 Solution

ID: past-exam-of-the-simulation-and-statistical-programming-course-of-the-university-of-oxford/2023/2/a/iii/solution

ρ

A_{i} := 1_{{V_{i} \leq w (Y_{i}) / M}} .

π (Y_{i}, V_{i}) / ρ (Y_{i}, V_{i}) = M A_{i}

. Therefore the requested augmented-space importance sampling estimator is

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

θ

, provided

E_{p} ∣ ϕ (X) ∣ < \infty

. This is not the ordinary estimator

n^{- 1} \sum_{i} w (Y_{i}) ϕ (Y_{i})

from question 1. Taking the conditional expectation of this augmented estimator given all

Y_{i}

produces that ordinary estimator, because

E [A_{i} ∣ Y_{i}] = w (Y_{i}) / M

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