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Edgeworth Series Approximation for Chi-Square Type Chance Constraints

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Commun.Fac.Sci.Univ.Ank.Series A1 Volume 56, Number 2, Pages 27–37 (2007) ISSN 1303–5991 EDGEWORTH SERIES APPROXIMATION FOR CHI-SQUARE TYPE CHANCE CONSTRAINTS MEHMET YILMAZ Abstract. We introduce two methods for approximation to distribution of weighted sum of chi-square random variables. These methods can be more use- ful than the known methods in literature to transform chi-square type chance constrained programming (CCP) problem into deterministic problem. There- fore, these are compared with Sengupta (1970)’s method. Some examples are illustrated for the purpose of comparing the solutions of these methods. 1. Introduction The distribution of positive weighted combination of chi-square random variables with any degrees of freedom arises in many application areas such as communication theory, reliability of systems, engineering, industry etc. Many authors are interested in obtaining such as above distribution. Literature review of this subject was given by Johnson et al. (1994). Recent work of Castaño et al. (2005) derived a Laquerre expansion which has been used to evaluate distribution function of the sum of weighted central chi-square random variables. But it is more complicated for using stochastic programming. The best known example of a skewed sum is the chi-square distribution, of course chi-square distribution is itself asymptotically normal, thus this led us to approximate the distribution of a sum of weighted chi-square random variables by an expansion method based on central limit theorem (see Kendall, 1945; Patnaik, 1949; Feller, 1966 and Lehmann, 1999). Next section can be viewed as recalling Castaño et al. (2005)’s work. Introduced methods will be compared with their work which can be seen a main tool of this paper for further discussions. In section three, we will give two methods based on normal approximation. These are called …rst and second edgeworth expansion respectively. We can adapt suggested methods to linear combination of independent random variables assumed to having …nite fourth central moments. Chi-square Received by the editors Sept. 28, 2007; Accepted: Dec. 25, 2007. 2000 Mathematics Subject Classi…cation. Primary 62E20, 65K99; Secondary 60E07, 90C15. Key words and phrases. weighted sum of chi-square random variables, expansion for distrib- utions, chance constrained stochastic model. c 2007 Ankara University 27 28 MEHMETYILMAZ random variables have important role in many application areas of stochastic model. On the other hand, methods can be applied to solve CCP problem. In other words, it should be found their convenient deterministic equivalents to solve such a CCP problem. Two methods have some advantages of …nding deterministic equivalents of chance constraints as these will be presented in fourth section. 2. Laquerre Expansion Let X1;X2; :::; Xn be independent chi-square random variables with vi (i = 1; 2; :::; n) n degrees of freedom, respectively. Consider aiXi; where ai > 0: Distribution of i=1 n P aiXi as a Laquerre series expansion is given by Castaño et al. (2005) as follows: i=1 P x v 2 2 v e x 1 k!mk ( 2 ) (v + 2)x F (x) = 1+ v v v Lk (1) (2 ) 2 ( + 1) ( + 1) 4 2 k=0 2 k 0 with P n v v = vi, p = 2 + 1 i=1 and P k 1 v n p 2 2 p p vi 1 ai 2 mk = k mjIk j; k 1, where m0 = p (1 + ( 1)) j=0 0 0 i=1 0 and P Q j a j 1 1 n 1 i I = + v ; j 1 j p i ai p 1 2 i=1 1 + ( 1) 0 ! 0 ! P v k r ( 2 ) r+v=2 (:) where Lk (:) = k r r! ; is the generalized Laguerre polynomial. They r=0 have obtained theP following bound for the truncation error of the distribution func- tion F x v v 2 2 e x m0 (v + 2)x 1 ( 2 + 1)k k "N v v j j exp( ) 1+ 2 (2 ) ( 2 + 1) 8 0 k=N k! a P 1 i where = maxi ai p : This expansion converges uniformly in any …nite 1+ ( 1) 0 interval, for all and conveniently chosen. They also showed with illustrative 0 examples that if 0 = p=10 and = (a1 + an) =2 are chosen, then truncation error bound is quite small. Therefore our results will be compared with each other for the same values of 0 and . CHI-SQUARETYPECHANCECONSTRAINTS 29 3. expansion for distributions In