An approximation to the distribution of quadratic forms in many normal ...

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Faculteit der Economische Wetenschappen en Econometrie

ET

Serie Research Memoranda 05348

An approximation to the distribution of quadratic forms in many normal variables

J.M. Sneek J. Smits

Research Memorandum 1990-49 October 1990

vrije Universiteit

amsterdam

AN APPROXIMATION TO THE DISTRIBUTION OF QUADRATIC FORMS IN MANY NORMAL VARIABLES

by J.M. Sneek J. Smits Free University of Amsterdam Faculty of Economics and Econometrics De Boelelaan 1105 1081 HV Amsterdam

Summary In this paper we seek to approximate to the distribution of

general

large quadratic forms in normal variables in at most ö(n ) arithmetic operations. The main idea is to split the quadratic form containing

the

dominant

approximated

using the

are

through

obtained

eigenvalues normal

and

distribution.

a

remaining The

part

dominant

a generalized power method,

expensive job of finding

in a part

thus

that

is

eigenvalues avoiding

the

all eigenvalues. In special cases the method

may involove only 0(n) operations.

1.

In

INTRODUCTION

this

paper

the

problem

of

approximating

the

probability

P(ÖiPft *

Ixfaej ,

(13)

where k is much smaller than n, in practice always less than 10 if warming up has

taken

place

approximately

long

satisfies

enough.

This

V

(13)

that

a fc-th order homogeneous

i.e. there exists a vector y = ( f t , ^ V " ! % ) '

We note that

implies

- fiY

sucn

(vector)

« 0.

there is a 1-1 correspondence

between

^ lv ..,i^ fc .

difference

{y }

equation,

tnat

w?

and the coefficients

'-...-

the sequence

The coëfficiënt

(14) the values Alv..,Afc in

vector

m+l,

but detmi^0.

If

the

(m-l)-th In practice

the difference equations are not holding exactly and furthermore the order of an approximate difference

equation will be lower as we progress through the

sequence {s t }, because in (13) the large lambdas become more and more dominant. In

BEGUIN,

and

GOURIEROUX

(1980) a detailed discussion (in the

MONTFORT

context of determining the order of ARMA(p,g) models) is given of the corner method, which can be used to determine the order m and the point in the sequence where the difference

equation starts to hold. Knowing the

mate) order m and possibly after renumbering the sequence {st}

(approxi-

the vector n / 2 + log(n/2 + 2) , i = l,...,n/2

.

, t = 1 , . . . , n.

as

percentile

points

uniform

distribution,

to

the

mirror

image

(l)-(3)

but

including

like

from

set (2) of

a

to

(2)

density, a

but

then

negative shifted

exponential

to

their

mirror

image

theorem

applies

if

set (1)

the

for

right.

negative

A's, which makes them symmetrie around zero. For

the

implies

sets

(l)-(6)

that

given

the k

central

the

use

limit of

k

dominant

n -> oo and

eigenvalues

this

becomes

less

influential if n-»oo. The sets (l)-(3) above were also shifted to the left and right by adding a constant to all the A's, but as in all cases the approximations improved

or remained

very similar we do not report

the results.

In table 1 we report results when Q in (3) is approximated by Q

in (4). The

table is organized as follows. The columns 2-11 correspond to the different sets {AJ and given n and

(ii)

Q2 =

X A i,nX?(l)

1 >A^(/X,(T2)

j'= k +1

then the approximation becomes arbitrary accurate: proof

If the first condition holds then Q2 becomes negligible compared to Q%

as one can prove that P ( | Q 2 | < e | Q i l ) - > 1 . for all e > 0 . If the second holds then there is nothing to prove.

