Lecture 18: L-estimators and trimmed sample mean

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1 Lecture 18: L-estimators and trimmed sample mean L-functional and L-estimator For a function J(t) on [0,1], define the L-functional as T (G) = xj(g(x))dg(x), G F. If X 1,...,X n are i.i.d. from F and T (F) is the parameter of interest, T (F n ) is called an L-estimator of T (F). T (F n ) is a linear function of order statistics: T (F n ) = xj(f n (x))df n (x) = 1 n since F n (X (i) ) = i/n, i = 1,...,n. Examples n i=1 When J(t) 1, T (F n ) = X, the sample mean. J ( i n) X(i), When J(t) = (1 2α) I (α,1 α) (t), T (F n ) = X α is the α-trimmed sample mean. UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

2 Lecture 18: L-estimators and trimmed sample mean L-functional and L-estimator For a function J(t) on [0,1], define the L-functional as T (G) = xj(g(x))dg(x), G F. If X 1,...,X n are i.i.d. from F and T (F) is the parameter of interest, T (F n ) is called an L-estimator of T (F). T (F n ) is a linear function of order statistics: T (F n ) = xj(f n (x))df n (x) = 1 n since F n (X (i) ) = i/n, i = 1,...,n. Examples n i=1 When J(t) 1, T (F n ) = X, the sample mean. J ( i n) X(i), When J(t) = (1 2α) I (α,1 α) (t), T (F n ) = X α is the α-trimmed sample mean. UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

3 Although the sample median is also a linear function of order statistics, it is not of the form T (F n ) with an L-functional T Asymptotic normality of L-estimators To establish the asymptotic normality for L-estimators T (F n ), we follow the following steps. Step 1. For x R, calculate φ F (x) = lim t 0 T (F + t(δ x F)) T (F) t (if it exists), where δ x is the point mass at x. The function φ F is called the influence function of T at F. The influence function is an important tool in the study of robuestness of estimators Also, verify that E[φ F (X 1 )] = φ F (x)df(x) = 0 UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

4 Although the sample median is also a linear function of order statistics, it is not of the form T (F n ) with an L-functional T Asymptotic normality of L-estimators To establish the asymptotic normality for L-estimators T (F n ), we follow the following steps. Step 1. For x R, calculate φ F (x) = lim t 0 T (F + t(δ x F)) T (F) t (if it exists), where δ x is the point mass at x. The function φ F is called the influence function of T at F. The influence function is an important tool in the study of robuestness of estimators Also, verify that E[φ F (X 1 )] = φ F (x)df(x) = 0 UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

5 Asymptotic normality of L-estimators Step 2. Verify that E[φ F (X 1 )] 2 < and obtain σf 2 = E[φ F (X 1 )] 2 = [φ F (x)] 2 df(x). Step 3. Verify that T (F n ) T (F) = 1 n n i=1 φ F (X i ) + o p ( 1 n ). This holds when T is differentiable in some sense ( 5.2.1). Then n[t (Fn ) T (F)] d N(0,σ 2 F ). Step 3 is the most difficult part. This approach can also be applied to other functionals ( 5.2). We now apply this approach to show the asymptotic normality of the trimmed sample mean. UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

6 Asymptotic normality of L-estimators Step 2. Verify that E[φ F (X 1 )] 2 < and obtain σf 2 = E[φ F (X 1 )] 2 = [φ F (x)] 2 df(x). Step 3. Verify that T (F n ) T (F) = 1 n n i=1 φ F (X i ) + o p ( 1 n ). This holds when T is differentiable in some sense ( 5.2.1). Then n[t (Fn ) T (F)] d N(0,σ 2 F ). Step 3 is the most difficult part. This approach can also be applied to other functionals ( 5.2). We now apply this approach to show the asymptotic normality of the trimmed sample mean. UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

7 Asymptotic normality of L-estimators Step 2. Verify that E[φ F (X 1 )] 2 < and obtain σf 2 = E[φ F (X 1 )] 2 = [φ F (x)] 2 df(x). Step 3. Verify that T (F n ) T (F) = 1 n n i=1 φ F (X i ) + o p ( 1 n ). This holds when T is differentiable in some sense ( 5.2.1). Then n[t (Fn ) T (F)] d N(0,σ 2 F ). Step 3 is the most difficult part. This approach can also be applied to other functionals ( 5.2). We now apply this approach to show the asymptotic normality of the trimmed sample mean. UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

8 Step 1: Derivation of the influence function φ F T (G) = xj(g(x))dg(x), G F For F and G in F, T (G) T (F) = xj(g(x))dg(x) xj(f(x))df (x) = = = = 1 [G (t) F (t)]j(t)dt 0 1 G (t) 0 F (t) F (x) G(x) dxj(t)dt J(t)dtdx [F(x) G(x)]J(F(x))dx U G (x)[g(x) F(x)]J(F(x))dx, UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

