,AD-R A MEASURE OF VARIABILITY BASED ON THE HARMONIC MEAN AND i/i ITS USE IN APPROXIMATIONS(U) CITY COIL NEW YORK DEPT OF MATHEMATICS M BROWN

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1 ,AD-R A MEASURE OF VARIABILITY BASED ON THE HARMONIC MEAN AND i/i ITS USE IN APPROXIMATIONS(U) CITY COIL NEW YORK DEPT OF MATHEMATICS M BROWN MAR 03 CUNY-MB3 AFOSR-TR UNCLASSIFIED RFOSR F/G 12/1 NL

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3 WFOSR-TR- 9,9 8 A Measure of Variability Based on the Harmonic Mean, and its Use in Approximations by Mark Brown The City College, CUNY March, 1983 APR A APPPOe 1o0pbI& r2el easej distr1bution u 0 ni.ited. Lai City College, CUNY Report No. MB3 AFOSR Technical Report No

4 UNCLASSIFIED SECURITY CLASSIFICATION OF THiS PAGE (Whm Date Enitred SPAGE READ INSTRUCTIONS REPORT DO MENTATION P EBEFORE COMPLETING FORM,, ~ ~.. TR a I. REPORT NUM ER 2. GOVT ACCESSION NO. 3. RECIPIENTOS CATALOG NUMeER -Zi 4. TITLE (m-d Iubl0e) b. TYPE OF REPORT & PERIOD COVERED A MEASURE OF VARIABILITY BASED ON THE HARMONIC TECHNICAL MEAN, AND ITS USE IN APPROXIMATIONS C 3 'r 6. PERFORMING ORG. REPORT NUMBER CU #_4B3-7. AUTmOR(e) 1. CONTRACTO GRANT NUMI1ER(a) Mark Brown AFOSR S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK Department of Mathematics AREA & WORK UNIT NUMBERS The Oity College, CUNY New York NY PE61102F; 2304/A5 II. CONTROLLING OFFICE NAME AND ADDRESS Ii. REPORT DATE Directorate of Mathematical & Information Sciences MAR 83 Air Force Office of Scientific Research 1. NUMBER OF PAGES Bolling AFB DC MONITORING AGENCY NAME A ADDRESS0t diffroent OMr Ceierelftni Office) IS. SECURITY CLASS. (of this report) UNCLASSIFIED OI. DECiASSIFICATIONiDOWNGRAOING IS. DISTRIBUTION STATEMENT (of *e Revrt) *l Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of as abstrace entered in &lock 0, it dffemt bm Rhp,-) IS. SUPPLEMENTARY NOTES IS. KEY WORDS (Continue on reveree side If necessary and identify by bock number) Harmonic mean; Laplace transforms; completely monotone functions; measures of variability. 20. puegyat.t fcontinue en revere. side If nec.eomy and Identify by blek number) Let X.be a positive variable and assme that both a * a EX - and - E X are finite. Define c 2 1-(a). -1. This quantity servs as a measure of variability for X which is reflected in the behavior of completely monotone functions of X. For 2 completely monotone with (O) < : 0 < I8(X)-S(fl) _ C 2 (O) Var 8(X) < c 2a2 (0) IJAN 1473 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (lhen Deo Entered)

5 7W Sumoy Let X be a positive random variable and assume that both a = X7 and P - E are finite. Define c 2 _ l-(api) l. This quantity serves as a measure of variability for X which is reflected In the behavior of completely monotone functions of X. For g completely monotone with S(O) < o < Eg()-s(EU)! C 2 S(O) Var S(X) c c (0) COPY - M~tC-.4 2

6 1. Introduction. Given a random variable X and a function S, crude approximations to the mean and variance of g(x) are obtainable by Taylor series arguments. The variance of X, a 2, is a key quantity under this approach, both in approximating the bias (Eg(X)-S(EX)) and the variance (Var(g(X))). For X positive and g rapidly decreasing, the bias and variance of g(x) should be relatively insensitive to the tail behavior of X, and a2 should therefore not play an important role. In practice, when a2 is very large the approximations for rapidly decreasing functions are often very poor. We take the point of view that a 2 is not measuring that aspect of variability which is relevant to the behavior of rapidly decreasing functions of X. For this purpose an often more informative measure of variability is- (1.1) c 2 U li(xl') -l. For X positive and g completely monotone we derive: (1.2) 0 < (Is(X)-s(Ex)]/s(o) < C2 (1.3) Var[S(X)/$(O)) I c2 Thus if EC - ' is close to (EU) - 1 (as measured by c 2 ) then for g completely monotone, g(x)/&(o) is close to a one point distribution at s(u)ls(o). 3

7 The quantity c 2 has an additional interpretation. For a positive random variable X with distribution F, consider a stationary renewal process on the whole real line, with interarrival time distribution F. Define T to be the length of the interval which covers {01, and V - T' -. Then: (1.4) a 2 - EES) - E 2 'V (El) 2 (1.) / - c 2 Ilil Moreover, defining h(z) - (EX)xg(x- ), it follows that g(ex) h(ev) and: (1.6) Eg(X) - zh(v) Use of (1.6) and a Taylor series argument yields: (1.7) 0 1 Eg(X)-g(X) < 2 Ix-lsup(x3g"(x)) Expressions (1.2) and (1.7) are competing inequalities. For X fixed with E-l < -, (1.2) will be better for some choices of g, and (1.7) for others ,-- --.,,

8 2. Definitions and preliminary results. A function g on [0,00) is defined to be completely monotone if it possesses derivitives of all orders and (-l)na(n)(x) > 0 for all X > O. In the above g(o).*- Someexamples are (z+a) Ā with a > O, k > O,e - x with a > O, and e " X ( ' - * z ) with X > 0. Law= (2.1) below is due to Bernstein. An interesting discussion and proof of Bernstein's theorem is given in Feller (2] p Liema 2.1. A function g on (0,) is completely monotone with s(o) - a <-= if and only if it is of the form ('4z (2.2) $(z) - 0 e dh(z) wbere 9 is a positive measure on [0,0w) with H([O,m)) - m. transform: Consider a probability distribution F on [0,m). Its Laplace f (2.3) (a) - e -ax df(x) is the survival function of a mixture of exponential distributions, i.e. if ZIXZx is exponential with parameter x, then Pr(Z >olix-x) a - " x and Pr(Z > ct) - (*). With this observation, Lemma 2.2 below follows from Brow [l, theorem 4.1 part (xii). 5 r"-f, '.'; ''.. ','>...'.. '"".,,;", ' r, ',T.. _...,. I. *...,...,., *..,,,,.,,...,,,...,,'.,.'. ".,*..'.,.

9 Lm 2.2. Let Z be the Laplace transform of a probability distribution on [0,w) with a f L(()dm < - and ui- -V(0) <0 Then: (.)0< <1-au' f or all a >0. 3.Derivation of inequalities. Consider a positive random variable X with a - EX- assumed finite, as veil as Ui - EX. Define c 2- l-(ai) Note that 0 < c 2 < 1 with equality if and only if X is a constant. Tbeorem 3.1. Let g be a completely monotone function on (0,00) with 8(O) <W~. Then: (3.2) 0 < Eg(x)-g(11) i c 2g(0) (3.3) Var(s(l)) 1 c 2 g2 (0) Proof. By Liem 2.2, (3.4) 0 ((Q-eO Cc 2 for all ax > 0 Since g is completely monotone, by Lemma 2.1 there exists a measure H on (0,M) with H[0,46) - g(0) and: (3.5) g(x) meaxdh(ci) 6

10 Now: (3.6) Z8() - if eaxdh(c)df(x) -ft()dh(a) (3.7) g (U) - f e-.(%dh(cl). Since g is convex, Eg(X) > g(ji). Thus from (3.4) and (3.6): (3.8) 0 1 Eg(g(() - To ( (c)-e-")dh(a) < c 2g(O) Since g is completely monotone so is g2 (Feller (2], p. 441). Applying (3.8) to g2 w obtain: (3.9) 0 < (1)-2 W t) e 2 g 2 (0) From (3.8) and (3.9), Var($(X)) - Eg 2(X)-(Ig(X)) 2 <(2()+c (0))-a 2 () - c 2 g 2 (0). This concludes the proof. Given X > 0 with distribution F, consider a stationary renewal process on the whole real line with interarrival time distribution F. Define T to be the length of the interval containing 0. It follows from Feller (21 p. 371 that: (3.10) dft(x), zdf(x)/u. From (3.10) we ee that is-1 U-1 and IT - 2. a& "1 where a- EI. Defining V - T 1 it follows that 2 - (&-1)U " 2 a c2aa- 1, while 7

11 2 /V2=C 2. Also from (3.10) we see that ES(X) -E(iiVg(V 1 )) Eh(V) where h(x) - ixs(xz ). Note that h(ev) -h(u± ) -g(ex). Finally since: (3.11) h(v) - h(ev) + (V-EV)h' (EV) + V-EV 2 h#'(v* 2 with V between EV and V, it follows that: a (3.12) Eg(X) S h(ev) + a' sup(h"(x)) _g(i.z) + C ap suli() 2 2 sph() But h"(x) -3 ix ( ), and thus sup h"(x). 4i sup(x g, (X)). Thus from (3.12) 12 3 (3.13) 0O1Eg(Z)-g(z)I 2 -a sup(x 9"(x)) 8

12 References Il] Brown, M. (1981). "Approximating IMRL distributions uy exponential distributions, with applications to first passage times." To appear in Ann. Probability. (2] Feller, W. (1971). An Introduction to Probability Theo"y and its Applications. Volume II, 2 n d Edition, John Wiley, New York.