Equivalence of likelihood ratio tests and obliquity

Statistics & Probability North-Holland

Letters

24 June 1992

14 (1992) 223-228

Equivalence of likelihood ratio tests and obliquity J.A. MenCndez and B. Salvador Universidad de Valiadolid, Spain Received September 1990 Revised October 1991

Abstract: This paper shows how in a normal population, when the mean holds restrictions given by a right cone, the likelihood ratio test (LRT) for testing a face of that cone is equivalent to another test which is also the LRT for testing the linear subspace associated to the above face against a right cone defined only by those restrictions taking part in the definition of the subspace. Obliqueness is introduced and the equivalence with strict acuteness of a cone is proved, becoming useful to explain the dominance or equivalence of the LRT for testing a face of a cone. AMS 1980 Subject Classifications: Primary

62FO3; Secondary

Keywords: Restricted

ratio test, right and acute cone, obliqueness.

inference,

likelihood

62H15.

and

1. Introduction

Consider X - N,(8, T) with nonsingular and known covariance T and a polyhedral closed cone c = {e E Rk I a)3 > 0, i = 1,. . .) m}. We are interested H,:

a:B=O, up >, 0,

H,:

in testing the hypotheses

i=l,..., i=r+l

I,

,*.e, m,

8 E c.

Let denote G =r-‘,

(x, y& =x’Gy

Ilxll~=(x’G~)

Research partially PB87-0905C02-01.

0167-7152/92/$05.00

supported

Departamento de EstadisUniversidad de Valladolid,

by the DGICYT

0 1992 - Elsevier

Science

under

grant

Publishers

,

the inner product and the norm associated with G, and denote by pJx/C) the G-projection of a point x onto a closed convex cone C, that is p,(x/C> is the solution to minimize IIx - y 11; subject to y E C. It is known that X’=po(X/H,) and Xa= &X/H,) are th e maximum likelihood estimators (MLE’s) of 0 under Ho and H, respectively, and the likelihood ratio statistic T(X) = - 2 In A(X) for testing Ho against H, - Ho is defined by T(X)

Correspondence to: J.A. Menendez, tica e LO., Facultad de Ciencias, 47071 Valladolid, Spain.

l/2

= 11x-x0

IIf+- 11X-X” II&

Now we give the definition of acute and right cone as it was introduced by Martin and Salvador (1988). Definition 1.1. A cone C is said to be G-acute (G-right) if a~G-‘aj < 0 (= 0) for all i #j, i, j E

B.V. All rights reserved

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STATISTICS

IL..., ml. We will say C is strictly the inequalities are strict.

acute

& PROBABILITY

when

Let us consider a nonsingular transformation given by a k X k matrix A and define the cone

where bi = (A-‘)ai, i = 1,. . ., m. After applying A, we have a new metric on Rk given by W = (A-‘)‘GA-’ in such a way that IIxII G= IIAXIIW and p,JAx/AC)= &,(x/C) for any x E [Wk. The acuteness or rightness of C is preserved in AC, since a~G-‘aj = b[W-‘bj. Next we will consider a nonsingular linear transformation Y =Ax, such that Y _ N,(n, Z) with n =A8 and covariance the identity, so that the G-metric becomes the usual identity Euclidean metric on [Wk. The hypotheses H, and H, are now, relative to the parameter 7, given by the cones AH, and AH,, and the MLE of q based on Y are defined by Y” =Ax” and Ya =Ax” respectively. The LRT remains invariant, since = T(Y)

= IIY-Y0

IV-

IIY-Y”

112,

the norm being here associated to the unit metric. By using this transformation we will show in Section 2 how, when C is a right cone, the LRT can be considered equivalent to another test which does not take into account some restrictions given in the original model. This equivalence is the counterpart to the dominance proved by MenCndez and Salvador (1990), when C is acute. In Section 3 the obliqueness concept is introduced, allowing to characterize the strictly acute cones, which becomes useful to explain the anomalies arisen in the use of the LRT for testing restricted hypotheses.

24June1992

N,(q, I), a rotation given by an orthogonal matrix A4 can be considered in such a way that H, becomes an orthant of Rk. Denote by B = m, Z = BX and I_L= B0. We have Z N N,(p., I) and A4 can be chosen in such a way that BH,:

AC = (7 E Rk I b[v > 0, i = 1,. . . , m},

T(X)

LETTERS

BH,:

Pu, = 0,

i=l

,...,r,

Pi 2 O7

i=r+l

Pi a O,

i=l

,...,m, >..*> m

The restricted, based on Z, MLE’s of EL are given by Z” = BX” and Za = BXa. The LR-statistic for testing Ho against H, - Ho, when C is right is given by T(X)

= 11x0 - xa 11; = I1z” - Z” II2.

