Asymptotic statistics using the Functional Delta Method

Transcription

1 Quantiles, Order Statistics and L-Statsitics TU Kaiserslautern 15. Februar 2015

2 Motivation Functional The delta method introduced in chapter 3 is an useful technique to turn the weak convergence of random vectors r n(t n θ) into weak convergence of transformations r n(φ(t n) φ(θ))

3 Motivation Functional The delta method introduced in chapter 3 is an useful technique to turn the weak convergence of random vectors r n(t n θ) into weak convergence of transformations r n(φ(t n) φ(θ)) Goal: Expand the technique to arbitrary normed spaces

4 Motivation Functional The delta method introduced in chapter 3 is an useful technique to turn the weak convergence of random vectors r n(t n θ) into weak convergence of transformations r n(φ(t n) φ(θ)) Goal: Expand the technique to arbitrary normed spaces Example: Apply the method to the empirical process n(fn F ), which converges in the space D[, ] to an F Brownian Bridge. Hence, we hope to derive the asymptotic distribution of transformations n(φ(fn) φ(f )).

5 Table of Contents Functional 1 Functional Heuristic Ansatz Types of differentiability 2 Asymptotic Normality Order Statistics Extreme Values 3 Definition and examples

6 Table of Contents Functional Heuristic Ansatz Types of differentiability 1 Functional Heuristic Ansatz Types of differentiability 2 Asymptotic Normality Order Statistics Extreme Values 3 Definition and examples

7 Heuristic Ansatz Types of differentiability For a statistic of the form φ(f n) we want to find out the asymptotic distribution with something like a Taylor expansion: φ(f n) φ(f ) = φ F (F n F ) + R n, where the remainder is hopefully small.

8 Heuristic Ansatz Types of differentiability For a statistic of the form φ(f n) we want to find out the asymptotic distribution with something like a Taylor expansion: φ(f n) φ(f ) = φ F (F n F ) + R n, where the remainder is hopefully small. If φ takes values in R, define φ F (G) as the ordinary derivative of the function t φ(f + tg) for a fixed pertubation G. Then, by Taylor s theorem φ(f + tg) φ(f ) = tφ F (G) + o(t 2 ).

9 Heuristic Ansatz Types of differentiability For a statistic of the form φ(f n) we want to find out the asymptotic distribution with something like a Taylor expansion: φ(f n) φ(f ) = φ F (F n F ) + R n, where the remainder is hopefully small. If φ takes values in R, define φ F (G) as the ordinary derivative of the function t φ(f + tg) for a fixed pertubation G. Then, by Taylor s theorem φ(f + tg) φ(f ) = tφ F (G) + o(t 2 ). Subtituting t = 1 n, G = G n, for the empirical process G n = n(f n F ) yields the von Mises expansion φ(f n) φ(f ) = 1 φ F (G n) n m! 1 φ(m) nm/2 F (Gn) +.

10 Heuristic Ansatz Types of differentiability For a statistic of the form φ(f n) we want to find out the asymptotic distribution with something like a Taylor expansion: φ(f n) φ(f ) = φ F (F n F ) + R n, where the remainder is hopefully small. If φ takes values in R, define φ F (G) as the ordinary derivative of the function t φ(f + tg) for a fixed pertubation G. Then, by Taylor s theorem φ(f + tg) φ(f ) = tφ F (G) + o(t 2 ). Subtituting t = 1 n, G = G n, for the empirical process G n = n(f n F ) yields the von Mises expansion φ(f n) φ(f ) = 1 φ F (G n) n m! 1 φ(m) nm/2 F (Gn) +. But: Because F n depends on n, this is not a legal choice for G and there is no guarentee that the remainder is small!

