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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Special Case: N(µ,σ 2 )Then µ 3 = 0 and µ 4 = 3σ 4 .Our <strong>ca</strong>lculation hasn 1/2 (s 2 −σ 2 ) ⇒ N(0,2σ 4 )You <strong>ca</strong>n divide through by σ 2 and getn 1/2 (s 2 /σ 2 −1) ⇒ N(0,2)In fact ns 2 /σ 2 has χ 2 n−1 distribution so usual CLT shows(n−1) −1/2 [ns 2 /σ 2 −(n−1)] ⇒ N(0,2)(us<strong>in</strong>g mean of χ 2 1 is 1 and variance is 2).Factor out n to get√ nn−1 n1/2 (s 2 /σ 2 −1)+(n−1) −1/2 ⇒ N(0,2)which is δ method <strong>ca</strong>lculation except for some constants.Difference is unimportant: Slutsky’s theorem.Richard 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 28 / 31
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