Changing C to an orthant is sensible because obtaining the MLE is very easy. In fact when 2 = (21,. . .) tk)’ is observed, where zi < 0 for i E ZUJ, icll,..., r1 and Jc(r + l,...,m) and zi 2 0 for i E (1,. . . , m) - (Z UJ), we have z” and za, the observed MLE’s, as follows: 0,

iE (l,...,r)

i zi,

otherwise,

0, i zi7

i=ZUJ, otherwise,

q!J =

2,” =

Now consider C*=

UJ,

the cone i= l,...,r}

{0ERkIajf3>0,

and the hypotheses H,*:

aiB=O,

H,*:

ew*.

The LRT given by T*(X) where

i=l,...,

r,

for testing

H,*

= IIx*O -X*a

11;

X*‘=pJx/H$)

against

and X*” =~o(x/H,*). for any x E (Wk.

Theorem 2.1. T(x) = T*(x) 2. Equivalence

of LRT’s

In this section we assume that C is a G-right cone. After performing A to get Y =AX _ 224

Proof. By applying

H,* - H,*

B we get p = Be,

BH,* :

pi=O,

i=l,...,

r,

BH,*:

pi>O,

i=l,...,

r,

is

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STATISTICS&PROBABILITY

and T*(x)= 11~*~-.~*~11~, where .z*~=Bx*~ and Zig = Bx*~. Consider now an observation x and z=Bx with zi
i=l ,...,r, otherwise,

z,*” = 0, i zi,

i EZ, otherwise.

z*” =

Ilz*O-z*aI12=

c

22

iE(l,...,r)--I

and the result

follows.

24June1992

dim(C) = k, and denote by C,, B c (1,. . . , ml, any face of C with associated subspace L, = {six = 0, i E B}. Proposition 3.1. C is acute iff aip(x/C) any x such that six < 0. q

= 0 for

Proposition 3.2. Zf C is acute (strictly acute) then C, is acute (strictly acute), for any face C, of C. 0

Therefore T*(x)=

LETTERS

.O

This theorem shows how the LRT for testing a face in a right cone against the cone is equivalent to another one which is the LRT for testing the least dimension face of a right cone against this one, which defines hypotheses on the parameters less restrictive than the considered in the former model. Shaked (1979) and Dykstra and Robertson (1983) studied the MLE’s under starshaped restrictions for the means of a normal model. These starshaped restrictions define a right cone, so that Theorem 2.1 can be used to show the equivalence between the LRT for testing a face of the cone and another LRT for testing the least dimension face of a cone defined by deleting some of the original starshaped restrictions. The null distribution of this LRT is given in Dykstra and Robertson (1983).

Proposition 3.3. Zf C is acute, then for any x in C and any face C, of Cp(x/C,) =p(x/L,). Moreouer p(x/C,) # 0, whenecer x # 0 and C is strictly acute. 0 Propositions 3.1 and 3.2 were proved by Martin and Salvador (1988). Proposition 3.3 is Lemma 2.1 in MenCndez and Salvador (1990). Next we give a definition of obliqueness as it was introduced by Warrack and Robertson (1984). Definition 3.1. Given any two cones H and C, with H c C, they are said to be oblique whenever there is a x for which p(p(x/C)/H) #p(x/H). in C the face C, ,,.,., rj = C n L,, ,,,,, rj where L,, (r) = {x E [WkI alx = 0, i = 1,. . . , r} is the subspace associated to C,, ,..,rj. Without loss of generality we can use the ‘identity metric on Rk, after performing the transformation shown at the Introduction. Through an orthogonal transformation we can . . map Ll,,...,,) in such a way that a point m L,,, , rl is characterized by having the first r coordinates zero, so that we can write Consider

C (l,....r) = {x E [Wk1x1 = . . . =x, = 0, 3. Obliqueness

and acute cones

b:+,x>O,...,b~x>o},

In this section we will show the relation between acuteness and obliqueness in a polyhedral closed cone. First let us give some known results which will be useful later. Consider a polyhedral closed cone C={xERkIa~x20,i=1

,...,

m},

br+l,. . . , b, being the rotated sidering ‘i

a,, . . . , a,. By con-

=P(bi/L{l,...,r))~

that is ci=

(O>...>O> bi,,+i,

bi,,+z,...,bi,k)‘, 225

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have

we

c,!cj = b;b, -

C I=l,...,r

i, j=r+l

bi/bjl,

7. . . 3m,

and C{i,_,,,,)={xERkIX,=

...

