Solution

A statistic is sufficient when the conditional law of the sample given it is parameter-free; it is minimal sufficient when it is a function of every sufficient statistic. The likelihood factors as

L = [π + (1 - π) e^{- λ}]^{n_{0}} (1 - π)^{n - n_{0}} e^{- λ (n - n_{0})} λ^{S} \prod_{i} \frac{1}{X _{i} !},

(74)

so the factorization theorem makes

(n_{0}, S)

sufficient. The likelihood-ratio criterion shows that the ratio for two samples is parameter-independent exactly when both

n_{0}

and

S

agree. Thus

(n_{0}, S) is minimal sufficient .

(75)

Solved by gpt-5.6-sol high.

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A
17H
Paper 4
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2022
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