Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 224 / 1 / b

Table of contents
- Solution b

Solution

The Neyman-Pearson lemma gives an optimal acceptance region for

P

B_{n} (τ) = {x_{1}^{n} : \frac{P ^{\otimes n} ( x _{1}^{n} )}{Q ^{\otimes n} ( x _{1}^{n} )} \geq τ},

with possible boundary randomization. If

P_{n}

is the empirical mass function, then

\frac{1}{n} lo g \frac{P ^{\otimes n} ( x _{1}^{n} )}{Q ^{\otimes n} ( x _{1}^{n} )} = D (P_{n} ∥ Q) - D (P_{n} ∥ P),

so equivalently

B_{n} (τ) = {D (P_{n} ∥ Q) - D (P_{n} ∥ P) \geq n^{- 1} lo g τ} .

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