Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 2 / d

Table of contents
- Solution d

Solution

The marginal predictor density is

x_{i} \sim N (μ, A), A = τ^{2} + σ_{x}^{2} .

Gaussian conditional expectation gives

ξ_{i} ∣ x_{i} \sim N (μ + κ (x_{i} - m u), v), κ = \frac{τ ^{2}}{A}, v = \frac{τ ^{2} σ _{x}^{2}}{A} .

Integrating this conditional distribution through the response model yields

y_{i} ∣ x_{i} \sim N (α + β [μ + κ (x_{i} - m u)], s^{2} + β^{2} v) .

When

τ ≫ σ_{x}

,

κ \to 1

,

v \to σ_{x}^{2}

, and

A \sim τ^{2}

. Hence

L ≃ L_{1} (α, β, σ^{2}) L_{2} (μ, τ^{2}),

where

L_{1} = \prod_{i} φ (y_{i}; α + β x_{i}, σ^{2} + σ_{y}^{2} + β^{2} σ_{x}^{2}), L_{2} = \prod_{i} φ (x_{i}; μ, τ^{2}) .

The approximate maximum-likelihood estimators are

μ = \overset{x}{ˉ}, τ^{2} = \frac{1}{N} \sum_{i} (x_{i} - \overset{x}{ˉ})^{2} .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version