Anderson-Darling Type Goodness-of-fit Statistic Based on a Multifold Integrated Empirical Distribution Function

Transcription

1 Anderson-Darling Type Goodness-of-fit Statistic Based on a Multifold Integrated Empirical Distribution Function S. Kuriki (Inst. Stat. Math., Tokyo) and H.-K. Hwang (Academia Sinica) Bernoulli Society Satellite Meeting to ISI23 Wed 4 Sept 23, Univ Tokyo / 27

2 Outline I. Anderson-Darling statistic and its extension II. Limiting null distribution III. Moment generating function IV. Statistical power V. Extension of Watson s statistic Summary 2 / 27

3 I. Anderson-Darling statistic and its extension 3 / 27

4 Goodness-of-fit tests X,..., X n : i.i.d. sequence from cdf F Goodness-of-fit test: (G is a given cdf) H : F = G vs. H : F G When G is continuous, we can assume G(x) = x (i.e., Unif(,)) WLOG. Empirical distribution function F n (x) = n n l(x i x) i= Test statistic is defined as a measure of discrepancy between F n (x) and G(x) = x. 4 / 27

5 Goodness-of-fit tests (cont) We focus on two integral-type test statistics. Anderson-Darling (952) statistic: A n = n x( x) (F n(x) x) 2 dx Watson (96) statistic (for testing uniformity on the unit sphere in R 2 ): { U n = n (F n (x) x) 2 dx n (F n (x) x)dx Limiting null distributions: Let ξ, ξ 2,... be i.i.d. sequence from N(, ). As n, A n d k= k(k + ) ξ2 k, U n d k= } 2 2π 2 k 2 (ξ2 2k + ξ2 2k ) 5 / 27

6 Closely looking at Anderson-Darling Anderson-Darling statistic Here, A n = B n (x) 2 x( x) dx where B n(x) = n(f n (x) x) B n (x) = n h [] (t; x)df n (t), h [] (t; x) = l(t x) x h [] (t; x) x t 6 / 27

7 An extension to Anderson-Darling To propose a new class of test statistics, instead of h [] ( ; x), we prepare different type h [m] ( ; x). Note first that h [] ( ; x) is piecewise constant s.t. h[] (t; x) dt = Define h [] ( ; x) to be continuous and piecewise linear s.t. h [] ( ; x) h[] (t; x) (at + b) dt =, a, b x t 7 / 27

8 An extension to Anderson-Darling (cont) General from of h [m] (t; x): h [m] (t; x) = m! (x t)m l(t x) m k= x m! (x u)m L k (u)du L k (t) where L k ( ) is the Legendre polynomial of degree k h [2] ( ; x) x t 8 / 27

9 An extension to Anderson-Darling (cont) We propose an extension of Anderson-Darling: where B [m] n A [m] n = B n [m] (x) 2 dx {x( x)} m+ (x) = n h [m] (t; x)df n (t) = { n F n (x )dx dx m x>x m > >x > m k= x>x m> >x > L k (x )dx dx m } L k (t)df n (t) (m-fold integral of empirical distribution function) A [m] n is well-defined (the integral exists) whenever X i (, ). A [] n is the original Anderson-Darling. 9 / 27

10 II. Limiting null distribution / 27

11 Main results Limiting null distribution W ( ) : the Winer process on [, ] B(x) = W (x) xw () : Brownian bridge Let B [m] n (x) = h[m] (t; x)db n (x). We can prove that as n, B [m] n ( ) d B [m] ( ) in L 2, where B [m] (x) = and hence (by continuous mapping) A [m] n = B n [m] (x) 2 {x( x)} m+ dx d A [m] := h [m] (t; x)db(t) B [m] (x) 2 dx {x( x)} m+ We will examine these limiting distributions B [m] ( ) and A [m]. / 27

12 Main results Limiting null distribution (cont) Theorem (Karhunen-Loève expansion) B [m] (x) {x( x)} (m+)/2 = k=m+ (uniformly in x, with prob. ), where ξ k = (k m )! (k + m + )! L(m+) k (x) ξ k L k (t)db(t), i.i.d. N(, ) L (m+) k is the associate Legendre function. Corollary (Limiting null distribution of A [m] n ) A [m] = B [m] (x) 2 dx = {x( x)} m+ k=m+ (k m )! (k + m + )! ξ2 k ξ 2 k χ2 () i.i.d. 2 / 27

13 III. Moment generating function 3 / 27

14 Moment generating function The moment generating function (Laplace transform) of is A [m] = k= λ k ξ 2 k, λ k = k(k + ) (k + 2m + ) [ E e sa[m]] = k= ( 2s λ k ) 2 Theorem Let x j (s) (j =,,..., 2m + ) be the solution of λ x 2s =, i.e., x(x + ) (x + 2m + ) 2s = Then [ E e sa[m]] = 2m+ j= Γ( x j (s)) j! 4 / 27

