Past exam of the statistics course of the University of Oxford / 2021 / 1 / a / ii

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- Solution ii

Solution

Using the raw moments from part (i), the variance of one observation is

Var_{θ} (X) = \frac{θ}{θ + 2} - \frac{θ ^{2}}{( θ + 1 ) ^{2}} = \frac{θ}{( θ + 1 ) ^{2} ( θ + 2 )} .

The central limit theorem for the independent statistical sample gives

n (\overset{ˉ}{X} - μ) d N (0, \frac{θ}{( θ + 1 ) ^{2} ( θ + 2 )}) .

For

g (x) = x / (1 - x)

, its derivative at the mean is

g^{'} (μ) = (1 - μ)^{- 2} = (θ + 1)^{2}

. The delta method therefore gives

n (T_{1} - θ) d N (0, \frac{θ ( θ + 1 ) ^{2}}{θ + 2}) .

Equivalently, the large-sample normal approximation is

T_{1} \tilde{˙} N (θ, \frac{θ ( θ + 1 ) ^{2}}{n ( θ + 2 )}), v (T_{1}) = \frac{θ ( θ + 1 ) ^{2}}{n ( θ + 2 )} .

Here

v

denotes the leading asymptotic variance, rather than an assertion about an exact finite-sample variance.

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