11 von Mises expansion for m=1 Heuristic Ansatz Types of differentiability However, it is reasonable to make the Ansatz φ(f n) φ(f ) 1 n φ F (G N ) = 1 n n φ F (δ Xi F ). If the random variables φ F (δ Xi F ) have zero mean and finite variance we may expect that n(φ(f n) φ(f )) is asymptotically normal. i=1

12 von Mises expansion for m=1 Heuristic Ansatz Types of differentiability However, it is reasonable to make the Ansatz φ(f n) φ(f ) 1 n φ F (G N ) = 1 n n φ F (δ Xi F ). If the random variables φ F (δ Xi F ) have zero mean and finite variance we may expect that n(φ(f n) φ(f )) is asymptotically normal. The function x φ F (δ X F ) is the so called influence function of the function φ: φ F (δ x F ) = d dt φ ((1 t)f + tδ x). t=0 In robust statistics we are looking for estimators with unbounded influence function. i=1

13 Example (Mean) Functional Heuristic Ansatz Types of differentiability Consider the sample mean X N and the mean function φ(f ) = x df. Then, φ(f n) = x df n = X n. Hence, φ F (δ x F ) = d dt t=0 x d[(1 t)f + tδ x](s) = x s df (s). Note, that the influence function is unbounded and the sample mean is not robust.

14 Example (Quantiles) Functional Heuristic Ansatz Types of differentiability For fixed p (0, 1) we want to estimate the p quantile of F with its empirical counterpart: ˆp = F 1 n (p) = φ(f n) with φ(f ) = F 1 (p).

15 Example (Quantiles) Functional Heuristic Ansatz Types of differentiability For fixed p (0, 1) we want to estimate the p quantile of F with its empirical counterpart: ˆp = F 1 n (p) = φ(f n) with φ(f ) = F 1 (p). To derive the influence function look at the identity p = F tf 1 t where F t = (1 t)f + tδ x. (p) = (1 t)f (F 1 (p)) + tδ x(f 1 (p)), Differentiation of both sides wrt. t yields 0 = f (F 1 (p)) d dt F t 1 (p) F (F 1 (p)) + δ x(f 1 (p)). t=0 t t

16 Heuristic Ansatz Types of differentiability Because d F 1 dt t (p) t=0 = φ 1 F (δx F ) we can solve this equation and derive the influence function φ 1 F (δ x F ) = p 1 [x, )(F 1 (p)) f (F 1 (p)) With the von Mises expansion we get the approximation n(f 1 n (p) F 1 (p)) 1 n what is asymptotically normal by the CLT. n i=1 p 1 [x, ) (F 1 (p)), f (F 1 (p))

17 Heuristic Ansatz Types of differentiability Definition (Gateaux differentiability) A map φ : D E is called Gateaux differentiable at θ D if there exists a continous, linear map φ θ : D E such that for every fixed h D φ(θ + th) φ(θ) φ θ(h) t 0, E as t 0.

18 Heuristic Ansatz Types of differentiability Definition (Gateaux differentiability) A map φ : D E is called Gateaux differentiable at θ D if there exists a continous, linear map φ θ : D E such that for every fixed h D φ(θ + th) φ(θ) φ θ(h) t 0, E as t 0. Definition (Hadamard differentiability) A map φ : D 0 E is called Hadamard differentiable at θ if there exists a continous, linear map φ 0 : D E such that φ(θ + tht) φ(θ) φ θ(h) t 0, E as t 0, for every h t h. If the map does not exist on the whole space D but on a subset D 0, then φ is called Hadamard differentiable tangentially to D 0.

19 Heuristic Ansatz Types of differentiability Definition (Frechet differentiability) A map φ : D E is called Frechet differentiable at θ D if there exists a continous, linear map φ θ : D E such that for every h D φ(θ + h) φ(θ) φ θ(h) = o( h ), E as h 0.

20 Heuristic Ansatz Types of differentiability Definition (Frechet differentiability) A map φ : D E is called Frechet differentiable at θ D if there exists a continous, linear map φ θ : D E such that for every h D φ(θ + h) φ(θ) φ θ(h) = o( h ), E as h 0. In statistical applications, Frechet differentiability may not hold, whereas Hadamard differentiability does. Hadamard and Frechet differentiability are equivalent when D = R k.

