
Monte Carlo Methods and Zero Variance Principle 4 67
TABLE 22.1
LZV estimators and their variances for the first thr e e moments of a univariate
normal target.
g(θ) ˜g(θ) Var [˜g(θ)]
θ µ 0
θ
2
θ
2
− 2µ(θ −µ) Var
θ
2
− 4µ
2
σ
2
θ
3
θ
3
− 3(µ
2
+ σ
2
)(θ − µ) Var
θ
3
− 9σ
2
(µ
2
+ σ
2
)
Gaussian target N(µ = 0, σ
2
), the LZV estimator has the same variance as the
original estimator. More generally, the further away the mean µ of the Normal
target is from zero, the greater is the variance reduction of the LZV estimator
of the second moment. The LZV estimator of the third order moment reduces
the variance of the corresponding original estimator by 9σ
2
(µ
2
+ σ
2
). As the
variance σ
2
of