Chapter 7 Statistical Functionals and the Delta Method 1 Estimates as Functionals of Fn or Pn Often the quantity we want to estimate can be viewed as a functional T(F)or T(P)of the underlying distribution function F or P generating the data.Then a simple nonparametric estimator is simply T(Fn)or T(Pn)where Fn and Pn denote the empirical distribution function and empirical measure of the data. Notation.Suppose that X1,...,Xn are i.i.d.P on (,A).We let n ∑dx,三the empirical measure of the sample, Pn三 i=1 where the measure with mass one at x (so 6(A)=1A(x)for AA.When =R,especially when k =1,we will write En(x)= a=P(-小F=P-,斗 Here is a list of examples. Example 1.1 The mean T(F)=fxdF(x).T(Fn)=fxdFn(x) Example 1.2 The r-th moment:for r an integer,T(F)=fx"dF(x),and T(Fn)=fr'dFn(). Example 1.3 The variance: TD=VarW=/e-∫FGYF(=∫fe-F(z)F(. T)=vam.x=/e-R.(aP证)=∫∫e-P亚.aE Example 1.4 The median:T(F)=F-1(1/2).T(Fn)=F1(1/2). Example 1.5 The a-trimmed mean:T(F)=(1-2a)-1fF-1(u)du for 0<a <1/2. T(Fn)=(1-2a)-1faFn(u)du. 3Chapter 7 Statistical Functionals and the Delta Method 1 Estimates as Functionals of Fn or Pn Often the quantity we want to estimate can be viewed as a functional T(F) or T(P) of the underlying distribution function F or P generating the data. Then a simple nonparametric estimator is simply T(Fn) or T(Pn) where Fn and Pn denote the empirical distribution function and empirical measure of the data. Notation. Suppose that X1, . . . , Xn are i.i.d. P on (X , A). We let Pn ≡ 1 n !n i=1 δXi ≡ the empirical measure of the sample, where δx ≡ the measure with mass one at x (so δx(A)=1A(x) for A ∈ A. When X = Rk, especially when k = 1, we will write Fn(x) = 1 n !n i=1 1(−∞,x](Xi) = Pn(−∞, x], F(x) = P(−∞, x]. Here is a list of examples. Example 1.1 The mean T(F) = " xdF(x). T(Fn) = " xdFn(x). Example 1.2 The r-th moment: for r an integer, T(F) = " xrdF(x), and T(Fn) = " xrdFn(x). Example 1.3 The variance: T(F) = V arF (X) = # (x − # xdF(x))2dF(x) = 1 2 # # (x − y) 2dF(x)dF(y), T(Fn) = V arFn(X) = # (x − # xdFn(x))2dFn(x) = 1 2 # # (x − y) 2dFn(x)dFn(y). Example 1.4 The median: T(F) = F −1(1/2). T(Fn) = F−1 n (1/2). Example 1.5 The α−trimmed mean: T(F) = (1 − 2α)−1 " 1−α α F −1(u)du for 0 < α < 1/2. T(Fn) = (1 − 2α)−1 " 1−α α F−1 n (u)du. 3