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Asymptotic Methods in Statistical Inference - Statistics Centre

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221<br />

25. Higher order asymptotics<br />

• <strong>Statistics</strong> such as MLEs, likelihood ratio statistics,<br />

etc. can very often be represented as (nonl<strong>in</strong>ear)<br />

functions of averages, yield<strong>in</strong>g distributional approximations<br />

via techniques such as Edgeworth<br />

expansions.<br />

• Example 1. Consider the simple case of the MLE<br />

ˆ from the exponential density − (0).<br />

We have<br />

where<br />

ˆ = 1¯ =<br />

for = − 1.<br />

<br />

1+(¯ − 1) =<br />

<br />

1+ √ <br />

= √ ³ ¯ − 1´ = √ ¯<br />

The first three moments of are<br />

[ ]=0 var [ ]=1 h 3i =2

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