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A random pilot coefficient remains conditionally unbiased

Example

Let the production draws be IID uniform on $[0, 1]$ , take $f (x) = x^{2}$ , $h (x) = x$ , and $θ = 1 / 2$ . From an independent pilot draw $V \sim Unif [0, 1]$ , set the random coefficient $\hat{β} = 2 V$ . The control estimator remains conditionally unbiased given the pilot.

Facts & Assumptions

Given: The pilot variable $V$ is independent of all production draws and $\hat{β} = 2 V$ has finite first moment.

[F1]

An independent pilot coefficient preserves conditional unbiasedness: An integrable pilot-measurable coefficient independent of the production sample gives conditional expectation $E_{P} f$ .

[F2]

Control-variate estimators: For a fixed scalar coefficient $b$ , the estimator is $\overset{ˉ}{f} - b (\overset{ˉ}{h} - θ)$ .

Verification

Given: The stated production and pilot laws.

1.1givenalgebra

Production integration gives $E_{P} f = \int_{0}^{1} x^{2} d x = 1 / 3$ and $E_{P} h = \int_{0}^{1} x d x = 1 / 2 = θ$ ; also $E ∣ \hat{β} ∣ = E [2 V] = 1 < \infty$ .

2.1F1F2givenstep 1.1algebra∎

Set $P = σ (V)$ . The independent pilot makes $P$ independent of the sigma-algebra of the entire production sample, and $\hat{β} = 2 V$ is $P$ -measurable and integrable by step 1.1. The functions $f, h$ are bounded and hence integrable. The random-coefficient estimator defined in [F1] is $\overset{ˉ}{f} - 2 V (\overset{ˉ}{h} - 1 / 2)$ , extending the fixed-coefficient expression in [F2]. Therefore [F1] gives $E [{\hat{μ}}_{\hat{β}} ∣ P] = 1 / 3$ almost surely.

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