Wilks Λ and Hotelling s T 2.

Transcription

1 Wilks Λ and. Steffen Lauritzen, University of Oxford BS2 Statistical Inference, Lecture 13, Hilary Term 2008 March 2, 2008

2 If X and Y are independent, X Γ(α x, γ), and Y Γ(α y, γ), then the ratio X /(X + Y ) follows a Beta distribution: B = X X + Y B(α x, α y ). A multivariate analogue of this result involves the Wishart distribution and asserts. If W 1 W d (f 1, Σ) and W 2 W d (f 2, Σ) with f 1 d, then the distribution of det(w 1 ) Λ = det(w 1 + W 2 ) does not depend on Σ and is denoted by Λ(d, f 1, f 2 ). distribution is known as Wilks distribution. The

3 To see that the distribution of Λ does not depend on Σ, we choose a matrix A such that AΣA = I d. Then and Λ = W i = AW i A W d (f i, I d ) det( W 1 ) det( W 1 + W 2 ) = det(a) det(w 1 ) det(a ) det(a) det(w 1 + W 2 ) det(a ) = Λ. Clearly, the distribution of Λ does not depend on Σ and as Λ = Λ this also holds for the latter.

4 Wilks distribution is closely related to the Beta distribution. It holds that d Λ = D where B i are independent and follow Beta distributions with B i B{(f i)/2, f 2 /2)}. Indeed the distribution of (W 1 + W 2 ) 1 W 1 is also known as the multivariate Beta distribution. i=1 B i

5 We first need a useful result about determinants of block matrices. If A is a d d symmetric matrix partitioned into blocks of dimension r r, r s, and s s as ( ) A11 A A = 12, A 21 A 22 it holds that det A = det(a 11 A 12 A 1 22 A 21) det(a 22 ). (1) Here the entire expression should be considered equal to 0 if A 22 is not invertible and det(a 22 ) = 0.

6 This follows from a simple calculation ( A11 A det(a) = det 12 A 21 A 22 = det ) det ( I r r A 1 ( A11 A 12 A 1 22 A 21 A 12 0 s r A 22 = det(a 11 A 12 A 1 22 A 21) det(a 22 ). 0 r s 22 A 21 I s s ) )

7 Consider a partitioning of W and Σ into blocks as ( ) ( W11 W W = 12 Σ11 Σ, Σ = 12 W 21 W 22 Σ 21 Σ 22 ), where Σ 11 is an r r matrix, Σ 22 is s s, etc. If W W d (f, Σ) and Σ 12 = Σ 21 = 0 then det(w ) Λ(r, f s, s) = Λ(s, f r, r). det(w 11 ) det(w 22 )

8 To see this is true we first use the matrix identity (1) to write det(w ) det(w 11 ) det(w 22 ) = det(w 1 2) det(w 11 ) = det(w 1 2 ) det(w W 12 W22 1 W 21), where W 1 2 = W 11 W 12 W22 1 W 21. Next we need to use that if Σ 12 = 0 and thus Σ 1 2 = Σ 11, it further holds that W 1 2 and W 12 W22 1 W 21 are independent and both Wishart distributed as W 1 2 W r (f s, Σ 11 ), W 12 W 1 22 W 21 W r (s, Σ 11 ). We abstain from giving further details.

9 Wilks distribution occurs as the likelihood ratio test for independence. Consider X 1,..., X n N d (0, Σ). The likelihood function is L(K) = (det K) n/2 e tr(kw )/2. As this is maximized by we have If Σ 12 = 0 we similarly have ˆK = nw 1 L( ˆK) = (det W ) n/2 e nd/2. L( ˆK 11, ˆK 22 ) = (det W 11 ) n/2 e nr/2 (det W 22 ) n/2 e ns/2. Hence the likelihood ratio statistic is { } L( ˆK 11, ˆK 22 ) det(w ) n/2 = = Λ L( ˆK) n/2. det(w 11 ) det(w 22 )

10 and relation to Wilks Λ Relation to Fisher s F Let Y N d (µ, cσ) and W W d (f, Σ) with f d, and Y W. Then T 2 = f (Y µ) W 1 (Y µ)/c is known as. This is the multivariate analogue of Student s t (or rather t 2 ). It is equivalent to the likelihood ratio statistic for testing µ = 0 from a sample X 1,..., X n where then Y = X, W = i (X i X ), f = n 1, and c = 1/n. It holds that T 2 /f Λ(d, f, 1) = Λ(1, f d + 1, d).

11 and relation to Wilks Λ Relation to Fisher s F To see this we exploit the matrix identity (1) and calculate a determinant in two different ways. We may without loss of generality let µ = 0. We have ( ) W Y / c det Y / = det(w + YY /c) 1, c 1 But we also have ( W Y / c det Y / c 1 ) = det(1 + Y W 1 Y /c) det W = (1 + Y W 1 Y /c) det W = (1 + T 2 /f ) det W.

12 and relation to Wilks Λ Relation to Fisher s F Hence T 2 /f = Y W 1 Y /c = det W det(w + YY /c). The result now follows by noting that Y N d (0, cσ) implies YY /c W d (1, Σ). Since Λ(d, f, 1) = Λ(1, f d + 1, d) and the latter is a Beta distribution, it also holds that f d + 1 T 2 F (d, f + 1 d) fd where F denotes Fisher s F -distribution.