21 Heuristic Ansatz Types of differentiability Theorem () Let D and E be normed linear spaces. Let φ : D φ D E be Hadamard differentiable at θ tangentially to D 0. Let T n : Ω D φ be maps such that r n(t n θ) T for some sequence of numbers r n and a random variable T with values in D 0. Then r n(φ(t n) φ(θ)) φ θ(t ). If φ θ can be extend to a continous map on the whole space D, then we also have r n(φ(t n) φ(θ)) = φ θ(r n(t n θ)) + o P (1).

22 Heuristic Ansatz Types of differentiability Theorem (Chain rule) Let φ : D φ E ψ and ψ : E ψ F be maps defined on subsets D φ and E ψ of normed spaces D and E. Let φ be Hadamard-differentiable at θ tangentially to D 0 and let ψ be Hadamard-differentiable at φ(θ) tangentially to φ 0(D 0). Then ψ φ : D φ F is Hadamard-differentiable at θ tangentially to D 0 with derivative ψ φ(θ) φ θ.

23 Nelson-Aalen estimator Functional Heuristic Ansatz Types of differentiability (1) Goal: Estimate the distribution function F of a random sample of failure times T 1,..., T n. (2) Problem: The obeserved data (X i, i ) are right-censored: X i = T i C i with the censoring time C i and i = 1 {Ti C i } records whether T i is censored or not. (3) The cumulative hazard function can be written as 1 1 Λ(t) = df = dh 1, 1 F 1 H [0,t] with H the distribution function of X i and H 1(x) = P(X i x, i = 1) (4) Estimating H, H 1 by their sample counterparts H n(x) = 1 n n i=1 1 {X i x}, H 1n = 1 n n i=1 1 {X i x, i =1} yields the Nelson-Aalen estimator ˆΛ n(t) = [0,t] [0,t] 1 1 H n dh 1n

24 Table of Contents Functional Asymptotic Normality Order Statistics Extreme Values 1 Functional Heuristic Ansatz Types of differentiability 2 Asymptotic Normality Order Statistics Extreme Values 3 Definition and examples

25 Asymptotic Normality Order Statistics Extreme Values The quantile function of a cumulative distribution function F is defined by F 1 : (0, 1) R, F 1 (p) = inf{x : F (x) p}.

26 Asymptotic Normality Order Statistics Extreme Values The quantile function of a cumulative distribution function F is defined by F 1 : (0, 1) R, F 1 (p) = inf{x : F (x) p}. The empirical quantile function is related to the order statistics through F 1 n (p) = X ( np )

27 Asymptotic Normality Order Statistics Extreme Values The quantile function of a cumulative distribution function F is defined by F 1 : (0, 1) R, F 1 (p) = inf{x : F (x) p}. The empirical quantile function is related to the order statistics through F 1 n (p) = X ( np ) Goal Show the asymptotic normality of the sequence n(f 1 n (p) F 1 (p)).

28 Asymptotic Normality Order Statistics Extreme Values To derive the asymptotic distribution, we have to investigate the Hadamard-differentiability of the map φ : F F 1 (p).

29 Asymptotic Normality Order Statistics Extreme Values To derive the asymptotic distribution, we have to investigate the Hadamard-differentiability of the map φ : F F 1 (p). The delta method is not restricted to empirical quantiles but can also be applied to other estimators of F.

30 Asymptotic Normality Order Statistics Extreme Values To derive the asymptotic distribution, we have to investigate the Hadamard-differentiability of the map φ : F F 1 (p). The delta method is not restricted to empirical quantiles but can also be applied to other estimators of F. For a nondecreasing F D[a, b] with [a, b] [, ], let φ(f ) [a, b] be an arbitrary point such that F (φ(f ) ) p F (φ(f )). The domain D φ of the resulting map φ is the set of all F such that there exists a solution to the inequalities.

