SYMMETRY AND LATTICE CONDITIONAL INDEPENDENCE IN A MULTIVARIATE NORMAL DISTRIBUTION

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1 The Annals of Statistics 1998, Vol 26, No 2, SYMMETRY AND LATTCE CONDTONAL NDEPENDENCE N A MULTVARATE NORMAL DSTRBUTON BY STEEN ANDERSSON 1 AND JESPER MADSEN 1,2 ndiana University and University of Copenhagen A class of multivariate normal models with symmetry restrictions given by a finite group and conditional independence restrictions given by a finite distributive lattice is defined and studied The statistical properties of these models including maximum likelihood inference, invariance and hypothesis testing are discussed 1 ntroduction Three of the most important concepts used in defining a statistical model are independence, conditional distribution and symmetry ŽThe assumption most often used in statistics is that of iid observations, that is, independent and identical distributed observations, which means independence between observations and symmetry under any permutation of the observations Statistical models given by a combination of two of these concepts, conditional distribution and independence, the so-called conditional independence Ž C models, have received increasing attention in recent years The models are defined in terms of directed graphs, undirected graphs, or the combination of the two, the so-called chain graphs See Whittaker Ž 1990 or Lauritzen Ž 1996 for an introduction to models of this type The special connections between statistical models and graphs have been the subject of many of the contributions to this area, see, for example, Andersson and Perlman Ž 1995b, Cox and Wermuth Ž 1993, Lauritzen Ž 1989, 1996, Lauritzen and Wermuth Ž 1989, or Frydenberg Ž 1990 The special class of C models where all distributions are assumed to be multivariate normal is of special interest Under this assumption, Andersson and Perlman Ž1993, 1995a hereafter abbreviated AP Ž 1993 and AP Ž 1995a, respectively introduced the so-called lattice conditional independence Ž LC models and presented a complete solution to their estimation and testing problems The relations between LC models Ž without the assumption of normality and other C models are studied in Andersson, Madigan, Perlman and Triggs Ž 1995a, b Received March 1996; revised January Supported in part by US National Security Agency rant MDA H-3083 and by NSF rant DMS Research carried out in part at the Department of Statistics, University of Washington AMS 1991 subject classifications Primary 62H12, 62H15; secondary 62H10, 62H20, 62A05 ey words and phrases roup symmetry, invariance, orthogonal group representation, quotient space, conditional independence, distributive lattice, join-irreducible elements, maximum likelihood estimator, likelihood ratio test, multivariate normal distribution 525

2 526 S ANDERSSON AND J MADSEN As part of a general development of the theory of the normal distribution, Brøns Ž 1969 presented a general definition of group symmetry Ž S models to S Andersson and S T Jensen n the years , Andersson, Brøns, and Jensen together developed an algebraic theory for these models containing a complete solution to the likelihood inference problem This basic theory, detailed in numerous Danish manuscripts eg, Andersson, Brøns and Jensen Ž 1975, Andersson Ž 1975a, 1976, Brøns Ž 1969 and Jensen Ž1973, 1974, 1977, 1983, has not yet been published Several manuscripts in English summarize the theory eg, Andersson Ž 1978, 1992 n Andersson Ž 1975b the structure of the models was explained and a solution to the estimation problem was given in a canonical form Perlman Ž 1987 reviews a small part of the theory n Andersson, Brøns and Jensen Ž 1983, the ten fundamental irreducible testing problems within this theory are discussed Andersson and Perlman Ž 1984 and Bertelsen Ž 1989 treat the noncentral distributions connected to two of these ten testing problems Since the present paper uses most of the basic theory of S models, a summary is presented in Appendix A The present paper combines the lattice conditional independence restrictions with the group symmetry restrictions to obtain the group symmetry lattice conditional independence Ž S-LC models The S models and the LC models then become special cases of the S-LC models n this paper we give necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood Ž ML estimator for an arbitrary observation, necessary and sufficient conditions for the existence and uniqueness of the ML estimator with probability 1, an explicit expression for the ML estimator, an explicit expression for the likelihood ratio statistic Q for testing one S-LC model against another and the central distribution of Q in terms of the moments EŽQ, 0 Andersen, Højbjerre, Sørensen and Eriksen Ž 1995 combine the symmetry given by the complex numbers, that is, the S condition given by the group 1, i 4, with C restrictions given by an undirected graph n Hylleberg, Jensen and Ørnbøl Ž 1993 a subgroup of the symmetric group is combined with C restrictions given by an undirected graph n both cases there is a nontrivial overlap with the models in the present paper These arise from the overlap between LC models and C models given by undirected graphs, as explained in Andersson, Madigan, Perlman and Triggs Ž 1995a, b However, in the case of Hylleberg, Jensen and Ørnbøl Ž 1993 the nontrivial overlap is also because the restriction of the interplay between the special group of permutations and the C conditions is relaxed compared to the restriction between the general S and LC conditions in the present paper Madsen Ž 1996 discusses ML estimation in a class of models which extends both the S-LC models and those of Andersen, Højbjerre, Sørensen and Eriksen Ž 1995 and Hylleberg, Jensen and Ørnbøl Ž 1993 We introduce the S-LC models by means of the following four simple examples

