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1 The Variance of the Product of Random Variables Author(s): Leo A. Goodman Source: Journal of the American Statistical Association, Vol. 57, No. 297 (Mar., 1962), pp Published by: American Statistical Association Stable URL: Accessed: 10/02/ :28 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association.

2 THE VARIANCE OF THE PRODUCT OF RANDOM VARIABLES' LEO A. GOODMAN University of Chicago An exact formula for the variance of the product of random variables, X1, X2, *, X, is given as a function of the means and the central product-moments of the xi. The usual approximate variance formula for i-1 is compared with the exact formula; e.g., we note, in the case where the xi are mutually independent, that the approximate variance is too small, and that the relative accuracy of the approximation depends on the magnitude of the coefficients of variations of the random variables. The case where the xi need not be mutually independent is also studied, and variance formulas are presented for two different "product estimators." (The relative magnitude of the variances of these estimators is shown to be related to the relative accuracy of the approximate variance formula for flx, i=1 as compared with the exact variance formula.) When the xi are mutually independent, simple unbiased estimators of these variances are suggested; in the more general case, consistent estimators are presented. 1. INTRODUCTION AND SUMMARY THE variance of the product of two random variables has been studied by Barnett [1] and Goodman [2] in the case where the random variables are independent, and by Goodman [2] in the case where they need not be inde- k pendent. Shellard [3] has studied the case where the distribution of 17 x, was (approximately) logarithmic-normal. In the present note, no assumptions will be made about the distribution of II xi. We shall discuss first the case where the random variables, X1, X2, *.*,* X, (? 2) are mutually independent, and then the case where they need not be. The usual formula, which has appeared in the statistical literature, for the variance of the product of two independent random variables is an approximation (see, for example, Yates [4, p. 198]). The relative accuracy of this approximation depends on the magnitude of the coefficients of variation of the random variables. In the present note, we shall present an exact formula for the vari- ance of ]J xi, which does not depend on any assumptions concerning the i=1 1 Part of this research was carried out at Columbia University, under sponsorship of the Office of Naval Research (Contract Number Nonr-266(33), Project Number NR , while the author was a Visiting Professor of Mathematical Statistics and Sociology there; and part at the Department of Statistics, University of Chicago, under sponsorship of the Logistics and Mathematical Statistics Branch, Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the United States Government. For their helpful comments, the author is indebted to G. J. Glasser, J. E. eilin, and H. L. Seal. 54

3 VARIANCE OF PRODUCT OF RANDOM VARIABLES 55 magnitudes of the coefficients of variation. It will be seen that, in the case where the xi are mutually independent, the "variance" obtained by using an approximate formula is less than the exact variance, and that the approximation may be satisfactory in some situations where or -1 of the coefficients of variations are relatively small. A simple unbiased estimator of the exact variance will also be presented. The exact formula for the variance will then be used to compute the variances of two different kinds of estimators of a parameter that is in fact equal to the product of other parameters. Finally, the situation where the random variables need not be mutually independent will be investigated, and variance formulas will be presented in this case along with consistent estimators of these variances. The results presented in the present note are generalizations of results presented in reference [2] for the special case where =2. A number of readers of reference [2] have written the author inquiring about the possibility of generalizing the results presented there, and so it seemed worthwhile to prepare the present note giving these generalizations. The methods used herein are rather simple and are quite similar to those in reference [2], although combinatorial considerations of the kind needed to justify equations (10) and (12) herein did not appear (and were not needed) in the earlier publication. 2. THE CASE WHERE THE RANDOM VARIABLES ARE INDEPENDENT Let us denote the expected value of the random variable xi(i= 1, 2, * * *, ) by E(xi) =Xi, the variance of xi by Vi, and the square of the coefficient of variation of xi by Vi/X = Gi. (For the sake of simplicity, we assume that Xi 7;z! 0, although some of the results presented do not require this assump- tion.) If the xi are mutually independent, we find that the variance of TI will be equal to V ( xi) E {,} HX V(II )=E{2i=} i= (1) = fl (Vi which can also be written as Xi) - IT xt = Xi v ~~~~ H Xi x Vi II Xi + E V'tlVi2 H xi i jfdi ~~il<i2 jxil.i2 2 1 [X H (Gi + 1) -j + E VilVi2Vi3 Xj + * + (V1V2... V) i1<i2<i3 ip&i1.i2.i3 (2) 2 ( =II xi E Gi + E Gi1Gi2 + E Gi+Gi2Gi +* il i il<i2 il<i2<is +... (G1G2 G)1