9 Step 1: Derivation of the influence function φ F where U G (x) = G(x) F(x) J(t)dt [G(x) F (x)]j(f (x)) 1 G(x) F(x),J(F(x)) 0 0 otherwise and the fourth equality follows from Fubini s theorem and the fact that the region in R 2 between curves F(x) and G(x) is the same as the region in R 2 between curves G (t) and F (t). Let G = F + t(δ x F), where δ x is the degenerated distribution at x. Since lim t 0 U F +t(δx F )(y) = 0, by the dominated convergence theorem, lim U F +t(δx F )(y)[δ x (y) F(y)]J(F(y))dy = 0. t 0 Hence lim t 0 T (F + t(δ x F)) T (F) t which is φ F (x), the influence function of T. = [δ x (y) F(y)]J(F(y))dy, UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

10 Step 1: Derivation of the influence function φ F By Fubini s theorem and the fact that δ x (y)df (x) = F(y), [ φ F (x)df (x) = (δ x F)(y)dF(x)] J(F(y))dy = 0, Consider now J(t) = () I (α,β) (t), φ F (x) = 1 F (β) [δ x (y) F(y)]dy. Assume that F is continuous at and F (β). F() = α and F(F (β)) = β. When x <, φ F (x) = 1 = F (β) y[1 F (y)] [1 F (y)]dy F (β) = (1 α) F (β)(1 β) 1 F (β) ydf(y) T (F) UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

11 Step 1: Derivation of the influence function φ F Similarly, when x > F (β), φ F (x) = 1 F (β) F(y)]dy = F (β)β α Finally, when x F (β), φ F (x) = 1 = yf(y) + x x y[1 F(y)] F(y)dy 1 1 F (β) x = x α F (β)(1 β) x T (F). F (β) x ydf(y) [1 F(y)]dy 1 F (β) ydf(y) x T (F). UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

12 Step 1: Derivation of the influence function φ F Hence, φ F (x) = (1 α) F (β)(1 β) β α T (F) x < x α F (β)(1 β) β α T (F ) x F (β) F (β)β α β α T (F) x > F (β). If F is symmetric about θ, J is symmetric about 1 2 (J(t) = J(1 t)), and 1 0 J(t)dt = 1, then F(x) = F 0(x θ), where F 0 is a c.d.f. that is symmetric about 0, i.e., F 0 (x) = 1 F 0 ( x), and xj(f 0 (x))df 0 (x) = xj(1 F 0 ( x))df 0 (x) = xj(f 0 ( x))df 0 (x) = yj(f 0 (y))df 0 (y), i.e., xj(f 0 (x))df 0 (x) = 0. UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

13 Step 1: Derivation of the influence function φ F Hence, T (F) = xj(f(x))df(x) = θ J(F(x))dF (x) + (x θ)j(f 0 (x θ))df 0 (x θ) 1 = θ J(t)dt + 0 = θ. yj(f 0 (y))df 0 (y) Assume that F is continuous at and F (1 α). When β = 1 α, J is symmetric about 1 2 and F 0 (α) 1 2α x < x θ φ F (x) = 1 2α x F (1 α) x > F (1 α), F 0 (1 α) 1 2α where + F (1 α) = 2θ, F 0 (α) = θ and F 0 (1 α) = F (1 α) θ. UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

14 Step 2: Calculation of σ 2 F = E[φ F (X 1 )] 2 Because F 0 [φ F (x)] 2 df(x) = (α) = F (1 α), we obtain that = 0 [F0 (α)]2 [F0 (1 α)]2 α + (1 2α) 2 (1 2α) 2 α + = σ 2 α. F (1 α) 2α[F0 (1 α)]2 (1 2α) 2 + (x θ) 2 df (x) (1 2α) 2 F 0 (1 α) x 2 F 0 (α) (1 2α) 2 df 0(x) Step 3: Asymptotic normality of the trimmed sample mean It can be shown that the L-functional T (G) is differentiable in some sense (see the textbook). Hence, for the α-trimmed sample mean X α, n( Xα θ) d N(0,σα). 2 UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10

15 Step 2: Calculation of σ 2 F = E[φ F (X 1 )] 2 Because F 0 [φ F (x)] 2 df(x) = (α) = F (1 α), we obtain that = 0 [F0 (α)]2 [F0 (1 α)]2 α + (1 2α) 2 (1 2α) 2 α + = σ 2 α. F (1 α) 2α[F0 (1 α)]2 (1 2α) 2 + (x θ) 2 df (x) (1 2α) 2 F 0 (1 α) x 2 F 0 (α) (1 2α) 2 df 0(x) Step 3: Asymptotic normality of the trimmed sample mean It can be shown that the L-functional T (G) is differentiable in some sense (see the textbook). Hence, for the α-trimmed sample mean X α, n( Xα θ) d N(0,σα). 2 UW-Madison (Statistics) Stat 710, Lecture 18 Jan / 10