3 SYMMETRY AND LATTCE MODELS 527 EXAMPLE 11 Let x Ž x, x, x Ž x, x and x Ž x, x a a1 a2 b b1 b2 c c1 c2 be three pairs of random observations with a joint normal distribution with mean zero and covariance matrix Ž l, k a, b, c;, 1, 2 l, k, that is, l, k is the covariance between the two observations x l and x k For example Ž x, x, x a b c could be measurements of three different variables a, b, and c on two symmetric objects, for example, two plants within the same plot Since the joint distribution should not depend on the Ž probably irrelevant numbering of the two plants, it should remain invariant under the simple linear transformation that corresponds to permutation of plant indices This implies that has the restriction lk,, H S : l, k ½ lk, where lk kl and lk kl are real numbers, l, k a, b, c Thus, under H, the six-dimensional variable x Ž x, x, x, x, x, x S a1 b1 c1 a2 b2 c2 has the 2 2 block covariance matrix 11, where Ž l, k a, b, c and Ž l, k a, b, c lk lk This co- variance structure is a special case of multivariate complete symmetry; compare Section A6 Next, consider the assumption that x a and x c are conditionally independent given x b, which we express in the familiar nota- tion H LC : x a x c x b ž / This restriction could occur if the three measured variables correspond to three sites on the plant where a is a neighbor to b, and b is a neighbor to c, in which case the dependence between the observations from site a and site c is indirect due only to their mutual dependence on the observations from site b The lattice Ž ring of subsets of the index set 1a,2a, 1b,2b,1c,2c 4, which defines this C restriction, is given by, 1b,2b 4, 1b,2b,1a,2a 4, 1b,2b,1c,2c 4, 4, compare AP Ž 1993, Example 25 The restriction imposed on by both H S and H can then be expressed as Ž 11 LC together with the additional restriction 1 ac ac ab ab bb bb bc bc ž ac ac / ž ab ab /ž bb bb / ž bc bc / H S-LC : We thus have four hypotheses for the covariance matrix, namely the unconstrained H, the two subhypotheses H S and H LC and their intersection H S-LC Now consider N iid observations x 1,, x N of the six-dimensional random observation x t is well known that under H the ML estimator exists

