Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 28K / b

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- Solution b

Solution

The maximum-likelihood estimator is

θ_{MLE} = \overline{X} .

Under the score-outer-product convention stated in the question,

i_{n} = \frac{1}{n} \sum_{j = 1}^{n} (X_{j} - \overline{X}) (X_{j} - \overline{X})^{T} .

This is the sample covariance matrix with divisor

n

. The strong law of large numbers, applied entrywise, gives

i_{n} ⟶ I_{p} almost surely .

Thus the empirical score outer-product information consistently estimates the per-observation Fisher information.

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