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An unbiased importance-sampling weight can have infinite second moment

Example

Let $Ω = {1, 2, \dots}$ , $P (k) = c_{2} / k^{2}$ , and $Q (k) = c_{3} / k^{3}$ , where $c_{s} = (\sum_{k = 1}^{\infty} k^{- s})^{- 1}$ . For $A_{m} = {k \geq m}$ , the ordinary importance-sampling contribution is unbiased and integrable, but its second moment is infinite.

Facts & Assumptions

Given: The probability mass functions above, the event $A_{m}$ for fixed $m \geq 1$ , and proposal law $Q$ .

[F1]

Ordinary importance sampling: The single-draw contribution is $1_{A_{m}} (X) P (X) / Q (X)$ for $X \sim Q$ .

[F2]

Ordinary importance sampling is unbiased: If the target event law is absolutely continuous with respect to the proposal and the weighted contribution is integrable, its expectation is $P (A_{m})$ .

[F3]

Ordinary importance sampling has finite variance and a CLT under a weighted second moment: A finite weighted second moment is a sufficient condition for finite variance.

[F4]

Rare-event efficiency and work criteria: Efficiency criteria record whether the second moment is finite; finite variance is not implied by unbiasedness alone.

Verification

technique · direct

1.1givenalgebra

Both constants are finite and positive because the defining series converge for exponents $2$ and $3$ , and $Q (k) > 0$ for every $k$ ; hence $P ≪ Q$ . The likelihood ratio is $P (k) / Q (k) = (c_{2} / c_{3}) k$ .

2.1F1F2step 1.1algebra

By [F1], $E_{Q} Z = \sum_{k = m}^{\infty} \frac{c_{3}}{k^{3}} \frac{c_{2}}{c_{3}} k = c_{2} \sum_{k = m}^{\infty} k^{- 2} = P (A_{m}) \in (0, \infty),$ so [F2] confirms unbiasedness.

3.1F3F4step 1.1algebra∎

The second moment is $E_{Q} Z^{2} = \sum_{k = m}^{\infty} \frac{c_{3}}{k^{3}} {(\frac{c_{2}}{c_{3}} k)}^{2} = \frac{c_{2}^{2}}{c_{3}} \sum_{k = m}^{\infty} \frac{1}{k} = \infty$ by divergence of the harmonic tail. Thus the weighted second-moment hypothesis in [F3] is absent and this unbiased estimator has infinite variance; [F4] therefore rules out describing its second-moment efficiency as finite.

Depends on

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Sources

Gino Biondini, An Introduction to Rare Event Simulation and Importance Sampling, Handbook of Statistics 33, Chapter 2, §§3.1–3.3 (standard reference, not scraped)