Large sample covariance matrices and the T 2 statistic

Transcription

1 Large sample covariance matrices and the T 2 statistic EURANDOM, the Netherlands Joint work with W. Zhou

2 Outline 1 2

3 Basic setting Let {X ij }, i, j =, be i.i.d. r.v. Write n s j = (X 1j,, X pj ) T and s = 1 n s j. The sample j=1 covariance matrix is defined by S = 1 n (s j s)(s j s). n j=1 Note in RMT, the matrix is S 1 = 1 n n s j s j. j=1 p Large dimension, that is, lim n n = c > 0. Here p is the dimension and n the sample size

4 Difference between fixed and large : I An example: An important statistic in multivariate analysis W n = ln(dets 1 ) = p ln λ j j=1 When p is fixed, by a Taylor expansion, n p W n D N(0, EX11 4 1) When p increases with cn with c (0, 1), what happens? However, it is not the case for large dimension. Indeed, n p W n.

5 Fig.1 Density of n p W n under different sample sizes with c = N=500 N=200 N=100 N=

6 Difference between fixed and large : II The spectral decomposition Σ = n λ i u i u T i. In classical multivariate analysis: and n(ˆλi λ i ) n(ûni u i ) i=1 D N(0, 2λ 2 i ) D N(0, Γ i ) But, what happens in the large dimensional RMT? Bai (1999), Johnstone and Lu(2007).

7 RMT in other subjects Finance (Potters, Bouchaud and Laloux 2005), Wireless communications (Turlino and Verdu 2004) Large linear systems (linear equations) and sparse problems. (Donoho 2004).

8 Tools (1): Stieltjes transform The Stieltjes transform for any distribution function G(x) 1 m G (z) = λ z dg(λ), z C+ {z C, Iz > 0}. Indeed, for any random matrix A n having real eigenvalues λ 1,, λ n m F An (z) = 1 n i 1 λ i z = 1 n tr(a n zi) 1 G n D G mgn (z) m G (z), for all z C +, which is due to G{[x 1, x 2 ]} = 1 π lim ε 0+ x2 x 1 Im(m G (x + iε))dx.

9 Tools (2): Moment method A simple fact is that the kth moment of F An (x) is β k = x k df An (x) = 1 n trak n. Thus, to show that F An (x) converges to F c (x), a limiting distribution, one need to verify that 1 n trak a.s. n x k df c (x). and that Carleman s condition k=1 β 1/2k 2k =.

10 Empirical spectral distribution the classical empirical spectral distribution F S (x) F S (x) = 1 n n I(λ i x), i=1 where λ 1,, λ n are eigenvalues of S Importance: for any function f(x), n f(λ j ) = j=1 o f(x)df S (x)

11 The M-P law Marcenko and Pastur (1967), Jonsson (1982) and Silverstein (1995) F S (x) a.s. F c (x). The M-P law: F c (x) has a density, f c (x) = { (x a(c))(b(c) x) 2πx, if a(c) < x < b(c) 0, otherwise and with an additional mass point at x = 0 for c > 1 f c (0) = 1 c, where a(c) = (1 c) 2 and b(c) = (1 + c) 2.

12 Extreme eigenvalues When 0 < c 1, Bai and Yin (1993) used a unified approach to prove that λ max a.s. (1 + c) 2, λ min a.s. (1 c) 2. (1.1) Further, Bai and Silverstein (1998, 1999) characterized the separation of eigenvalues of general sample covariance matrices.

13 Asymptotic distribution of λ max For the asymptotic distribution of λ max, if S 1 is the Wishart matrix, Johnstone (2001) showed that where λ max µ np σ np D W1 F 1 (1.2) µ np = ( p 1+ n) 2, σ np = ( p 1+ n)( 1 p + 1 n ) 1/3 and F 1, the Tracy-Widom law of order 1, is given by F 1 (s) = exp{ 1 2 s q(x) + (x s)q 2 (x)dx} with q(x) being the solution of Painleve II differential equation.

14 CLT of eigenvalues / and eigenvectors Bai and Silverstein (2004) and Pan and Zhou (2007a): n j=1 f(λ j ) f(x)df cn (x) N(µ, σ 2 ). Eigenvectors: Silverstein (1989, 1990), Johnstone and Lu (2007), Bai, Miao and Pan (2007) and Pan and Zhou (2007a).

15 Motivation 1 Noth that the sample mean s and the sample covariance matrix S are independent when X ij are i.i.d N(0, 1). Conjecture: s and S might be in some sense asymptotically independent under general distributions instead of Gaussian.

16 Motivation 2 Hotelling s T 2 statistic (1931) defined by T 2 = n( s µ 0 ) T S 1 ( s µ 0 ). (1.3) When X ij are i.i.d. N(0, 1), s is independent of S 1. So, in this case it is easy to obtain the limiting distribution of T 2. But, what happens in general cases?

17 Ideas How to formulate it if s and S have some asymptotic independence? Instead, investigate the statistics s T f(s) s, (2.4) where f(s) = U T diag(f(λ 1 ),, f(λ n ))U and f(x) is a smooth function. For example, when f(x) = 1/x or x m, m 1, then, (2.4) becomes s T S m s or s T S 1 s. We hope s T f(s) s s 2 is asymptotically independent of s.

18 Theorem 2 (Pan and Zhou (2008)) Suppose that: (1) X ij, i, j = 1, 2,, are i.i.d. real r.v. with EX 11 = 0, E X 11 2 = 1 and E X 11 4 <. (2) p n = c n c (0, ) as n. (3) g(x) is a continuous function with first derivative and f(x) an analytic function. Then, ( n[ s T f(s) s s 2 f(x)df cn (x)], ) n(g( s T D s) g(c n )) (X, Y ), where Y N(0, σ 2 ), X N(0, σ1 2 ), Y independent of X.

19 Remark Indeed, Theorem indicates that ( n[ s T lim P f(s) s n s 2 f(x)df cn (x)] t ) n(g( s T s) g(c n ) is equal to lim P n ( n[x T n f(s)x n ) f(x)df cn (x)] t, x n = 1

20 The asymptotic distribution of the T 2 statistic Corollary 1. In addition to the assumptions of Theorem 1, suppose that EX 11 = µ, µ = µ 0, 0 < c < 1 then T 2 1 n( n c n x df c n (x)) D 1 N(0, 2c x 2 df c(x)). 0 < c < 1 follows directly from Theorem 1.

21 CLT for linear spectral statistics of S Corollary 2. Under the assumptions of Theorem 1, n j=1 f(λ j ) f(x)df cn (x) N(µ 2, σ 2 ). Note: The mean µ 2 for S is different from the mean µ for S 1.

22 Thank you