=x,=0,

c ;+,Xzo,...,c~>o}.

24 June 1992

LETTERS

closed cone in Rk-’ preserving the acute character, by vanishing the first r coordinates in the definition of C. The non-positiveness of bj,l,. . . , bj r for j = r + 13. . . 2m could be interpreted as C is located to ‘one side’ of L{,,,,,,,, and those vectors defining the restrictions in C that do not intervene in the definition of L,,, , rJ are to the ‘other side’ (see Figure 1).

If C is acute then b,!bj
i, je(l,...,

m},

Theorem 3.1. C is strictly acute iff every face C, and every subface of C, are oblique for any B c 11,. . . , m} except B = (1,. . . , m) and B = @.

which can be written (1, o,..., j=r+l

O)‘bj< (0 ,...,

1, 0,..., O)‘bj
,***> m,

so that

bj,i GO,..*, bj,,gO,

j=r+

l,...,

m,

hence C*kj< 0,

i, j=r+l

,...,

m,

and Ccl ,,,,r, is acute, as was to be expected from Proposition 3.2. We could consider Ctl,,,.,r) to be a polyhedral

L,,

, m). Also p(a,/C)

# 0, because if p(a,/C)= 0 then C would be contained in L,, in contradiction with dim(C) = k. Consider a face of C, C,, B={l,..., r), 1 Q r < m. From Proposition 3.3 we have

p( p(ai/CVG)

Fig. 1. The face CtI ,..., rI of the acute cone C. 226

=p( &/CV&)

# 0.

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24 June 1992

LETTERS

On the other hand p(a,/L,) = 0, hence a, can be taken to satisfy the condition of obliquity between C, and C. From Proposition 3.2 we get the same result for any face and subfaces of C. Sufficiency : Assume that c,, = {X E Rk I a;x 2 0, a;x > o} and L, n C,, are oblique, and consider x satisfying P(P(X/C,,VL,

n C,,) #P(X/L,

fJ Ci,)

(3.1)

(see Figure 2). Whenever x satisfies (3.11, x P C,, and 64 L,, since P(X/C,,) G L,. Also p(x/C,,) p(x/C,,> EL,, implies p(x/C,,)

=P(x/~,)

=P(x/L,

the last inequality holds since x @ C,,. Therefore tZ C,, which implies that

n C,,),

p(x/L,)

hence x would not satisfy (3.1). As a consequence ~(x/Ci,)

=P(x/J%)

and

4~(x/L,)

PWL,

> 0.

On the other hand any point between x and sati sfies (3.11, therefore we can take x satisfying u;x > 0, so that there is some A > 0 for which x -p(x/L,) = Au,, since x -p(x/L,) E

p(x/C,,)

Lf. Now suppose C,, is not strictly acute. Then

u;( x -p(

x/L,))

= hu;u,

2 0

or u;p( x/L,)

Fig. 3. The cone C = {x1 > 0, xj > 0, x1 - x2 + x3 > 0).

n Cd

In contradiction with (3.1). The same arguments can be applied to any c,,, i, j E (1,. . .) m}, therefore C is strictly acute. 0 Remark. In the theorem,

assuming only that C is oblique with any face C,, B Z (1,. . . , ml, is not sufficient to get the acuteness of C as we can see in the following example: Consider the cone in R3, C={x,>O,

< u;x < 0,

=~(x/Ld~

x,>o,

x,-x,+x+0}

and the linear subspaces L,={x,=O}, L3

=

1x1

L,={x,=O)

and

-x,+x,=0}.

It is easy to prove that C is oblique with every face, but Cij is non-oblique with every subface Li n Cij. The result becomes apparent through Figure 3, by noting how C, = L, n C is ‘acute’ although this acuteness cannot be expressed in terms of ui since +zj 2 0, i # j. We could say that every two faces determining Cij are obtuse.

References

Fig. 2.

Dykstra, R.L. and T. Robertson (1983), On testing tendencies, J. Amer Statist. Assoc. 78, 342-350.

monotone

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Martin, M. and B. Salvador (1988), Validity of the Pool-adjacent-violator algorithm, Statist. Probab. Left. 6, 143-145. Menendez, J.A. and B. Salvador (1990), Anomalies of the likelihood ratio test for testing restricted hyphoteses, Ann. Statist. 19(2), 889-898.

228

LETTERS

24June1992

Shaked, M. (19791, Estimation of starshaped sequences of Poisson and normal means, Ann. Statist. 7, 729-741. Warrack, G. and T. Robertson (1984), A likelihood ratio test regarding two nested but oblique order restricted hypotheses, J. Amer. Statist. Assoc. 79, 881-886.