Write

p = p^{*}

μ_{z} = E (Y ∣ Z = z)

, and

τ^{*} = μ_{1} - μ_{0}

. The summand has finite variance, so the central limit theorem gives

n (τ - τ^{*}) d N (0, V_{known}),

where

V_{known} = \frac{E ( Y ^{2} ∣ Z = 1 )}{p} + \frac{E ( Y ^{2} ∣ Z = 0 )}{1 - p} - (μ_{1} - μ_{0})^{2} .

b

Solution

For

ϑ = (q, t)

ψ (Y, Z; ϑ) = (Z Y / q - ( 1 - Z ) Y / ( 1 - q ) - t Z - q) .

Its empirical mean vanishes exactly at

(p, τ)

(p, τ^{*})

, so the Z-estimator is consistent. Linearizing the equation, or simplifying the corresponding sandwich covariance, gives the influence function

ϕ (Y, Z) = \frac{Z}{p} (Y - μ_{1}) - \frac{1 - Z}{1 - p} (Y - μ_{0}) .

Hence

n (τ - τ^{*}) d N (0, V_{estimated}), V_{estimated} = \frac{Var ( Y ∣ Z = 1 )}{p} + \frac{Var ( Y ∣ Z = 0 )}{1 - p} .

Subtracting the two asymptotic variances and expanding the conditional second moments gives

V_{known} - V_{estimated} = \frac{( ( 1 - p ) μ _{1} + p μ _{0} ) ^{2}}{p ( 1 - p )} \geq 0.

Estimating the randomized treatment probability therefore projects out the component of the known-probability influence function proportional to

Z - p

, weakly improving asymptotic efficiency.

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