this section, we will introduce two known expansions for sum of independent and identical random variables related to the central limit theorem. These are called as …rst and second edgeworth expansion in the literature. First edgeworth expansion which is well known yields more approximated probability than normal approximation does. Also, second edgeworth expansion could be better than the …rst for the sum of chi-square random variables (see Kendall, 1945; Patnaik, 1949 and Lehmann, 1999). Hence, extensions of two methods for the weighted chi-square random variables, are set up the following subsection. Theorem 3.1. Let X1;X2; :::; Xn be independent continuous random variables such 3 that E Xj < (j = 1; 2; :::; n), for large values of n, then j j 1 n n Xj E Xj (n) 2 j=1 j=1 ! 3 (1 x ) n P 1 x = (x) + 3 (x) + o( 3 ) (2) 0 (n) 2 1 (n) 2 (n) 2 P 2 P 6 2 2 B C is used approximation@ to standardA normal distribution. Here and stand for standard normal cumulative distribution function, its density function respectively n (n) k and k denotes E (Xj EXj) (Feller, 1966 Chapter XVI). j=1 P The …rst term given in (2) is known as …rst term edgeworth expansion in the literature. The second term edgeworth expansion can be de…ned without loss of generality as follows: Subtract from the right side of (2) the term K K2 4 (x3 3x) + 3 (x5 10x3 + 15x) (x) (3) 24 72 n (Xj E(Xj )) where K and K are third and fourth cumulants of j=1 (see Wallace, 3 4 n P 2 (Xj E(Xj )) v uj=1 u 1958). If the random variables are identically distributedtP then the quantity given in (3) can be written as 2 (n) 2 (n) 3(2 ) (n) 4 n 3 3 5 3 2 (x 3x) + 3 (x 10x + 15x) (x): (4) 24 (n) 72 (n) " 2 2 # 30 MEHMETYILMAZ 3.1. Set-up for Weighted Chi-Square Random Variables. Consider random variables Xi (i = 1; 2; :::; n) distributed as chi-square with vi (i = 1; 2; :::; n) degrees of freedom. The probability density function of Xi (i = 1; 2; :::; n) is given by vi 1 2 1 xi f(x ; v ) = v x exp( ); x > 0; v > 0 (i = 1; 2; :::; n): i i v i i i i i 2 2 ( 2 )2 (1) The central moments l (l = 1; 2; 3; 4) can be obtained by evaluating the following integral, (1) 1 1 vi x k 2 1 = v (x v ) x exp( )dx k v i i i 2 2 ( 2 )2 0 k R k j vi j k ( vi) ( + j)2 = 2 ; k = 1; 2; 3; ::: j ( vi ) j=0 2 (1) (1) P (1) (1) 2 Hence 1 = 0; 2 = 2vi; 3 = 8vi and 4 = 12vi + 48vi are calculated. From now on, we can set up (2) and (3) for the weighted sum of chi-square variables. Let n n n n 2 3 4 M1= aivi;M 2= 2 ai vi;M 3= 8 ai vi;M 4= 48 ai vi: (5) i=1 i=1 i=1 i=1 P P n P Pn Mk (k 1) denote cumulants of ai(Xi E(Xi)), then P ( aiXi t) is i=1 i=1 approximated as P P 2 M3(1 x ) (x) + 3 (x) (6) 6(M2) 2 M (1 x2) M (x3 3x) M 2(x5 10x3+15x) 3 4 3 (x) + 3 24M2 72M 3 (x) (7) 6M 2 2 2 2 3M2 2 0 2 3 2 5 3 M3(1 x ) (M )(x 3x) M (x 10x +15x) (7) 4 n 3 (x) + 3 24M 2 72M 3 (x) 6M 2 2 2 2 n where x = t M1 , and M 0 = a4 12v2 + 48v . First and second edgeworth pM 4 i i i 2 i=1 approximations for distributionP of weighted sum of chi-square random variables can be given respectively in (6) and (7). Here the second expression (7*) can be viewed as modi…ed and extended version for weighted sum, this modi…ed expansion allows to get better results than the other when the chi-square random variables have di¤erent degrees of freedom. Otherwise, (7) and (7*) give same result for v1 = v2 = ::: = vn: In Table 1, we …rst compare our results for P 22 + 22 t , n = 2 and (1) (1) a1 = a2, v1 = v2. Here FE, SE and LE denote …rst term edgeworth expansion, second term edgeworth expansion and Laquerre expansion. These are calculated from Matlab. CHI-SQUARETYPECHANCECONSTRAINTS 31 Notes: (i) Since the weights are equal and random variables are independent and identically distributed, if the quantity P (22 + 22 t) can be arranged as (1) (1) P 2 t , then it is easy to compare FE and SE by checking chi-square tables. (2) 2 Table 1 and Table 3 are specially presented because of these probabilities are easily checked from chi-square tables. So that tabulations could be useful to …nd an answer how close FE and SE to exact values. (ii) Various approximation methods of distribution of weighted sum of chi-square random variables can be seen in Johnson et al.
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