4 528 S ANDERSSON AND J MADSEN and is unique with probability 1 if and only if N 6 Moreover it is well known from classical multivariate analysis that in the models H S and H LC, the required conditions are N 3 and N 4, respectively n all three cases, an explicit expression for the ML estimator is easily obtained n the case of the model H S-LC, the results in the present paper applied to this simple case show that the condition for existence and uniqueness of the ML estimator with probability 1 is N 2 The ML estimator can be found using a combination of the techniques applied for S models and LC models First, one determines the ML estimator / ˆ ˆ ˆ S ž, ˆ ˆ for under H, where ˆ Ž l, k a, b, c and ˆ Ž l, k a, b, c S ˆlk ˆlk Under H, the likelihood function Ž LF LC factorizes into the product of the conditional LF of x a given x b, the conditional LF of x c given x b and the marginal LF of x These factors then contain two 2 2 regression parameb ters R a and R c, two 2 2 conditional covariance matrices a and c and one 2 2 marginal covariance matrix The ML estimator ˆ for under H S-LC, is then determined by 1 lb lb /ž bb bb / 1 ˆ ll ˆ ll ˆ lb ˆ lb ˆ bb ˆ bb ˆ bl ˆ bl / ž /ž / ž / ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ll ll lb lb bb bb bl bl ˆ lb ˆ lb ˆ bb ˆ bb ˆR l, ž ˆ ˆ ˆ ˆ ˆ, l ž for l a, c, respectively, and ˆ bb ˆ bb ˆ b ž ˆ ˆ b bb bb / S-LC Of the five possible testing problems within the design of the models given by H, H S, H LC and H S-LC, the three involving H S-LC seem to be new The likelihood ratio statistic and its central distribution for these tests can easily be obtained from the general theory presented in this paper EXAMPLE 12 n Example 11, instead of H, consider the assumption LC H : x x, LC a c that is, x a and x c are marginally independent The interpretation of this restriction is that the actual measurements of plants on site a do not contain any information about the measurements on site c and vice versa The lattice Ž ring of subsets of the index set 1a,2a,1b,1c,2c4 which defines this C restriction is given by, 1a,2a 4, 1c,2c 4, 1a,2a,1c,2c 4, 4;

5 SYMMETRY AND LATTCE MODELS 529 compare AP Ž 1993, Example 24 The restriction imposed on by both H S and H can then be expressed as Ž 11 LC together with the additional restriction H S-LC : ac ac 0 Note that H and H LC LC are nonnested and have a nontrivial intersection We thus again consider four hypotheses for the covariance matrix, namely the unconstrained H, the two subhypotheses H and H S LC and their inter- section H S-LC Consider N iid observations x 1,, x N of the six-dimensional variable x From Example 24 in AP Ž 1993 it follows that under H, the ML estimator ance matrices, The ML estimator ˆ a c S-LC for under H S-LC, is then determined by and ˆ aa ˆ ac ˆ aa ˆ ac ˆ ab ˆ bc ˆ ab ˆ bc ˆ ac ˆ cc ˆ ac ˆ cc ˆR b, ž ˆ ab ˆ bc ˆ ab ˆ bc / ˆ aa ˆ ac ˆ aa ˆ ac ˆ ˆ ˆ ˆ ˆ b ˆ ab ˆ ˆ ˆ ˆ ˆ Rˆb ˆ ˆ ˆ ˆ ˆ ˆ ž / exists and is unique with probability 1 if and only if N 6 The results in the present paper shows that under H S-LC, the required condition is N 3 n this case, the ML estimator can be determined in the same way as in Example 11 Under H, the likelihood function Ž LF LC factorizes into the conditional LF of x given Ž x, x b a c and the marginal LFs of x a and x c, respectively These factors then contain one 2 4 regression parameter R b, one 2 2 conditional covariance matrix and two 2 2 marginal covarib LC 0 0 ac cc ac cc ab bb bb bc bc bb bb ab ab ˆ ll ˆ ll ˆ l, ž ˆ ˆ ll ll / bc for l a, c, respectively As in Example 11, the three testing problems involving H S-LC of the possible five within the design of the models given by H, H, H, and H S LC S-LC, seem to be new The likelihood ratio statistic and its central distribution for these tests can be obtained from the general theory presented in this paper EXAMPLE 13 Let x Ž x, x,, x, x Ž x, x,, x a a1 a2 an b b1 b2 bn and a b x Ž x, x,, x c c1 c2 cn be three families of n a, nb and nc multivariate c random observations, respectively The dimensions of the multivariate observations within each of the families are p, p and p, respectively The bc a b c 1

