Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 37 / 3 / b

Table of contents
- Solution b

Solution

Independence gives the log-likelihood

ℓ (β) = \sum_{i} [y_{i} β^{T} x_{i} - lo g (1 + e^{β^{T} x_{i}})] .

Differentiating yields the score equations

i \sum x_{i} [y_{i} - p_{i} (\hat{β})] = 0.

When a finite maximum-likelihood estimator exists, multiplying these equations by

\hat{β}^{T}

gives

\sum_{i} y_{i} logit p_{i} (\hat{β}) = \sum_{i} p_{i} (\hat{β}) logit p_{i} (\hat{β}) .

The qualification about a finite estimator matters under complete separation in logistic regression, where coefficient estimates can escape to infinity.

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