31 Asymptotic Normality Order Statistics Extreme Values Lemma Let F D φ be differentiable at a point ζ p (a, b) such that F (ζ p) = p, with positive derivative. Then, φ : D φ D[a, b] R is Hadamard-differentiable at F tangentially to the set of functions h D[a, b] that are continous at ζ p with derivative φ F (h) = h(ζ p)/f (ζ p).

32 Asymptotic Normality Order Statistics Extreme Values Lemma Let F D φ be differentiable at a point ζ p (a, b) such that F (ζ p) = p, with positive derivative. Then, φ : D φ D[a, b] R is Hadamard-differentiable at F tangentially to the set of functions h D[a, b] that are continous at ζ p with derivative φ F (h) = h(ζ p)/f (ζ p). Corollary Let 0 < p < 1. If F is differentiable at F 1 (p) with positive derivative f (F 1 (p)), then n(f 1 n (p) F 1 (p)) = 1 n n 1 {Xi F 1 (p)} p + o p(1). f (F 1 (p)) Hence, the asymptotic distribution of the sequence is N (0, σ 2 ) with variance σ 2 = p(1 p)/f 2 (F 1 (p)). i=1

33 Asymptotic Normality Order Statistics Extreme Values Example We estimate the 0.1 quantil of a Weibull(200, 2) distribution N=100 N=500 Histogram of estimate Histogram of estimate Density Density estimate estimate Normal Q Q Plot Normal Q Q Plot Sample Quantiles Sample Quantiles Theoretical Quantiles Theoretical Quantiles

34 Asymptotic Normality Order Statistics Extreme Values Consider not only a single quantile, but the function F (F 1 (p)) p1 <p<p 2 for fixed numbers 0 p 1 < p 2 1.

35 Asymptotic Normality Order Statistics Extreme Values Consider not only a single quantile, but the function F (F 1 (p)) p1 <p<p 2 for fixed numbers 0 p 1 < p 2 1. Because any quantile function is bounded on [p 1, p 2] (0, 1) we may hope to strenghen the result to convergence in l (p 1, p 2). If the distribution has compact support we are able to show weak convergence in l (0, 1).

36 Asymptotic Normality Order Statistics Extreme Values Lemma Given an interval [a, b] R, let D 1 be the set of all restrictions of distribution functions on R to [a, b]. Let D 2 be the subset of D 1 of distribution functions of measures that give mass 1 to (a, b]. (i) Let 0 < p 1 < p 2 < 1 and let F be continously differentiable on the interval [a, b] = [F 1 (p 1) ɛ, F 1 (p 2) + ɛ] for some ɛ > 0, with strictly positive derivative f. Then the inverse map G G 1 as a map D 1 D[a, b] l [p 1, p 2] is Hadamard differentiable at F tangentially to C[a, b]. (ii) Let F have compact support [a, b] and be continously differentiable on its support with strictly positive derivative f. Then G 1 as a map D 2 D[a, b] l (0, 1) is Hadamard differentiable at F tangentially to C[a, b]. In both cases the derivative is the map h (h/f ) F 1.

37 Asymptotic Normality Order Statistics Extreme Values Corollary (i) Let 0 < p 1 < p 2 < 1. If F is differentiable on the interval [a, b] = [F 1 (p 1) ɛ, F 1 (p 2) + ɛ] for some ɛ > 0 with positive derivative f. Then, the sequence n(f 1 n (p) F 1 (p)) converges weakly in l (p 1, p 2). (ii) If F has compact support and is continously differentiable on its support with stricly positive derivative f. Then, the result can be strenghten to weak convergence in l (0, 1). In both cases the limit process is G λ /f (F 1 (p)), where G λ is a standard Brownian bridge.

38 Order Statistics Functional Asymptotic Normality Order Statistics Extreme Values In estimating a quantile, we could also use the order statistics directly.

39 Order Statistics Functional Asymptotic Normality Order Statistics Extreme Values In estimating a quantile, we could also use the order statistics directly. For X (kn) be a consistent estimator for F 1 one needs minimally kn p. n To ensure that F 1 n (p) and X (kn) are asymptotic equivalent we also need k nn p faster than 1 n.

