Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

42*3Similarly, for the fourth moment05t> = < - 2 0 > = -^< 2*M- 2 2 ^ 1Vn , / n 4 V, , i ;i (2 + 22 2 xfxf) - 'r (x 4 > +Inserting the above expressions into Equation (6.284), the quantity to be minimized becomeQ(n)N(x 4 ) + (N ~ 2 - n + — — W > ?\ N — n/(6.285)where we have inserted k — N/n. Obviously, Q(n) is minimum with n = 1, i.e., if evervbatch consists of a single history, as stated.We have thus shown that when estimating the theoretical variance of a random variable(e.g., of the final score in a history) from the realizations X 1(i - 1, 2,...,N) the mostefficient estimate isN1s = ~ 2 (x, - m) 2 (6,286)as follows from Equation (6.280) with k = N. In other words, for variance estimation,batchwise evaluation is not efficient.In the proof above, we have also established the variance of the sample variance s. Notethat the derivation was performed for a zero-expectation random variable, i.e., in the genera!case, x -- m is to be inserted instead of x and the variance of s in Equation (6.286) isD 2 [S] = Q(I) ~ D 4 [x] = -J-K(X - m) 4 ) - D 2 Ix]] + - 7 - - - - 7 : D 2 [x] (6.287,N N(N — 1)The efficiency of batchwise variance estimation was examined by Dubi. 12the statement of Theorem 6.14 for the estimateHe provesNI ,s = 7; 2 (x, ^ m) 2Nand shows thatD 2 [s] - -J- |((x - m) 4 ) - D 4 [x]] (6.288)NComparing Equations (6.287) and (6.288), it is seen that the variance of the estimatedvariance is lower by D 4[x]/[N(N - 1)] if the expectation of the random variable x is knownand need not be estimated by m.