STAT 830 Convergence in Distribution - People.stat.sfu.ca - Simon ...
STAT 830 Convergence in Distribution - People.stat.sfu.ca - Simon ...
STAT 830 Convergence in Distribution - People.stat.sfu.ca - Simon ...
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Delta Method Cont<strong>in</strong>uesUs<strong>in</strong>g b<strong>in</strong>omial expansion:E(X 4 ) = E{(X −µ+µ) 4 }= µ 4 +4µµ 3 +6µ 2 σ 2 +4µ 3 E(X −µ)+µ 4 .So Σ 11 = µ 4 −σ 4 +4µµ 3 +4µ 2 σ 2 .Top right entry is expectation ofwhich isSimilar to 4th moment get(X 2 −µ 2 −σ 2 )(X −µ)E(X 3 )−µE(X 2 )µ 3 +2µσ 2Lower right entry is σ 2 .So [µ4 −σΣ =4 +4µµ 3 +4µ 2 σ 2 µ 3 +2µσ 2 ]µ 3 +2µσ 2 σ 2Richard Lockhart (<strong>Simon</strong> Fraser University) <strong>STAT</strong> <strong>830</strong> <strong>Convergence</strong> <strong>in</strong> <strong>Distribution</strong> <strong>STAT</strong> <strong>830</strong> — Fall 2011 25 / 31