The expectation of products of quadratic forms in normal variables: the practice

Transcription

1 The expectation of products of quadratic forms in normal variables: the practice by JAN R. MAGNUS* Summary A table is presented to simplify the computation of the expectation of a product of quadratic forms in normal variables. Some peculiarities of the table are discussed. 1 Introduction The reader of this journal has recently been exposed to two papers on the expectation of the product of an arbitrary number of quadratic forms in normally distributed variables. Let u N Nn(O,I), and E N N,(O, V), then the problem is to determine the expectation of n;= lu Aju, and fl3= I~ A,~, where the Aj (j = 1...s) are symmetric (n,n) matrices. In [3] I proved that this expectation is what I called an A@) polynomial. DON [I] presented an alternative proof of the main result, based on polarization. Also he showed that the covariance matrix of EV may be singular, without affecting the results, and he derived a simple formula for the coefficients of the A(s)polynomial. In an earlier paper (unknown to me until recently), KUMAR [2] studied the same problem.** He obtained an equivalent expression using the joint moment generating function of the s quadratic forms. It may, however, not be all to clear for the practical statistician how to compute these expectations. In the present note I shall present a table by which the expectation of fir=,e A~E is straightforwardly computed up to s = 8. The unfortunate statistician who needs the expectation for s 2 9 will have to extend the table, which is somewhat tedious but not difficult. First the reader should understand the concept of an A(s)polynomial, and I must repeat some definitions fro3 my earlier article. 2 The A(s)polynomial Let A,, A,,..., A, be real symmetric matrices of the same order. The following four definitions describe the A(s)polynomial. DeJinition 2.1 (A (s)form) Divide the index set {I, 2,..., s} into mutually exclusive and exhaustive subsets. Within each subset, take the trace of the matrix product of the Aj s corresponding ~. Department of Economics, The University of British Columbia, Vancouver, Canada. This paper was written at the Institute of Actuarial Science and Econometrics, University of Amsterdam. ** I am very grateful to Pascal Mazodier (Unite de Recherche, INSEE, Paris) for bringing Kumar s paper to my attention. 131

2 with indices from the subset. The product of all these traces will be called an A(s) form. Examples of A(3)forms: Definition 2.2 (similarity class) Two belong to the same similarity class iff their corresponding subsets (see definition 2.1) differ only by a permutation of indices. EXAMPLES: tr(a,)tr(a,a,) is equal to tr(a,a,)tr(a),, but not necessarily equal to tr(a2)tr(aia3) or tr(a,)tr(a,a,). However, all four A(3)forms belong to the same similarity class. On the other hand, tr(a, A,A,) belongs to a different similarity class. Definition 2.3 (A(s)sum) The sum of all nonequal (i.e. not necessarily equal) &)forms within a similarity class is called an A(s)sum. EXAMPLES: The three A(3)sums are: tr(a,a,a3), (tra,)(tra,)(tr A,), and tr(a 411. [(tr A 1) tr(a2a3) + (tr 4) tr(a I Ad + (tr Ad Definition 2.4 (A(s)polynomial) Any linear combination of A(s)sums is called an A(s)polynomial. EXAMPLES: The 42) and A(3)polynomial are: The importance of the A(s)polynomial is clear from the following theorem (see MAGNUS [3, p and DON [I, sec.51). Theorem Let A,, A,,..., As be real symmtric (n,n)matrices, u Nn(O,I), and E N N,(O, V), V positive semidefinite (possibly singular). Then, S (i) E n u Aju is an A@)polynomial; i= I (ii) S E fl & Aj& is the same A@)polynomial with each A, replaced by A,V. j= I 132

3 It is no simple exercise to write down the A(s)polynomial for s 2 4. Furthermore, in order to compute the desired expectation one has to determine the coefficients of the A($)polynomial. The following table will prove useful. structure of similarity class c number of nonequal A(s)forms ~ ~ I 10 7 ~ ~ How to use the table For s < 8 the table informs the statistician of: (i) (ii) (iii) the number and structure of the similarity classes (first 8 columns); the number of nonequal A@)forms within each similarity class (last 8 colunms); the coefficient of each A@)sum (middle column). 133

4 The working of the table is best explained by means of an example. Let s = 4, i.e. the desired expectation is En:=,E A~E, where E N N(0, V), V positive semidefinite (possibly singular). Step I : From the left part of the table (under structure of similarity class ), copy the northeast rectangle with 4 (s, in general) columns and 5 rows. The last row thus consists of a single 4 (s, in general); from the middle part of the table, copy the 5 numbers under c; from the right side, copy the 5 numbers under 4 (s, in general). This gives : similarity class C # &)forms Step 2: If the number k appears under similarity class, this means that the typical A@)form contains the trace of a product of k matrices. This leads to the following derived table : typical &)form 2c # A(s)forms (tr A) (tr B) (tr C) (tr D) (tr A) (tr B) (tr CD) (tr A) (tr BCD) (tr AB) (tr CD) tr (A BCD) Note that the second column displays 2 rather than c. Step 3: For s = 4 there are 5 similarity classes and therefore 5 A@)sums. Of each &)sum, we know the typical &)form, the coefficient (27, and the number of non equal A(s)forms. Thus, the 5 A(#)sums D,, a2,..., c5 are A($)sum coefficient 134

5 Step 4: By the theorem of section 2, the desired expectation is 4 E fl (u Aju) = o,+2~~+8~~~+4a~+16a~. j= 1 Moreover, the expectation of f14=l(~ Aj~) is the same expression with each A, replaced by A V. 4 Some peculiarities of the table The left part of the table gives all possiblepartitions (in DON S words) of the numbers, in a logical ordering. For s = 4, these partitions are (1, 1, 1, l), (I, 1,2), (1, 3), (2,2), and (4). For any partition of s let nj (j = 1...s) be the number of times thatj appears in the partition. Again for s = 4, this gives partition nl n2 4 c = sn, n, P S Of course jrij = s. DON showed that the coefficients of the A@)polynomial are j= 1 which is ca in his terminology (see DON [ 1, p. 781). Also he proved that the number of nonequal A(s)forms in a similarity class (the number of standard tableaux in his words, see [l, p. 771) equals Note the special role of n, and n2 in both formulas. Because of the way the similarity classes are ordered, two peculiarities of the table arise. First, there is only one (rather than s) column c. Thus, for example, the partition (1,3) for s = 4 and (1, 1, 3) for s = 5 have the same coefficient, namely 23 = 8. The reason is that n, increases with s, so that sn, remains unaltered. Secondly, there is a simple connection between the numbers in each row of the right side of the table (with the exception of the first row which contains only ones). Let cps (1) denote any of the underlined values in the right side of the table in column s, e.g. (ps (1) may be 15, 45, 10 or 60. Let (~~(2) be its right neighbour (105, 315, 70, or 420), etc. Then (j),(2) = (s+ l)qs(l), and in general 135

6 where This provides a simple means for extending the table. References [l] DON, F. J. H., The expectation of products of quadratic forms in normal variables. Statistica Neerlandica (1979), 33, [2] KUMAR, A., Expectation of product of quadratic forms. Sankhya (1973), Series B, 35, [3] MAGNUS, J. R., The moments of products of quadratic forms in normal variables. Statistica Neerlandica (1978), 32,