Tham khảo tài liệu 'simulation and the monte carlo method episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 130 CONTROLLING THE VARIANCE Suppose that X can be generated via the composition method. Thus we assume that there exists a random variable y taking values in 1 . m say with known probabilities Pi i 1 . m and we assume that it is easy to sample from the conditional distribution of X given y. The events y i i 1 . m form disjoint subregions or strata singular stratum of the sample space Q hence the name stratification. Using the conditioning formula we can write E E H X y PiE K X y i . i i This representation suggests that we can estimate Í via the following stratified sampling estimator. m - Ni t i j i where Xj3 is the j-th observation from the conditional distribution of X given y i. Here Ni is the sample size assigned to the i-th stratum. The variance of the stratified sampling estimator is given by m 2 m 2 2 Var g-Var Jff X y i E - i i 1 1 i i where a2 Var 7 X I y ỉ . How the strata should be chosen depends very much on the problem at hand. However for a given particular choice of the strata the sample sizes Ni can be obtained in an optimal manner as given in the next theorem. Theorem Stratified Sampling Assuming that a maximum number of N samples can be collected that is Nị N the optimal value of Ni is given by N n which gives a minimal variance of Var i 3 Proof. The theorem is straightforwardly proved using Lagrange multipliers and is left as an exercise to the reader see Problem . Theorem asserts that the minimal variance of f s is attained for sample sizes Ni that are proportional to Pj Tj. A difficulty is that although the probabilities Pi are assumed to be known the standard deviations ơi are usually unknown. In practice one would estimate the ơi from pilot runs and then proceed to estimate the optimal sample sizes N from 5 36 . A simple stratification procedure which can achieve variance reduction without requiring prior knowledge of Tt2 and H X is presented next. IMPORTANCE SAMPLING 131 Proposition Let .
