Sufficiency

Objective. Use factorisation to identify sufficient statistics.

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Learn: the key idea

Sufficiency identifies a statistic T(X) that captures all the information in the sample about a parameter θ, so that the conditional distribution of the data given T(X) does not depend on θ. The Fisher–Neyman factorisation theorem states T is sufficient if and only if the likelihood factorises as L(θ) = g(T(x),θ)h(x), where h does not involve θ. For n Bernoulli(p) trials, the likelihood p^k(1−p)^(n−k) depends on the data only through the total count k, so T = Σxᵢ is sufficient for p.

Worked example

Using the factorisation theorem, identify a sufficient statistic for λ based on n i.i.d. Exponential(λ) observations x1,...,xn.

1 $Write the likelihood L(λ) = λⁿ e^(-λΣxᵢ)$
2 $Identify this as a function of λ and the sum Σxᵢ only$
3 $No remaining factor depends on the individual xᵢ beyond their sum$
4 $State the sufficient statistic by the factorisation theorem$

Answer: T = Σxᵢ

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Common mistake. Students sometimes think any statistic appearing in the likelihood is sufficient, without checking that the remaining factor h(x) is genuinely free of θ. Always verify the full factorisation before declaring sufficiency.

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