 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2010/o/o4/a/solution

θ \in arg max_{θ} L_{n} (θ), L_{n} (θ) = \prod_{i = 1}^{n} f (X_{i}; θ) .

ℓ_{n} = lo g L_{n}

and the score

S_{n} = \nabla_{θ} ℓ_{n}

I_{n} (θ) = E_{θ} [S_{n} S_{n}^{T}] = - E_{θ} [\nabla_{θ}^{2} ℓ_{n}],

where the equality uses the usual regularity conditions. For independent identically distributed observations,

I_{n} = n I_{1}

Under the standard identifiability, interior-parameter, differentiability and nonsingular-information assumptions, the large-sample distribution is

n (θ - θ) d N (0, I_{1} (θ)^{- 1}),

or approximately

θ \sim N (θ, I_{n} (θ)^{- 1})

. Boundary parameters need not obey this regular approximation.

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