) 1/q (y 1 y). Differentiation under the integral sign is permitted because g(y) (y 1 y )Mf(y), < y < y 1, where M = sup y> f(y). We know M < because (2) implies that f(x 1/p ) is log-concave in x R +. In the Appendix, we prove Claim 1. For each y (, y ), there exists a unique ỹ (y, y 1 ) such that y p + x p (y) = ỹ p + x p (ỹ). (8) Henceforth let y and ỹ be related by (8). Direct calculation using the implicit function theorem gives dỹ dy = (βy)p/q ((1 β)(y 1 y)) p/q (βỹ) p/q ((1 β)(y 1 ỹ)) p/q. A change of variables y ỹ in y g(y) dy yields q(1 β) 1/p β 1/q+1 h (β) = g(y) dy y1 dỹ dỹ + g(ỹ)dỹ, A where A (y, y 1 ) is the image of the interval (, y ) under the mapping y ỹ. Note that g(ỹ) for y < ỹ < y 1. Hence q(1 β) 1/p β 1/q+1 h (β) A A ( ) g(ỹ) + g(y) dy dỹ dỹ [ (ỹ y ) f(x(ỹ))f(ỹ) y ( ) x(y)y δ f(x(y))f(y)] dỹ, (9) x(ỹ)ỹ where δ = min{, 2 p}. The inequality (9) is deduced from Claim 2, which we prove in the appendix. Claim 2. We have In the appendix we also show ( ) dỹ x(ỹ)ỹ δ ( ) dy y y, < y < y. (1) x(y)y ỹ y 8

9 Claim 3. For < y < y, we have βỹ (1 β)(y 1 y); (11) βy (1 β)(y 1 ỹ). (12) For < y < y, (12) yields y min{ỹ, y 1 ỹ}, i.e., (ỹ, y 1 ỹ) (y, y 1 y). Thus, y(y 1 y) ỹ(y 1 ỹ), or, equivalently, y p x p (y) ỹ p x p (ỹ). By (8), this implies the relation (ỹ p, x p (ỹ)) (y p, x p (y)), < y < y. The assumption (2) then yields (δ = min{, 2 p}) (x(ỹ)ỹ) δ f(x(ỹ))f(ỹ) (x(y)y) δ f(x(y))f(y), ỹ A. It follows that the integrand in (9) is nonnegative, and h (β), β [1/2, 1), as required. Remark. The main complication in the proof of Theorem 2 is that the mapping y ỹ is not in closed form. In the special case p = q = 2, where ỹ is explicitly available, the proof can be simpler. Appendix: proofs of Claims 1 3 It is convenient to prove Claims 1, 3, 2 in that order. We emphasize that no circular argument is involved. Proof of Claim 1. Define L(y) = β p/q y p + (1 β) p/q (y 1 y) p, y y 1, (13) where y 1 is given by (7). We have L (y) = pβ p/q y p 1 p(1 β) p/q (y 1 y) p 1, and the unique solution of L (y) = is y = t(1 β) 1/p. Moreover, [ L (y) = p(p 1) β p/q y p 2 + (1 β) p/q (y 1 y) p 2] >. 9

10 Hence L(y) strictly decreases on the interval (, y ) and strictly increases on (y, y 1 ). We have L() L(y 1 ) because β [1/2, 1). By continuity, for any < y < y there exists a unique ỹ (y, y 1 ) that satisfies L(y) = L(ỹ), which reduces to (8) after routine algebra. Proof of Claim 3. We only prove (11); the proof of (12) is similar. For < y < y, define D(y) = L((β 1 1)(y 1 y)) L(y). Direct calculation using (13) gives D(y) =(1 β) p/q [( y 1 (β 1 1)(y 1 y) ) p (β 1 1)(βy/(1 β)) p (2 β 1 )(y 1 y) p], where the inequality follows from Jensen s inequality (αu + (1 α)v) p αu p + (1 α)v p, p > 1, with That is, α = β 1 1, u = βy 1 β, v = y 1 y. L((β 1 1)(y 1 y)) L(y) = L(ỹ), < y < y. (14) By the strict monotonicity of L( ) on the interval (y, y 1 ), if (β 1 1)(y 1 y) > ỹ, then L((β 1 1)(y 1 y)) > L(ỹ), which contradicts (14). Hence (β 1 1)(y 1 y) ỹ, as required. To prove Claim 2, we use Proposition 2 below. Define u α v α u v, u, v >, u v, Q α (u, v) = αu α 1, u = v >. 1

11 Proposition 2. If < α 1, then Q α (u, v) decreases in each of u, v > ; if α > 1, then Q α (u, v) increases in each of u, v >. Proposition 2 follows from basic properties of the generalized logarithmic mean (Bullen, 23, pp ). Proof of Claim 2. For < y < y, define u = βy, v = (1 β)(y 1 y), ũ = βỹ, ṽ = (1 β)(y 1 ỹ). Claim 3 says that v ũ and u ṽ. Applying Proposition 2, we obtain Q p/q (u, v) Q p/q (ṽ,ũ), 1 < p 2, (15) and Q p/q (u 1, v 1 ) Q p/q (ṽ 1, ũ 1 ), p > 2. (16) After routine algebra, (15) (for 1 < p 2) and (16) (for p > 2) reduce to (1). References [1] Bock, M. E., Diaconis, P., Huffer, H. W. and Perlman, M. D. (1987). Inequalities for linear combinations of gamma random variables, Canad. J. Statist. 15, [2] Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Schur properties of convolutions of exponential and geometric random variables. J. Multivariate Anal. 48, [3] Bon, J. L. and Paltanea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5, [4] Bullen, P. S. (23). Handbook of Means and Their Inequalities, Kluwer Acad. Publ., Dordrecht. [5] Hitczenko, P. (1998). A note on a distribution of weighted sums of i.i.d. Rayleigh random variables. Sankhya Ser. A 6, [6] Hu, C.-Y. and Lin, G.D. (2). On an inequality for the Rayleigh distribution. Sankhya Ser. A 62,

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