40 Order Statistics Functional Asymptotic Normality Order Statistics Extreme Values In estimating a quantile, we could also use the order statistics directly. For X (kn) be a consistent estimator for F 1 one needs minimally kn p. n To ensure that F 1 n (p) and X (kn) are asymptotic equivalent we also need k nn p faster than 1 n. Lemma Let F be differentiable at F 1 (p) with positive derivative and let k nn = p + c n + o( 1 n ). Then n(x(kn) F 1 n (p)) p c f (F 1 (p).

41 Asymptotic Normality Order Statistics Extreme Values Example Goal: Estimate a confidence interval using the order statistics, e.g. determine k N, l N N such that P(X (k) < F 1 (p) < X (l) ) 1 α

42 Asymptotic Normality Order Statistics Extreme Values Example Goal: Estimate a confidence interval using the order statistics, e.g. determine k N, l N N such that P(X (k) < F 1 (p) < X (l) ) 1 α Ansatz: P(X (k) < F 1 (p) < X (l) ) = P(U (k) < p < U (l) ).

43 Asymptotic Normality Order Statistics Extreme Values Example Goal: Estimate a confidence interval using the order statistics, e.g. determine k N, l N N such that P(X (k) < F 1 (p) < X (l) ) 1 α Ansatz: Set P(X (k) < F 1 (p) < X (l) ) = P(U (k) < p < U (l) ). k, l p(1 p) N = p ± z 1 α. 2 N

44 Asymptotic Normality Order Statistics Extreme Values Example Goal: Estimate a confidence interval using the order statistics, e.g. determine k N, l N N such that P(X (k) < F 1 (p) < X (l) ) 1 α Ansatz: P(X (k) < F 1 (p) < X (l) ) = P(U (k) < p < U (l) ). Set k, l p(1 p) N = p ± z 1 α. 2 N Using the preceeding Lemma to obtain ( ) p(1 p) U (k) = G 1 1 N (p) ± z α + o P N N

45 Asymptotic Normality Order Statistics Extreme Values Example Goal: Estimate a confidence interval using the order statistics, e.g. determine k N, l N N such that P(X (k) < F 1 (p) < X (l) ) 1 α Ansatz: P(X (k) < F 1 (p) < X (l) ) = P(U (k) < p < U (l) ). Set k, l p(1 p) N = p ± z 1 α. 2 N Using the preceeding Lemma to obtain ( ) p(1 p) U (k) = G 1 1 N (p) ± z α + o P N N Hence, U (k) < p < U (l) is asymptotically equivalent to G N 1 N (p) p z 1 α p(1 p) 2. The probability of this event converges to 1 α.

46 Asymptotic Normality Order Statistics Extreme Values Example (Weibull distribution) Estimate a confidence interval for the 0.1 quantil of a Weibull(200, 2) distribution. Simulation of 1000 intervals (theoretical quantil: 32.46): N empirical interval Coverage probability 50 [12.65,45.98] 97.2% 100 [21.86,40.99] 93.5% 1000 [29.12,35.45] 94.9% [31.46,33.48] 95.3%

47 Extreme Values Functional Asymptotic Normality Order Statistics Extreme Values Study the asymptotic behaviour of extreme order statistics (for example X (kn) with kn 0 or 1). n

48 Extreme Values Functional Asymptotic Normality Order Statistics Extreme Values Study the asymptotic behaviour of extreme order statistics (for example X (kn) with kn 0 or 1). n For x R it holds P(X (kn) x) = P(Y n k n), where Y Binomial(n, p n) with p n = P(X i > x).

49 Extreme Values Functional Asymptotic Normality Order Statistics Extreme Values Study the asymptotic behaviour of extreme order statistics (for example X (kn) with kn 0 or 1). n For x R it holds P(X (kn) x) = P(Y n k n), where Y Binomial(n, p n) with p n = P(X i > x). Hence, the limit distributions of general order statistics can be derived from approximations to the binomial distribution.

