5J (Statistical Modelling)

f (y; θ) = h (y) exp {θT (y) - A (θ)},

with support independent of the natural parameter

θ

. In the present canonical setting the natural statistic is

T (y) = y

b

Independence gives the joint density

\prod_{i = 1}^{n} h (y_{i}) exp {θ \sum_{i = 1}^{n} y_{i} - n A (θ)} .

By the factorization criterion, its dependence on the sample relevant to

θ

is only through

\sum_{i} Y_{i}

, equivalently through

\overline{Y}

. Thus

\overline{Y}

θ

ℓ^{'} (θ) = \sum_{i} Y_{i} - n A^{'} (θ) .

Differentiating the normalizing identity for the family gives

A^{'} (θ) = E_{θ} Y_{1} = μ (θ), A^{''} (θ) = Var_{θ} (Y_{1}) \geq 0.

Hence an interior maximum satisfies, and under nondegeneracy uniquely satisfies,

μ (θ_{MLE}) = \overline{Y} .

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