6 530 S ANDERSSON AND J MADSEN simultaneous distribution of these na pa nbpb nc pc real observations is assumed to be normal with mean vector zero and Ž n n n Ž a b c na n n block covariance matrix Ž b c l, k l, k a, b, c; 1,, n l; 1,, n k ; that is, l, k is the pl pk covariance matrix between the two multivariate observations x l and x k For example, x a, x b and x c could be multivariate measurements on plants from three different varieties a, b and c, respectively Since the joint distribution should not depend on the numbering of plants within a variety, it must remain invariant under any linear transformation of the sample space that corresponds to renumbering of plants within varieties This implies that the covariance matrix has the restrictions given by l, l k,, H S: l, k l, l k,, lk, l k, where is a p p matrix, is a p p matrix, and l l l l l l l l lk kl is a pl pk matrix, l, k a, b, c; l k Thus, under H S the random vector x Ž x, x,, x, x, x,, x, x, x,, x a1 a2 ana b1 b2 bnb c1 c2 cnc of real dimension na pa nbpb nc pc has the block covariance matrix aa ab ac Ž 12 ba bb bc, ca cb cc where l l l l l l l Ž 13 l ll 0 0 l l l l and lk lk Ž 14 lk, 0 lk for l, k a, b, c; l k This is an example of what we could call multivariate compound symmetry, first considered by Votaw Ž 1948 in the univariate case; that is, p p p 1 Ž see Section A6 a b c Next consider the assumption that the families x a and x c are conditionally independent given the family x b, which we express in the familiar notation H : x x x LC a c b lk

7 SYMMETRY AND LATTCE MODELS 531 This restriction could occur if the three families of variables correspond to three plots a, b and c where a is neighbor to b, and b is a neighbor to c, in which case the dependence between the observations from plot a and plot c is due only to the observations from plot b The lattice Ž ring of subsets of the index set a1, a2,, an a, b1, b2,, bn b, c1, c2,, cnc4 which defines this conditional independence is given by,,,,, ½ b a b b c 5 where l1, l2,, ln 4, l a, b, c; compare AP Ž 1993 l l, Example 25 The restriction imposed on by both H S and H LC can then be expressed as Ž 12, Ž 13 and Ž 14 together with the additional restriction H S-LC : ac ac 0 ac or equivalently, ac b b b b b b b b ab ab bc bc ab ab bc bc b b b b 1 S-LC ac b ab b Ž b b bc H : n n 1 We thus again have four hypotheses in the covariance matrix, namely the unconstrained H, the two subhypotheses H S and H LC, and the intersection H S-LC Let x, x,, x be N iid observations of the Ž n p n p n p 1 2 N a a b b c c - dimensional random vector x t is well known that under H the ML estimator exists and is unique with probability 1 if and only if N na pa nbpb nc p c The model given by H S is well known when pa pb pc 1; com- pare Votaw Ž 1948 t follows from the theory of S models presented in Appendix A that in the general case, the ML estimator for exists and is unique with probability 1 if and only if N p p p, Nn Ž 1 a b c a p, Nn Ž 1 p, and Nn Ž 1 a b b c p c; see Section A4 n the familiar model given by H, the condition is N maxn p n p, n p n p 4 LC a a b b c c b b ; compare AP Ž 1993, Example 25 For both models, the ML estimator is easily obtained n the case of the model H S-LC, the theory presented in the present paper shows that the conditions for existence and uniqueness of the ML estimator with probability 1 become N p p, N p p, Nn Ž a b c b a 1 p, Nn Ž 1 p and Nn Ž 1 a b b c p c The ML estimator can be found using a combination of the techniques from S models and LC models First 1,

8 532 S ANDERSSON AND J MADSEN one finds the ML estimator Ž ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ a b c a b c ab ac bc under H S Let y Ž x, x,, x 1 2 N be the N observation matrix and let yl be the p N submatrix ŽŽ x, Ž x,, Ž x l 1 l 2 l N l of y, 1,, n l, l a, b, c We then obtain that 1 l l l l ˆ Ý Ž y y 1,, n, n l 1 ˆ l Ý Ž yl yl, 1,, n l,, n Ž n 1 l l 1 lk l k l k nn l k Ý ˆ y y 1,, n, 1,, n, where l, k a, b, c, l k Under H the likelihood function Ž LF LC can be factorized into the conditional LF of x a given x b, the conditional LF of x c given x b and the marginal LF of x These factors then contain two multivariate regression parameters b R and R, of dimensions n p n p and n p n p, respectively; ab cb a a b b c c b b two multivariate conditional covariance matrices a and c of dimensions n p n p and n p n p, respectively and one marginal covariance a a a a c c c c matrix b of dimension nbpb nbp b Under H S-LC, the regression parame- ters R, l a, c and the variance parameters, l a, b, c have the form lb Ž 14 and Ž 13, respectively Thus, 0 l 0 l l l T T l l l lb lb l R and l lb l, Tlb Tlb l l l l where T is a p p matrix, l a, c and, lb l b l l l l are pl pl matrices, l a, b, c The ML estimator ˆ for under H S-LC is then determined by setting ˆ, ˆ and b b b b 1 T ˆ ˆ Ž n 1 ˆ, lb lb b b b 1 ˆ ˆ Ž ˆ ˆ ˆ l l b lb b b b bl 1 n n 1, ˆ n ˆ ˆ Ž n 1 ˆ ˆ, l l b lb b b b bl for l a, c, respectively Of the five possible testing problems within the design of the models given by H, H S, H LC, and H S-LC, the problem of testing H LC versus H is well known from the literature; compare AP Ž 1995a, and the problem of testing H versus H follows from the theory covered in Appendix A The three tests S