50 Asymptotic Normality Order Statistics Extreme Values Consider the extreme cases k n = n k for fixed k and start with the maximum X (n).

51 Asymptotic Normality Order Statistics Extreme Values Consider the extreme cases k n = n k for fixed k and start with the maximum X (n). The distribution of the maximum is P(X (n) x) = F (x n) n = ( 1 ns(x) ) n, n where S is the survival function S(x) = P(X i > x).

52 Asymptotic Normality Order Statistics Extreme Values Consider the extreme cases k n = n k for fixed k and start with the maximum X (n). The distribution of the maximum is P(X (n) x) = F (x n) n = ( 1 ns(x) ) n, n where S is the survival function S(x) = P(X i > x). Lemma For any sequence x n and any τ > 0, we have P(X (n) x n) e τ if and only if ns(x n) τ.

53 Asymptotic Normality Order Statistics Extreme Values Goal: Find constants a n and b n > 0 such that bn 1 (X (n) a n) converges to a nontrivial limit. In view of the lemma we must choose a n and b n such that S(a n + b nx) = O( 1 n ). The set of possible limit distributions is extremely small:

54 Asymptotic Normality Order Statistics Extreme Values Goal: Find constants a n and b n > 0 such that bn 1 (X (n) a n) converges to a nontrivial limit. In view of the lemma we must choose a n and b n such that S(a n + b nx) = O( 1 n ). The set of possible limit distributions is extremely small: Theorem (Extreme Value Distributions) Let b 1 N (X (N) a N ) G for a nondegenerate distribution G. Then G belongs to the location-scale family of a distribution of one of the following forms: (i) e e x with support R. (ii) e ( 1 x α ) with support [0, ) and α > 0. (iii) e ( x)α with support (, 0] and α > 0.

55 Asymptotic Normality Order Statistics Extreme Values Example (Normal-distribution) Set a n = 2log(n) 1 log(log(n))+log(4π) and b 2 n = 1. Using Mill s ratio, 2 log(n) 2 log(n) which asserts that ψ(t) ψ(t) t as t to see that nψ(a n + b nx) e x for every x. Hence, 2 log(n)(x (n) an ) converges to a limit of type (i).

56 Asymptotic Normality Order Statistics Extreme Values Theorem Let τ F = sup{t : F (t) < 1}. Then, there exist constants a N and b N such that the sequence b 1 N (X (N) a N ) converges in distribution if and only if, as t τ F, (i) There exists a strictly positive function g on R such that S(t+g(t)x) S(t) e x, for every x R; (ii) τ F = and S(tx) S(t) x α, for every x > 0; (iii) τ F < and S(τ F (τ F t)x) S(t) x α, for every x > 0. The constants (a N, b N ) can be taken equal to (u N, g(u N )), (0, u N ) and (τ F, τ F u N ), for u N = F 1 (1 1 N ).

57 Asymptotic Normality Order Statistics Extreme Values The convergence of the maximum X (i) implies the weak convergence of X (n k). Theorem Let b 1 N (X (N) a N ) G. Then b 1 (X (N k) a N ) H for the distribution function H(x) = G(x) k i=0 N ( logg(x)) i. i! The limit distribution follows from the Poisson approximation to the binomial distribution.

58 Table of Contents Functional Definition and examples 1 Functional Heuristic Ansatz Types of differentiability 2 Asymptotic Normality Order Statistics Extreme Values 3 Definition and examples

59 Definition and examples Definition (L-statistics) Let X (1),..., X (n) be the order statistics of a sample of real-valued random variables. A linear combination of (transformed) order statistics n c ni a(x (i) ) is called L-statisic with coefficients c ni and score function a. i=1

60 Definition and examples Example (Trimmed and Winsorized means) (i) The α trimmed mean is the average of the middle (1 2α) th fraction of the observations: X T,α 1 n = n 2 αn n αn i=1+ αn X (i)

61 Definition and examples Example (Trimmed and Winsorized means) (i) The α trimmed mean is the average of the middle (1 2α) th fraction of the observations: X T,α 1 n = n 2 αn n αn i=1+ αn X (i) (ii) The α Winsorized mean replaces the αth fraction of smallest and largest data and next takes the average: X W,α n = 1 n [ αn X ( αn ) + n αn i=1+ αn X (i) + αn X (n αn +1) ]

62 Definition and examples Abbildung : Asymptotic variances of the α trimmed mean.

63 Definition and examples The order statistics can be expressed in their empirical distribution through F 1 n (p) = X ( pn ) = X (i) for i 1 < p i n n.