19

condition

lemma 3

Let

^ E-\j,n—*°°5

n

the —*°°

sequencess tnen

f°

r

{^n}jmi an

be

normalized

such

y bounded sequence k(n)ax

S

s e t

0.0113 0.0107 0.0103 0.0098 0.0095 0.0093 0.0090 0.0088 0.0086 0.0082 0.0078 0.0074

0.0115 0.0135 0.0117 0.0102 0.0093 0.0087 0.0081 0.0075 0.0070 0.0060 0.0052 0.0045

8 1 1 1 3 3 3 3 3 3 3 3

e

max

5

set

0.0113 0.0100 0.0087 0.0080 0.0078 0.0076 0.0074 0.0072 0.0070 0.0067 0.0065 0.0063

0.0115 0.0084 0.0062 0.0096 0.0089 0.0083 0.0077 0.0071 0.0066 0.0057 0.0049 0.0046

8 8 8 1 1 1 1 1 1 1 3 3

e

e m a x always a t p t a r g e t - 0 . 0 5

24

TABLE 3 34 iterations *i

0 0 0 0 0 0 0 0 0 0 0 0

368435 310321 276326 252207 233498 218212 205288 194092 184217 175384 167393 160098

30 iterations

K-K

*i-*i

*i

0.0000008 0.0000520 0.0034475 0.0138658 -0.0037015 0.0105486 0.0000990 -0.0109916 0.0129975 0.0067024 0.0211043 0.0338830

0 138388 0 137991 0 137589 0 137181 0 136769 0 136350 0 135926 0 135497 0 135061 0 134619 0 134171 0 .133717

0.001798 0.001896 0.001497 0.007521 0.008028 0.008010 0.019393 0.021067 0.021893 0.033813 0.036207 0.037429

39 iterations Ai-Xi *i

-0 191243 0 191243 -0 186402 0 186402 -0 181560 0 181560 -0 176719 0 176719 0 171877 -0 171877 -0 167035 0 .167035

-0.000301 0.002490 -0.000983 0.000612 -0.005685 0.004358 -0.007053 -0.000209 0.004311 -0.005306 -0.011238 0.007160

dominating eigenvalues Aj^ for sets (2) , (3) and (4) for n=80 difference AA-XL between exact and approximated values

25

TABLE 4A n-40 Ptarg

0.005 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 0.995

k-O 0.0001 0.0010 0.0079 0.0282 0.0827 0.8938 0.9375 0.9618 0.9792 0.9865

set (1)

k-4 0.0004 0.0020 0.0112 0.0339 0.0887 0.8944 0.9414 0.9671 0.9845 0.9912

k-8 0.0004 0.0019 0.0108 0.0333 0.0881 0.8944 0.9411 0.9666 0.9840 0.9908 n-80

0.005 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 0.995

0.0008 0.0028 0.0128 0.0355 0.0892 0.8947 0.9403 0.9651 0.9821 0.9890

0.0011 0.0035 0.0143 0.0377 0.0913 0.8952 0.9420 0.9672 0.9842 0.9908

0 0 0 0 0 0 0 0 0 0

0.0010 0.0034 0.0147 0.0390 0.0933 0.8959 0.9447 0.9705 0.9871 0.9931

•fc-12

0.0008 0.0031 0.0141 0.0381 0.0925 0.8958 0.9442 0.9699 0.9867 0.9928

0.0017 0.0051 0.0181 0.0431 0.0965 0.8974 0.9469 0.9725 0.9885 0.9940

0.0013 0.0042 0.0164 0.0410 0.0947 0.8969 0.9459 0.9715 0.9877 0.9935

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

set (1)

