Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 2 / b

Table of contents
- Solution b

Solution

For

ϑ = (q, t)

use the estimating equation

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

Its empirical mean vanishes exactly at

(p, τ)

. The population equation has the unique root

(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} .

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