64 Definition and examples The order statistics can be expressed in their empirical distribution through F 1 n (p) = X ( pn ) = X (i) for i 1 < p i n n. Hence, we may hope to write the L-statistic in the form φ(f n) and next apply the delta method to derive the asymptotic distribution.

65 Definition and examples The order statistics can be expressed in their empirical distribution through F 1 n (p) = X ( pn ) = X (i) for i 1 < p i n n. Hence, we may hope to write the L-statistic in the form φ(f n) and next apply the delta method to derive the asymptotic distribution. For a fixed function a and a signed measure K on (0, 1), consider the function Hence, φ(f n) = φ(f ) = i=1 1 0 a(f 1 ) dk. n ( i 1 K n, i ] a(x (i) ), n which is an L-statistic with coefficients c ni = K ( i 1 n, i n ]. Not all, but most of the arrays of coefficients c ni can be generated through a measure K.

66 Definition and examples Example (Trimmed- and Winsorized mean) (i) Let a be the identity function and K the uniform distribution on (α, 1 α). The corresponding L-statistic is 1 α 1 F 1 n (s) ds 1 2α α 1 = ( αn αn)x ( αn ) + n 2αn n αn i=1+ αn X (i) + ( αn αn)x (n αn +1). The difference of this L-statistic and the α trimmed mean can be seen to be O P ( 1 n ).

67 Definition and examples Example (Trimmed- and Winsorized mean) (i) Let a be the identity function and K the uniform distribution on (α, 1 α). The corresponding L-statistic is 1 α 1 F 1 n (s) ds 1 2α α 1 = ( αn αn)x ( αn ) + n 2αn n αn i=1+ αn X (i) + ( αn αn)x (n αn +1). The difference of this L-statistic and the α trimmed mean can be seen to be O P ( 1 n ). (ii) For the α Winsorized mean K is the sum of the Lebesgue measure on (α, 1 α) and the pointmass of size α at the points α and 1 α. φ(f n) = 1 αx ( αn ) + n n αn i=1+ αn X (i) + (αx (n αn )+1 )

68 Definition and examples Consider a L Statistic of the form 1 0 a(f 1 ) dk. There are two ways to prove asymptotic normality by applying the delta method:

69 Definition and examples Consider a L Statistic of the form 1 0 a(f 1 ) dk. There are two ways to prove asymptotic normality by applying the delta method: (1) View the L-statistic as a function of the empirical quantiles φ(f 1 n ) = 1 0 a (F 1 n ) dk.

70 Definition and examples Consider a L Statistic of the form 1 0 a(f 1 ) dk. There are two ways to prove asymptotic normality by applying the delta method: (1) View the L-statistic as a function of the empirical quantiles φ(f 1 n ) = 1 0 a (F 1 n ) dk. (2) Express the L-statistic through the empirical process φ(f n) = 1 0 a (F 1 n ) dk.

71 Definition and examples Consider a L Statistic of the form 1 0 a(f 1 ) dk. There are two ways to prove asymptotic normality by applying the delta method: (1) View the L-statistic as a function of the empirical quantiles φ(f 1 n ) = 1 0 a (F 1 n ) dk. (2) Express the L-statistic through the empirical process φ(f n) = 1 0 a (F 1 n ) dk. Both approaches are valid under different sets of conditions on K, a and F. Often, we have to combine both approaches!

72 First approach Functional Definition and examples Goal: Apply the delta method to the sequence n(φ(f 1 n ) φ(f 1 )), where φ(f 1 n ) = 1 a 0 (F 1 n ) dk.

