slides_10_asymptotic

# N as avar n n dg d n 2 c n and then we define an

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̂ n as Avar n ̂ n  dg d ̂ n 2 c ̂ n , and then we define an estimator of the asymptotic variance of ̂ n as Avar ̂ n dg d ̂ n 2 c ̂ n / n 50

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EXAMPLE : Returning to the Poisson example with exp we take ̂ n X ̄ n and ̂ n exp X ̄ n . Now g exp and g 1 exp . It follows that Avar n ̂ n  exp  2 Avar n X ̄ n  exp 2 and so the asymptotic standard deviation of ̂ n is exp / n . It is not true that Var n ̂ n  exp 2 . We do not know Var n ̂ n  . 51
The delta method can be applied to vectors of parameters. Let g : Θ m be continuously differentiable. We can write g g 1 g 2 g m where g j g j 1 , 2 ,..., p . 52

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Let g denote its m p Jacobian, which we can write g g 1 g 2 g m where g j is the 1 p gradient of g j . 53