9 SYMMETRY AND LATTCE MODELS 533 involving the hypothesis HS-LC seem to be new The likelihood ratio test statistics and a representation of the corresponding central distributions can easily be obtained from the general theory in the present paper EXAMPLE 14 Analogously to the construction of Example 12, consider the assumption H LC : x a x c, instead of H LC in Example 13 This restriction could occur if the plants on plot a are assumed not to influence the plants on plot c and vice versa The lattice Ž ring of subsets of the index set which defines this C restriction is given by,,,, ; 4 a c a c compare AP Ž 1993, Example 24 The restriction imposed on by both H S and H can then be expressed as Ž 12, Ž 13 and Ž 14 LC together with the additional restriction H S-LC : ac 0 As in Example 13, consider N iid observations x,, x of the Ž 1 N na pa n p n p -dimensional variable x From Example 24 in AP Ž 1993 b b c c it follows that under H LC, the ML estimator exists and is unique with probability 1 if and only if N na pa nbpb nc p c, that is, the same as in the unconstrained case The results in the present paper shows that un- der H, the required conditions are N p p p, Nn Ž 1 S-LC a b c a p, Nn Ž 1 p and Nn Ž 1 a b b c p c, respectively, that is, the same as in the case of H S n this case, the ML estimator can easily be determined in the same way as in the previous examples Similarly, the three testing problems involving H S-LC of the possible five within the design of the models given by the unconstrained H, H, H, and H S LC S-LC, seem to be new, and the likelihood ratio statistic and its central distribution for these tests can be obtained from the general theory presented in this paper n general the observation space is where is a finite index set The general definition of a S-LC model is stated in terms of an orthogonal group representation of a finite group on together with a ring Ž lattice of subsets of the index set The S-LC model is then defined by imposing symmetry conditions given by and conditional independence conditions given by A condition on the interplay between the group representation and the ring is required to ensure the complete solution of the S-LC model n Section 2 the S-LC models are defined Ž Section 24, the fundamental factorization of the parameter space P Ž, of all covariance matrices determined by the S and LC restrictions is obtained Ž Theorem 21 and the fundamental invariance group L Ž, is defined together with its transitive action on P Ž Ž Theorem 22, The distribution results for the likelihood ratio statistics are greatly facilitated by this transitive action The derivations of these distributions which generalize and improve the corresponding derivations for LC models are presented in Appendix B; compare