73 First approach Functional Definition and examples Goal: Apply the delta method to the sequence n(φ(f 1 n ) φ(f 1 )), where φ(f 1 n ) = 1 a 0 (F 1 n ) dk. Lemma Let a : R R be continously differentiable with a bounded derivative. Let K be a signed measure on the interval (α, β) (0, 1). Then the map Q a(q) dk from l (α, β) to R is Hadamard-differentiable at every Q with derivative H a (Q)H dk.

74 First approach Functional Definition and examples Goal: Apply the delta method to the sequence n(φ(f 1 n ) φ(f 1 )), where φ(f 1 n ) = 1 a 0 (F 1 n ) dk. Lemma Let a : R R be continously differentiable with a bounded derivative. Let K be a signed measure on the interval (α, β) (0, 1). Then the map Q a(q) dk from l (α, β) to R is Hadamard-differentiable at every Q with derivative H a (Q)H dk. Assumptions for asymptotic normality To ensure the convergence of the empirical quantile process in l (α, β), F has to have positive density between its α and β quantiles. This smoothness of F is unnecessary if we assume K is smooth.

75 Definition and examples Goal: Apply the delta method to the sequence n(φ(f 1 n ) φ(f 1 )), where φ(f n) = 1 a 0 (F 1 n ) dk.

76 Definition and examples Goal: Apply the delta method to the sequence n(φ(f 1 n ) φ(f 1 )), where φ(f n) = 1 a 0 (F 1 n ) dk. Lemma Let a : R R be of bounded variation on bounded intervals with (a + + a ) d K F < and a(0) = 0. Let K be a signed measure on (0, 1) whose distribution function K is differentiable at F (x) for almost-every x and satisfies K(u + h) K(u) M(u)h for every sufficiently small h, and some function M such that M(F ) d a <. Then, the map F a F 1 dk from DF [, ] D[, ] to R is Hadamard differentiable at F, with derivative H (K F )H da.

77 Definition and examples Goal: Apply the delta method to the sequence n(φ(f 1 n ) φ(f 1 )), where φ(f n) = 1 a 0 (F 1 n ) dk. Lemma Let a : R R be of bounded variation on bounded intervals with (a + + a ) d K F < and a(0) = 0. Let K be a signed measure on (0, 1) whose distribution function K is differentiable at F (x) for almost-every x and satisfies K(u + h) K(u) M(u)h for every sufficiently small h, and some function M such that M(F ) d a <. Then, the map F a F 1 dk from DF [, ] D[, ] to R is Hadamard differentiable at F, with derivative H (K F )H da. Assumptions for asymptotic normality We have to assume that K is sufficiently smooth but the lemma does not require that F is smooth.

78 Definition and examples Example (Trimmed mean) Consider the α trimmed mean, where K is the distribution function of the Uniform distribution on (α, 1 α). Assume that F has Lebesgue measure zero on the set {x : F (x) = α, or 1 α}. Then, the trimmed mean is asymptotically normal with zero-mean and variance 1 (1 2α) 2 F 1 (1 α) F 1 (1 α)) F 1 (α) F 1 (α) (F (x y) F (x)f (y)) dx dy

79 Definition and examples Example (Trimmed mean) Consider the α trimmed mean, where K is the distribution function of the Uniform distribution on (α, 1 α). Assume that F has Lebesgue measure zero on the set {x : F (x) = α, or 1 α}. Then, the trimmed mean is asymptotically normal with zero-mean and variance 1 (1 2α) 2 F 1 (1 α) F 1 (1 α)) F 1 (α) F 1 (α) (F (x y) F (x)f (y)) dx dy Example (Winsorized mean) The generating measure of the α Winsorized mean is the sum of the Lebesgue measure on (α, 1 α) and the pointmass of size α at the points α and 1 α. Hence, we can decompose K (and the Winsorized mean) in a discrete and a continous part. Combining the two approaches yields the asymptotic normality.