15 Moment generating function (m = ) When m = (Anderson-Darling), λ k = k(k + ) The equation x(x + ) 2s = has a solution x (s), x (s) = 2 ± + 8s Hence, [ E e sa[]] = Γ( x (s))γ( x (s)) = 2πs cos π 2 + 8s (Anderson and Darling, 952) Euler s reflection formula Γ(z)Γ( z) = π/ sin(πz) is used. 5 / 27

16 Moment generating function (m = ) When m =. λ k = k(k + )(k + 2)(k + 3). The equation x(x + )(x + 2)(x + 3) 2s = has the explicit solution, because by letting x = y 3/2, LHS =(y 3/2)(y /2)(y + /2)(y + 3/2) 2s = { y 2 (3/2) 2}{ y 2 (/2) 2} 2s is a quadratic equation in y 2 = (x + 3/2) 2. As a result, x (s), x (s), x 2 (s), x 3 (s) = ( ± 5 ± 4 ) 2s + 3, 2 [ E e sa[]] πs = 3 cos( π s) cos( π s) 6 / 27

17 Moment generating function (m = 2) When m = 2, [ E e sa[2]] = (πs) 3/2 432 cos(π η ) cosh(π η 2 ) cosh(π η 3 ) where η = 3 27s s s 728 and η = ( 4η η + 2 ) 2η η 2 = ( 4e πi/3 η 2 35η + 2e πi/3) 2η η 3 = ( 4e πi/3 η 2 35η + 2e πi/3) 2η 7 / 27

18 Calculation of upper prob. Finite representation is useful in numerical calculation. P ( A [m] > x ) = ( ) k k= π λ2k /2 λ 2k /2 (Smirnov-Slepian technique, see Slepian (958)). The case m = : s e xs E [e sa[m] ] ds Upper Prob x 8 / 27

19 IV. Statistical power 9 / 27

20 Statistical power We have the sample counterpart of the KL-expansion: B n [m] (x) {x( x)} (m+)/2 = (k m )! (k + m + )! L(m+) k (x) ξ k where ξ k = k m+ L k (x)db n (x) = n n L k (X i ), k The (extended) Anderson-Darling statistics are also written in terms of ξ k s as A [] n A [] n = k = k 2 i= k(k + ) ξ 2 k = 2 ξ ξ ξ (k )k(k + )(k + 2) ξ 2 k = 24 ξ ξ / 27

21 Statistical power (cont) First two components: ξ = 2nm, ξ2 = 6 5n (m 2 /2), where m k = n n i= (X i /2) k (the sample kth moment around /2) A [] n and A [] n = ξ 2 /2 + has much power for mean-shift alternative, = ξ 2 2 /24 + has much power for dispersion-change alternative 2 / 27

22 V. Extension of Watson s statistic 22 / 27

23 Watson s statistic and its extension Similar extensions are possible for Watson s statistic. Let where U [m] n = h [m] (t; x) = C [m] n (x) 2 dx, C [m] n = h [m] (t; x)df n (t), (t x)m l(t x ) + m! (m + )! b m+(t x) b m (y) is the Bernoulli polynomial, which satisfies U [] n b m (y + ) = b m (y) + my m. is the original Watson statistic. U [] n is proposed by Henze and Nikitin (22). 23 / 27

24 Watson s statistic and its extension (cont) h [] ( ; x) -x h [] ( ; x) and h [2] ( ; x) x t x t x t 24 / 27

25 Limiting null distribution Let C [m] n ( ) d C [m] ( ) and U n [m] U [m]. KL-expansion of C [m] (x): C [m] { } (x) = (2kπ) m+ l [m] 2k (x)ξ 2k + l [m] 2k (x)ξ 2k, where k= l [m] 2k (x) = sin ( ξ 2k = Consequently, 2kπx m + 2 sin(2kπx)db(x), ξ 2k = U [m] = k= ) ( π, l [m] 2k (x) = cos d 2kπx m + 2 cos(2kπx)db(x). ) π, (2kπ) 2(m+) {ξ2 2k + ξ2 2k }. 25 / 27

26 Summary We proposed a class of extended Anderson-Darling statistics A [m] n (m ) based on m-fold integrated empirical distribution function. The limiting null distribution A [m] is explicitly derived as weighted infinite sums of chi-square random variables. We provided moment generating function of A [m] without using infinite product. The same-type extension for Watson s statistic U n is possible. Acknowledgment: The authors thank Y. Nishiyama of ISM. 26 / 27

27 References Anderson, T. W. and Darling, D. A. (952). Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Ann. Math. Statist., 23 (2), Henze, N. and Nikitin, Ya. Yu. (22). Watson-type goodness-of-fit tests based on the integrated empirical process. Math. Methods. Stat., (2), Slepian, D. (957). Fluctuations of random noise power. Bell. Syst. Tech. J., 37, Watson, G. S. (96). Goodness-of-fit tests on a circle. Biometrika, 48 (,2), / 27