10 534 S ANDERSSON AND J MADSEN the Appendix of AP Ž 1995a n Section 3 a necessary and sufficient condition for the existence of the ML estimator and a necessary and sufficient condition for the uniqueness of the ML estimator for a fixed observation x is obtained together with an almost explicit expression for the ML estimator Ž Theorem 31 n Proposition 32 the necessary and sufficient algebraic condition for the existence and uniqueness of the ML estimator with probability 1 is obtained The structure constants for a S-LC model are then introduced n terms of these, another, very useful, necessary and sufficient condition for the existence and uniqueness of the ML estimator with probability 1 is obtained Ž Proposition 33 Section 4 presents the general testing problem, and the likelihood ratio test statistic Q is derived The central distribution of Q in terms of the moments EŽQ, 0, is given as a function of the structure constants n Section 5, it is established that independent repetitions Ž iid of a S-LC model is again a S-LC model, except for a trivial reparametrization Furthermore, it is shown how estimators and structure constants for the iid S-LC model are obtained in terms of the original S-LC model Ž Section 51 n Section 52 it is demonstrated how to construct new examples ad libitum based on well-known examples of S models Ž cf Section A6 and the examples of LC models in AP Ž 1993 Finally, in Section 6, we indicate how the S-LC models can be extended in various ways 2 Mathematical formulation n this section we explain the mathematical set-up for the combined S-LC models to be investigated Furthermore, we present some fundamental theorems describing the structure of the set P Ž, of covariance matrices that satisfy the S-LC restrictions We have tried as much as possible to use the same type of notation as in AP Ž 1993 and Ž 1995a n the following, let and J denote finite index sets and let denote the number of elements in 21 Notation Let be the vector space of all families x Ž x i i of real numbers indexed by For, let p : be the canonical projection and u : the canonical imbedding; that is, p ŽŽ x i i Ž ŽŽ Ž x i and u x i x i i i i, where x i x i for i and x 0 otherwise For x, let x denote p x Note that 0 4 i J Let M J denote the vector space of all J matrices The algebra MŽ is denoted by MŽ For A MŽ J let A MŽ J denote the transposed matrix The group of all nonsingular matrices, the group of all orthogonal matrices, the cone of all positive semidefinite matrices and the cone of all positive definite matrices are denoted by LŽ,OŽ,PSŽ, and PŽ, respectively The action of the group LŽ on PŽ given by Ž 21 LŽ PŽ PŽ, Ž A, A A

11 SYMMETRY AND LATTCE MODELS 535 is well known to be transitive and proper The identity matrix is denoted by 1 For A Ža Ž i, i MŽ ii and, let A denote the submatrix of A; that is, A Ža Ž i, i MŽ ii f A is non- 1 singular, then A denotes the inverse matrix A 1 For any subspace U, let U denote the orthogonal complement to U wrt the usual inner product in ; that is, U x z U: xz 04 and denote by P MŽ U the corresponding orthogonal projection matrix For and P let N, denote the normal distribution on with expectation and covariance matrix Let N denote NŽ 0, The overall normal model ŽN PŽ is invariant under the action of L on the observation space given by Ž 22 LŽ, Ž A, x Ax, and the transitive action of L on the parameter space P given by The lattice conditional independence model Let be a subring of the ring DŽ of all subsets of ; that is, is closed under union and intersection Since is a distributive lattice wrt these operations, we usually refer to as a lattice of subsets of Without loss of generality we assume that, A matrix A M is called -preserving if for every and x, Ž Ax A x, or equivalently, if Au Ž u Ž Let M Ž be the algebra of all -preserving matrices in AP Ž 1993, M Ž was denoted MŽ, and let L Ž be the group of all nonsingular -preserving matrices in AP Ž 1993, L Ž was denoted LŽ Define the subset P Ž PŽ as follows: P Ž if and only if x L and x M are conditionally independent given x L M for every L, M whenever x follows N in AP Ž 1993, P Ž was denoted by PŽ The statistical model Ž 23 N P Ž with observation space and parameter space P Ž is called the lattice conditional independence Ž LC model determined by For, define ² : Ž and ² :, so that Ž 24 ² :, where indicates that the union is disjoint Let Ž denote the set of join-irreducible elements of, that is, Ž if and only if ² :, or equivalently, if The subsets of, Ž, are all disjoint, and Ž,,

