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An independent pilot coefficient preserves conditional unbiasedness

Statement

Let $n, q \geq 1$ , let $X_{1}, \dots, X_{n}$ be IID with law $P$ , let $f, h_{1}, \dots, h_{q}$ be integrable, and let $θ = E_{P} h$ be known. Write $X_{n}$ for the sigma-algebra generated by the production sample, namely the smallest sigma-algebra containing each event ${X_{i} \in B}$ for measurable $B$ . Let $P$ be a sigma-algebra generated by pilot data and assume $P$ and $X_{n}$ are independent in the sense of Independent sigma-algebras and independent events. Let the $P$ -measurable coefficient $\hat{β} \in R^{q}$ satisfy $E ∥ \hat{β} ∥ < \infty$ . Then the random-coefficient control estimator

${\hat{μ}}_{\hat{β}} = \frac{1}{n} \sum_{i} [f (X_{i}) - {\hat{β}}^{T} (h (X_{i}) - θ)]$

is integrable and $E [{\hat{μ}}_{\hat{β}} ∣ P] = E_{P} f$ almost surely. The first-moment condition on $\hat{β}$ ensures the displayed estimator is integrable; no claim is made for an arbitrary nonintegrable pilot coefficient.

Facts & Assumptions

Given: The IID production sample, integrable target and controls, known control mean, independent pilot sigma-algebra, and integrable pilot coefficient in the Statement.

[F1]

Control-variate estimators: With sample means, ${\hat{μ}}_{β} = \overset{ˉ}{f} - β^{T} (\overset{ˉ}{h} - θ)$ .

[F2]

Identical distribution and IID families: The production sample is mutually independent and identically distributed.

[F3]

Conditional expectation given a sigma algebra: Conditional expectation is characterized by its integral equality on every event in the conditioning sigma-algebra.

[F4]

Expectations factor over finite products of independent random variables: Expectations of products of independent integrable real variables factor.

[F5]

Linearity, monotonicity, and the modulus bound for expectation: Expectation is linear on integrable finite sums.

[F6]

Change of variables for expectation: Expectations of functions of a variable with law $P$ are integrals against $P$ .

[F7]

Independent sigma-algebras and independent events: For events $A \in P$ and $B \in X_{n}$ , independence means $P (A \cap B) = P (A) P (B)$ .

Proof

Given: The objects and hypotheses in the Statement.

1.1F1F2F4F5F6F7

The production sample means are $X_{n}$ -measurable and satisfy $E \overset{ˉ}{f} = E_{P} f$ and $E (\overset{ˉ}{h} - θ) = 0$ by [F2], [F5], and [F6]. Each ${\hat{β}}_{j}$ is $P$ -measurable; for any Borel sets, its preimage event belongs to $P$ while the corresponding production-mean event belongs to $X_{n}$ , so [F7] makes it independent of each production mean. Since $E ∥ \hat{β} ∥ < \infty$ , [F4] applied to the absolute product makes every ${\hat{β}}_{j} ({\overset{ˉ}{h}}_{j} - θ_{j})$ integrable; the full estimator is therefore integrable.

2.1F1F4F5F7step 1.1

Let $A \in P$ . The random variable $\overset{ˉ}{f}$ is $X_{n}$ -measurable, so its Borel preimage events are independent of $A$ by [F7]; factorization [F4] gives $E [1_{A} \overset{ˉ}{f}] = P (A) E_{P} f$ . Also $1_{A} {\hat{β}}_{j}$ is $P$ -measurable and independent of ${\overset{ˉ}{h}}_{j} - θ_{j}$ , which is $X_{n}$ -measurable, so [F4] gives $E [1_{A} {\hat{β}}_{j} ({\overset{ˉ}{h}}_{j} - θ_{j})] = E [1_{A} {\hat{β}}_{j}] E ({\overset{ˉ}{h}}_{j} - θ_{j}) = 0$ . By [F1] and [F5], $E [1_{A} {\hat{μ}}_{\hat{β}}] = P (A) E_{P} f$ .

3.1F3step 1.1step 2.1∎

The constant $E_{P} f$ is $P$ -measurable and its integral on every $A \in P$ equals the integral of the estimator on $A$ by step 2.1. The defining characterization [F3] therefore gives $E [{\hat{μ}}_{\hat{β}} ∣ P] = E_{P} f$ almost surely.

Depends on

Used by

A random pilot coefficient remains conditionally unbiased Example

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Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 8, §8.9, p. 32 (standard reference, not scraped)
Paul Glasserman, Monte Carlo Methods in Financial Engineering, Chapter 4, §4.1.3, p. 200 (standard reference, not scraped)