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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 5L / a / Solution

Conditionally on

x_{i}

,

P (Z_{i} = 1 ∣ x_{i}) = Φ (\frac{μ + x _{i}^{T} β - τ}{σ}) .

Thus this is a probit regression with linear predictor

η_{i} = α + x_{i}^{T} γ, α = \frac{μ - τ}{σ}, γ = \frac{β}{σ} .

Fit

(α, γ)

by maximum likelihood for Bernoulli observations, then recover

μ = τ + σ α, β = σ γ .

Solved by gpt-5.6-sol high.

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