0010 0034 0142 0375 0911 8952 9419 9670 9840 9906

0.0014 0.0042 0.0158 0.0397 0.0931 0.8958 0.9436 0.9690 0.9857 0.9920

0013 0040 0155 0394 0928 8957 9433 9686 9854 9918

0017 0049 0172 0416 0947 8965 9449 9704 9868 9929

0015 0044 0163 0404 0936 8961 9441 9694 9861 9923

n-160 set (1) 0.005 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 0.995

0.0017 0.0046 0.0165 0.0403 0.0931 0.8958 0.9427 0.9677 0.9844 0.9908

0.0018 0.0050 0.0171 0.0411 0.0938 0.8961 0.9434 0.9685 0.9851 0.9914

0.0018 0.0049 0.0170 0.0410 0.0937 0.8960 0.9433 0.9684 0.9850 0.9913

0.0020 0.0053 0.0177 0.0419 0.0944 0.8963 0.9440 0.9692 0.9857 0.9919

0.0020 0.0052 0.0175 0.0416 0.0942 0.8963 0.9438 0.9690 0.9855 0.9917

0.0022 0.0056 0.0183 0.0426 0.0951 0.8966 0.9446 0.9699 0.9863 0.9923

0.0020 0.0053 0.0178 0.0420 0.0945 0.8964 0.9441 0.9693 0.9858 0.9920

tailprobabilities p a e f c u a j_ for k exact and approximated A's

26

TABLE 4B n-40 set (2) Ptarg

k-O

0.005 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 0.995

0.0000 0.0002 0.0035 0.0192 0.0720 0.8945 0.9355 0.9587 0.9758 0.9834

k-4 0.0005 0.0022 0.0124 0.0363 0.0920 0.8953 0.9450 0.9713 0.9880 0.9938

k-8

0.0004 0.0022 0.0122 0.0360 0.0918 0.8952 0.9449 0.9712 0.9879 0.9938

0.0016 0.0050 0.0182 0.0438 0.0977 0.8979 0.9481 0.9737 0.9893 0.9946

k-12

0.0015 0.0046 0.0176 0.0430 0.0970 0.8978 0.9478 0.9734 0.9892 0.9945

0.0029 0.0072 0.0217 0.0474 0.0995 0.8991 0.9492 0.9745 0.9897 0.9948

0.0024 0.0064 0.0204 0.0460 0.0985 0.8988 0.9488 0.9741 0.9895 0.9947

0.0016 0.0048 0.0174 0.0422 0.0957 0.8968 0.9464 0.9722 0.9883 0.9940

0.0024 0.0062 0.0199 0.0452 0.0978 0.8980 0.9479 0.9734 0.9891 0.9945

0.0021 0.0057 0.0190 0.0441 0.0970 0.8976 0.9474 0.9730 0.9888 0.9943

0.0020 0.0053 0.0180 0.0425 0.0954 0.8966 0.9455 0.9712 0.9875 0.9934

0.0025 0.0062 0.0195 0.0444 0.0968 0.8974 0.9468 0.9723 0.9884 0.9940

0.0023 0.0060 0.0191 0.0439 0.0964 0.8973 0.9465 0.9720 0.9881 0.9938

11-80 set (2) 0.005 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 0.995

0.0002 0.0012 0.0082 0.0282 0.0819 0.8946 0.9378 0.9618 0.9789 0.9861

0.0009 0.0032 0.0141 0.0379 0.0922 0.8952 0.9439 0.9699 0.9869 0.9931

0.0009 0.0032 0.0141 0.0379 0.0922 0.8952 0.9439 0.9699 0.9869 0.9930

0.0017 0.0049 0.0176 0.0425 0.0960 0.8969 0.9465 0.9723 0.9884 0.9940

n-160 set (2) 0.005 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 0.995

0.0008 0.0028 0.0125 0.0348 0.0881 0.8952 0.9402 0.9647 0.9816 0.9885

0.0015 0.0043 0.0161 0.0401 0.0933 0.8958 0.9438 0.9694 0.9862 0.9925

0.0015 0.0043 0.0161 0.0400 0.0933 0.8958 0.9438 0.9694 0.9862 0.9925

0.0020 0.0054 0.0181 0.0427 0.0955 0.8967 0.9457 0.9713 0.9876 0.9935

tailprobabilities p a c t u a i for k exact and approximated A's

27

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