4 56 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1962 where E denotes summation over all values of il<i2<. < is il<i2< *<is ranging over 1, 2, ***,, and where H denotes the product over j il, i2 ***is the -S values different from the S values il <i2 <... is. From equation (2) we note that the approximation F (fti x) = HI xx ~ ~i= Gil3 E ai, (3) is less than the exact variance. We also observe that ( H x may be a satis- factory approximation for V ( I if all of the coefficients of variation, G1, G2, ***, G, are small. In addition, we note that the approximation V \ ft xi) may be satisfactory even in some cases where only -1 of these coefficients are small, contrary to what has usually been suggested in the literature. (See related comments in reference [2].) Equation (3) is a generalization of the usual approximate formula for the variance of a product (see, for example, Yates, [4, p. 198]). Equation (2) is a generalization of equation (a) given by Barlnett [1] and equations (2) and (15) given by Goodman [2]. Equation (1) here was presented earlier by Shellard [3]. Let us now consider the situation where a sample of ni observations (xil, xi2,..., xij) is drawn from a population having mean Xi and variance Vi(i= 1, 2,..., ). Let xi and vi denote the sample mean and sample variance computed from the sample of ni observations. Then E {x fi } =Xi and E{vi} =Vi. Writing hi=xi-v/ni and w = E xi2/ni, we note that E Ih} =h X, and E { wi } = E { X } = E { x }. When mutually independent drawn (i= 1, 2, - -*, ), we find that j=1 samples are C \ v Hxi) = wi--h hi = H (vi + h)-h hi (4) \ / is an unbiased estimator of V ( ft x The statistic J vx is an unbiased estimator of I Xi. By applying equa $~~~~~~~~~=1

5 VARIANCE OF PRODUCT OF RANDOM VARIABLES 57 tion (1), we find that the variance of I xi is where ;i Gil/ni. An unbiased estimator of V (I i) is 2 S IIz t-ii hi. (6) We now consider the situation where a sample of n observations (Xlj 2 *,** Xj) is obtained (j=1, 2,, n), where E{xii} =Xi and where the variance of xii is Vi (i = 1, 2, n *, ). Then E Z j=1 i=1 xij/n=z is an unbiased estimator of H Xi in the special case where the xij are mutually i=1 independent. By applying equations (1) and (4), the variance of z is and an unbiased estimator of V(z) is where n V(z) = xi n, (7) v(z) = v Xi n, (8) xx-e xi,/n, Vi= (Xi,j - )2/(n -1), j=1 j=1 n hi = Xi -v/n, = xij/n. j=l1 In the situation where a sample of n observations (xlj, X2j,.., Xj) is drawn (j= 1, 2, *, n), and where the xii are mutually independent, the statistic \ X i will also be an unbiased estimator of I V (I H i of H xi can be computed. I i-=1 i =1 Xi. The variance from equation (5). We note that the limit of nv (lixi) is i (u xsi as n-, and that V (t x/ V(z) ap- p ( / V H proaches, VH I xi / v ( xi. Thus, we see that, when the xii are mutually

