N example random sampling from the normal

Info icon This preview shows pages 26–32. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 26

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 27
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
Image of page 28

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 29
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.)
Image of page 30

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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  = , ... ... ... ...
Image of page 31
Image of page 32
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern