N example random sampling from the normal

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n Example: Random sampling from the normal distribution, n Sample mean is asymptotically normal[μ,σ2/n] n Median is asymptotically normal [μ,(π/2)σ2/n] n Mean is asymptotically more efficient ™  25/42
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Part 11: Asymptotic Distribution  Theory The Delta Method The delta method (combines most of these concepts) Nonlinear transformation of a random variable: f(xn) such that plim xn =  but n (xn - ) is asymptotically normally distributed. What is the asymptotic behavior of f(xn)? Taylor series approximation: f(xn)  f() + f() (xn - ) By Slutsky theorem, plim f(xn) = f() n[f(xn) - f()]  f() [n (xn - )] Large sample behaviors of the LHS and RHS sides are the same (generally - requires f(.) to be nicely behaved. RHS is a constant times something familiar. Large sample variance is [f()]2 times large sample Var[n (xn - )] Return to asymptotic variance of xn Gives us the VIR for the asymptotic distribution of a function. ™  26/42
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Part 11: Asymptotic Distribution  Theory Delta Method ™  27/42 a 2 n n a 2 2 n If x   N[ , / n] and  f(x ) is a continuous and continuously differentiable function that does not involve n, then f(x )  N{ f( ),[f '( )] / n}   → μ σ → μ μ σ
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Part 11: Asymptotic Distribution  Theory Delta Method - Applications ™  28/42 a 2 n n n n n n x N[ , / n] What is the asymptotic distribution of  f(x )= exp(x )  or  f(x )= 1/x (1) Normal since x  is asymptotically normally distributed (2) Asymptotic mean is f( )= exp( )  or  1/ . (3) For the var → μ σ μ μ μ 2 2 2 4 2 n iance, we need f'( ) = exp( )  or  -1/        Asy.Var[f(x )]=  [exp( )] / n  or  [1/ ] / n μ μ μ μ σ μ σ
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Part 11: Asymptotic Distribution  Theory Krinsky and Robb vs. the Delta Method ™  29/42 n Krinsky and Robb, ReStat, 1986.  The delta method doesn't work very well when f(.) is highly nonlinear.  Alternative approach based on the law of large numbers: (1)  Compute x  =  the estimator of  , and  θ n n n n estimate the asymptotic      variance (standard deviation), v  (s ). (2)  Compute f(x ) to estimate f( ). (3)  Estimate the asymptotic variance of f(x ) by using a random       number generator to draw a  θ 2 n n 1 R n,1 n,R normal random sample from the       asymptotic population N[x ,s ]. Compute a sample of function       values  f  ... f =  f(x ),...,f(x ).  Use the sample variance       of these draws to estimate t n he asymptotic variance of f(x ). (Krinsky and Robb, ReStat, 1991. Programming error. Retraction.)
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Part 11: Asymptotic Distribution  Theory Delta Method – More than One Parameter ™  30/42 1 1 K 11 12 1K 21 22 2K 1 2 K K1 K2 KK 1 1 K ˆ ˆ ˆ If   , ,...,  are K consistent estimators of K parameters  v v ... v v v ... v , ,...,  with asymptotic covariance matrix  = , ... ... ... ...
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