 Solution

ID: past-exam-of-the-simulation-and-statistical-programming-course-of-the-university-of-oxford/2023/1/c/i/solution

Y_{1}, \dots, Y_{n} \sim q

w = p / q

θ_{n}^{IS} = \frac{1}{n} i = 1 \sum n w (Y_{i}) ϕ (Y_{i}) .

E_{q} [w (Y) ϕ (Y)] = E_{p} ϕ (X) = θ

, this is an unbiased estimator. Moreover,

0 \leq w \leq M

, so

E_{q} [w (Y)^{2} ϕ (Y)^{2}] = \int w (x) ϕ (x)^{2} p (x) d x = E_{p} [w (X) ϕ (X)^{2}] \leq M E_{p} [ϕ (X)^{2}] = M (1 + θ^{2}) .

Here

E_{p} ϕ^{2} = Var_{p} (ϕ) + θ^{2} = 1 + θ^{2}

. Subtracting the squared expected value and dividing by

n

Var (θ_{n}^{IS}) = \frac{E _{p} [ w ϕ ^{2} ] - θ ^{2}}{n} \leq \frac{M + ( M - 1 ) θ ^{2}}{n} .

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