6 58 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1962 independent, the effect of using the approximate formula ( r xi rather than the exact formula V (i xi is comparable to the effect of using the sta- tistic z rather than tj xi as an estimator of JJ Xi (when n-* oo). 3. THE CASE WHERE THE RANDOM VARIABLES NEED NOT BE INDEPENDENT In the case where the xi need not be mutually independent, we see that the variance of H xi is i=1 Xi E r (x= + 1)2} - B2} where Ai=xi-X;, Si= (xi-xi)/xi, M=E{Hxi}, B=M/ TXiS. Equa- f tion (9) can be written in a number of different forms. Two such forms, which are convenient for computational purposes are the following equations (10) and (12). Writing B-1 =A, we have v(fi- ) {E{[ft( i= 1A} ]} =H Xi{E{[ 3i + E aila$2+ l E ilt +* ~~i il<i2 il<i2<i3 +( S)]} -A2} =H 2 Xi E C(s1, 82, *,S) - A2; (10) i= where C(S1, S2,, S) = D(u, m)e { 8i:}) D(u, m)= 2u - 2 form =0, u t2u for m> l, si = O, 1, or 2 for i = 1, 2, ** *,, u = number of l's in (Si, S2,, S), and where E #1.., o m = number of 2's in (si, 82,, 8), denotes summation over the 3--i sets of (si, s2...,s)

7 VARIANCE OF PRODUCT OF RANDOM VARIABLES 59 where 2mn+u>1. Formula (10) can be derived by straightforward combina- torial considerations; E I } will appear in the expansion of E {[TI (3 +l)-1]} a total of 2u times when m> 1, and it will appear a total of 2u-2 times when m=o (u>l). We note that ( ( A =E411(i+1) - ~~~~~ 1} E= ~t tlbt... tr i (11) where ti=o or 1, for i=1, 2,,, and where * denotes summation ti 't over the 2&-1 sets of (tl, t2,, t) where t,= 1 for at least two values of i. From (10) and (11), we have V Hx)-r x F(s1, s2,,s) + A(2 - A), (12) ' where F(s1, s2,,s) =H(u, m) E { II 6 } H(u, m) = {2u - 4 for m = 0, yt 2u ~ for om21 m >1. The usual approximate formula for V t xi) is Xi = llx2{xg, +2ZEaiSj}}. (13) i31 t1 i iv j Here we have a summation of +(-1)/2=(+1)72 terms within the brackets, while in equation (10) there was a summation within the brackets of 3--1-(+1)/2=3-1-(+3)/2 terms and a subtraction of A2 in addition to the summation of the (+1)/2 terms of equation (13). Thus, formula (13) will be a satisfactory approximation for V ( I x only if the summation of the additional 3-1- A2 can be neglected. (+3)/2 terms and the subtraction of n We now consider briefly the product estimators z= E I j=1 i31 J xj/n and y-ii xi, based on a sample of n observations (xlj, x2j,..., Xj) where i-1, 2, *. * n, and where i= se ji1 xii/n and E{xXj} =Xi. The statistic z

8 60 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1962 is an unbiased estimator of M = E { H x4}, and y is a consistent estimator of \ TI Xi. The variance V(z) of z is V (t xi)/ n. Thus, nv(z) is equal to v ( I xj) given by equations (10) or (12). Writing V(y) for the variance of y, we see that nv(y) approaches V ( t xj) given by equation (13) when n-*. Con- sistent estimators of nv(y) and nv(z) can be obtained by replacing the various population moments in equations (10) and (13), respectively, by the corresponding sample moments. Finally, we take note of the fact that V(y)/ V(z) approaches Z ( fi x3i / v ( xi xs) as n-* X, which indicates that the effect usual approximate formula of using the ( f x x.) rather than the exact formula V ( l xi) is comparable to the relative difference between the variance of the estimator z of Ill and the variance of the estimator y of fi Xi (when n-> oc). i1 REFERENCES [1] Barnett, H. A. R., "The variance of the product of two independent variables and its application to an investigation based on sample data," Journal of the Institute of Actuaries, 81 (1955), 190. [2] Goodman, Leo A., "On the exact variance of products," Journal of the American Statistical Association, 55 (1960), [3] Shellard, G. D., "Estimating the product of several random variables," Journal of the American Statistical Association, 47 (1952), [4] Yates, Frank, Sampling Methods for Censuses and Surveys, Second Edition. London: Charles Griffin and Co. Ltd., 1953.

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