by

Morris L. Eaton & Robb J. Muirhead

   School of Statistics             Statistical Research and Consulting Center 
           University of Minnesota 

Pfizer Inc.

Abstract

Let Γ be a Haar distributed random matrix on the group p O of p p × real orthogonal matrices. Partition Γ into four blocks, ( ) ( ) 1 1 : , 1 1 : ,1 1 : 21 12 11 × − Γ − × Γ × Γ p p and ( ) ( ), 1 1 : 22 − × − Γ p p so

. 22 21 12 11 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Γ Γ Γ Γ

Γ

The marginal distribution of 11 Γ is well known. In this paper, we give the conditional distribution of ( ) 12 21,Γ Γ given 11 Γ , and the conditional distribution of 22 Γ given ( ). , , 11 12 21 Γ Γ Γ This conditional specification uniquely determines the Haar distribution on p O . The two conditional distributions involve well known probability distributions – namely, the uniform distribution on the unit sphere { } 1 1 1

∈

− − x R x p p S and the Haar distribution on 2 − p O . Our results show how to construct the Haar distribution on p O from the Haar distribution on 2 − p O coupled with the uniform distribution on . 1 − p S

1. Introduction and Summary

The focus of this paper is the Haar probability distribution on the group p O of p p × real orthogonal matrices. The use of this group and the Haar distribution in multivariate statistical analysis has a long history, with James (1954) and Wijsman (1957) being two important early contributions. A standard description of the Haar distribution on p O is in terms of invariant differential forms – see Farrell (1985) for a systematic development and excellent history of this approach in multivariate analysis. A useful alternative is the use of random matrices, the multivariate normal distribution, and invariance properties of the objects under study. For example, see Eaton (1983) and Eaton (1989, Chapter 7). The

2 primary technical tools used in this paper stem from the invariance considerations discussed at length in Eaton (1989).

To describe the problem under consideration in this paper, suppose the random p p ×

orthogonal matrix Γ has the Haar distribution. This distribution is characterized by its invariance. To be more precise, let ( )⋅ L denote the distribution (or probability law) of ," “⋅ where " “⋅ can be a random variable, a random vector, a random matrix, etc. Using the L -notation, the Haar probability distribution is characterized by

( ) ( ) ( ) 2 1 g g Γ

Γ

Γ L L L

for all . , 2 1 p g g O ∈ In other words, the Haar distribution is the unique invariant (right or left) probability distribution on p O .

In all that follows, we will assume that + ∈ Γ p O , where

( ) . 1 ,1 , 11 22 21 12 11 ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ − ∈ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

∈

h h h h h h h p p O O

Note that + − p p O O is a set of Haar probability zero. In the arguments below, this set of probability zero has been removed from the sample space of Γ.

To describe the results in this paper, partition + ∈ Γ p O as

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Γ Γ Γ Γ

Γ 22 21 12 11

where ( ) ( ) 1 1 is , 1 1 is ,1 1 is 21 12 11 × − Γ − × Γ × Γ p p

and ( ) ( ). 1 1 is 22 − × − Γ p p

The marginal distribution of 11 Γ is well known – see below following Theorem 1.1. (It is well known even if 11 Γ is t s × with p t s ≤ + ; see Mitra (1970), Khatri (1970) and Eaton (1989, Chapter 7).) Thus, we will proceed with ( ) 11 Γ L being specified. In what follows, the notation ( ) ∗ ⋅ L is used for the conditional distribution of " “⋅ given “. “∗ The basic results in this paper provide a complete description of the two conditional distributions

( )
11 12 21, Γ Γ Γ L (1.1)

and
( ) . 11 12 21 22 , , Γ Γ Γ Γ L (1.2)

3 A moment’s reflection will convince the reader that knowing ( ) 11 Γ L , (1.1) and (1.2) determines the Haar distribution on p O and conversely.

Here is a rigorous specification of (1.1). Let 1 U and 2 U be independent, identically distributed (iid) and uniform on the unit sphere { } 1 1 1

∈

− − x R x p p S .

Theorem 1.1 In the notation above and with ( )1 ,1 11 − ∈ Γ fixed,

( ) ( ) ( ) ( )

2 2 / 1 2 11 1 2 / 1 2 11 11 12 21 1 , 1 , U U ′ Γ − Γ −

Γ Γ Γ L L (1.3)

where 2 U′ is the transpose of . 2 U

The above result asserts that ( ) 11 12 21 , , Γ Γ Γ L can be generated as follows:

(i) First, draw 11 Γ from the density (see Eaton (1989, Proposition 7.3))

( ) ( ) ( ) ( ) ( )( ) ( ) 1 1 1 2 / 3 2 2 1 2 1 2 1 < − − Γ Γ Γ

− x x p p p x f p

,

(ii) Next, draw 1 U and 2 U which are iid uniform on . 1 − p S Then use (1.3) to specify the conditional distribution of ( ) 12 21,Γ Γ given . 11 Γ

It is obvious that (i) an

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