TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Unbiasedness and optimal coefficients for fixed controls

Statement

Let $n \geq 1$ , assume $P f^{2} < \infty$ and $P h_{j}^{2} < \infty$ for each component, and set $μ = P f$ , $θ = P h$ . Define the vector covariance matrix and cross-covariance vector by $Σ_{j k} = E_{P} [(h_{j} - θ_{j}) (h_{k} - θ_{k})]$ and $c_{j} = E_{P} [(h_{j} - θ_{j}) (f - μ)]$ . A matrix $Σ$ is positive definite when $v^{T} Σ v > 0$ for every nonzero $v$ . For an IID sample $X_{1}, \dots, X_{n} \sim P$ and a fixed $β \in R^{q}$ , the estimator in Control-variate estimators is unbiased and

$Var ({\hat{μ}}_{β}) = \frac{1}{n} {Var}_{P} (f - β^{T} h) .$

If $Σ$ is positive definite, the unique variance-minimizing vector coefficient and the minimum variance are

$β^{*} = Σ^{- 1} c, Var ({\hat{μ}}_{β^{*}}) = \frac{1}{n} ({Var}_{P} (f) - c^{T} Σ^{- 1} c) .$

For a scalar control with ${Var}_{P} (h) > 0$ , this reads $β^{*} = {Cov}_{P} (f, h) / {Var}_{P} (h)$ . If controls are redundant, delete redundant directions or use the theorem only on a positive-definite covariance subspace; no inverse is asserted for a singular $Σ$ .

Facts & Assumptions

Given: The measurable functions, probability law, known control mean, IID sample, fixed coefficient, and second-moment hypotheses in the Statement.

[F1]

Control-variate estimators: The estimator averages $f (X_{i}) - β^{T} (h (X_{i}) - θ)$ , with $θ = P h$ known.

[F2]

Identical distribution and IID families: IID means identical law and mutual independence; no moments are implicit.

[F3]

Linearity, monotonicity, and the modulus bound for expectation: Expectation of an integrable finite linear combination is the same linear combination of expectations.

[F4]

Measurable coordinatewise functions preserve independence: Applying the same measurable function coordinatewise to an independent family preserves independence.

[F5]

Expectations factor over finite products of independent random variables: For independent integrable real variables, expectation of their product is the product of expectations.

[F6]

Change of variables for expectation: If $X$ has law $P$ and $g (X)$ is integrable, then $E [g (X)] = \int g d P$ .

[F7]

Moments, variance, and covariance on a probability space: Variance is $E [(Y - E Y)^{2}]$ and covariance is $E [(Y - E Y) (Z - E Z)]$ for square-integrable real variables.

[F8]

Covariance is symmetric and bilinear in finite linear combinations: Covariance is symmetric and bilinear on finite linear combinations of square-integrable variables.

Proof

Given: The objects and hypotheses in the Statement.

1.1F1F3F6

For $Z_{i} = f (X_{i}) - β^{T} (h (X_{i}) - θ)$ , the expectation convention [F6] and linearity [F3] give $E Z_{i} = P f - β^{T} (P h - θ) = μ$ . Thus $E {\hat{μ}}_{β} = μ$ .

1.2F2F4

The $Z_{i}$ are the same measurable function of the IID variables $X_{i}$ , so [F2] and [F4] make them IID.

1.3F7F8

By [F7]–[F8], the variance as a function of $β$ is ${Var}_{P} (f) - 2 β^{T} c + β^{T} Σ β$ .

2.1F5F7step 1.2algebra

Expanding the centered square of $n^{- 1} \sum_{i} Z_{i}$ , the diagonal terms contribute $n^{- 2} n Var (Z_{1})$ . For $i \neq j$ , $Z_{i} - E Z_{i}$ and $Z_{j} - E Z_{j}$ are independent integrable variables, so [F5] factors their product expectation into zero. Hence $Var ({\hat{μ}}_{β}) = n^{- 1} Var (Z_{1}) = n^{- 1} {Var}_{P} (f - β^{T} h)$ .

3.1step 1.3algebra∎

If $Σ$ is positive definite, substituting $c = Σ β^{*}$ and completing the square gives ${Var}_{P} (f) - c^{T} Σ^{- 1} c + (β - β^{*})^{T} Σ (β - β^{*})$ . The last term is nonnegative and vanishes only at $β = β^{*}$ ; for $q = 1$ the displayed scalar formula follows.

Depends on

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Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 8, §8.9, pp. 28–32 (standard reference, not scraped)
Paul Glasserman, Monte Carlo Methods in Financial Engineering, Chapter 4, §4.1, pp. 185–200 (standard reference, not scraped)
Jonathan Goodman, Lecture Notes on Monte Carlo Methods, Chapter 3, §2, pp. 1–3 (standard reference, not scraped)