12 536 S ANDERSSON AND J MADSEN n particular, Ž 25 Ž Ž see AP Ž 1993, Proposition 21 For every Ž and A MŽ, partition A according to the decomposition Ž 24 as follows: A² : A² A, A A ž / : so A MŽ² :, A MŽ² :, A MŽ ² : ² : ² : and A 1 M For P and, define : ² : ² The following five results are the main tools in solving the estimation and testing problems for models of the type Ž 23 1 The mapping P ² : Ž Ž MŽ PŽ Ž, 1 ž Ž : ² :, Ž /, is bijective AP Ž 1993, Theorem 22 ; 2 The covariance matrix P Ž if and only if tr xx tr x : ² : x² :, ž / Ý ž / for all x AP Ž 1993, Theorem 21 ; 3 For P Ž and L, Ž 27 detž L Ł ž detž Ž, L/ AP 1993, Lemma 25 n particular, Ł ž Ž / Ž 28 det det Ž ; 4 The action of the group L Ž on P Ž given by restriction of Ž 21 is well defined, transitive and proper; 5 The model Ž 23 is invariant under the action of L Ž on the observation space given by the restriction of the action Ž 22 and the transitive action of L Ž on the parameter space P Ž 23 The group symmetry model Let be a finite group and : OŽ an orthogonal group representation of on, that is, Ž 1 1 and Ž g g Ž g Ž g, for all g, g Let M Ž denote the subalgebra of all matrices A MŽ that commute with Ž, that is, AŽ g Ž g A for all g The group of all nonsingular matrices and the cone of all positive definite matrices in M Ž are denoted by L Ž and P Ž, respectively Note that P Ž if and only if PŽ and is -invariant, that is, g g Thus if x follows the distribution NŽ, where P Ž, then Ž g x follows the same distribution for all g The statistical

13 SYMMETRY AND LATTCE MODELS 537 model Ž 29 Ž N P Ž with observation space and parameter space P Ž is thus called the group symmetry Ž S model given by A summary of the basic theory of these models is presented in Appendix A; see the ntroduction The smoothing Ž averaging mapping Ž 210 :PSŽ PS Ž, 1 S Ý Ž Ž g S Ž g g, where PS Ž denoting the cone of all positive semidefinite -invariant -matrices is fundamental for likelihood inference for group symmetry models When, or both are subsumed, we denote by, and, respectively Similar to Ž 4 in Section 21, the action of the group L Ž on P Ž given by restriction of Ž 21 is well defined, transitive and proper Ž see Appendix A The model Ž 29 is invariant under the action of L Ž on the observation space given by the restriction of the action Ž 22 and the transitive action of L Ž on the parameter space P Ž 24 Models having both S and LC restrictions Let DŽ be a lattice of subsets of and : OŽ an orthogonal group representation of on The intersection P Ž P Ž is denoted by P Ž,, that is, P Ž, is the set of covariance matrices having both symmetry restrictions wrt and conditional independence restrictions given by The corresponding statistical model with observation space and parameter space P Ž is thus, Ž 211 N P Ž n the present paper we shall assume that all matrices Ž g, g, are -preserving Thus for, the mapping : OŽ given by Ž g g, g, is a well-defined orthogonal group representation of on Under this assumption, the model Ž 211 is called the group symmetry lattice conditional independence Ž S-LC model determined by and The statistical interpretation of the above condition is that all the marginal distributions NŽ, are -invariant themselves As a consequence Ž cf Lemma 21, all the matrices Ž g, g, are block diagonal wrt to the decomposition Ž 25 Symmetry conditions are thus only allowed to operate within each of the marginal variables x,,of x Example 61 presents a model of type Ž 211 where the matrices Ž g, g, are not -preserving; that is, the model has S and LC restrictions but it is not a S-LC model LEMMA 21 Let g The matrix Ž g is -preserving if and only if Ž Ž Ž Ž u is a -subspace, that is, g u u, Ž,

14 538 S ANDERSSON AND J MADSEN PROOF First assume that Ž g is -preserving Then Ž g Ž g 1 Ž 1 g is also -preserving t then follows that Ž g is block-diagonal wrt the decomposition Ž 25 see AP Ž 1993, Remark 21 This establishes the only if claim The converse is a consequence of Ž 25 From Lemma 21, it follows that for, the mapping : OŽ given by Ž g Ž g, g, is a well-defined orthogonal group repre- sentation of on Thus for g, Ž g diagž Ž g Ž,, n particular, for Ž, Ž 212 Ž g diag Ž g, Ž g ² : For Ž, denote by M Ž ² : the vector space of all ² : matrices R that commute with, that is, Ž g R R Ž g : : : ² :, g The following theorem is a generalization of Theorem 22 in AP Ž 1993 THEOREM 21 The mapping P M ² : P, Ž Ž, Ž ž Ž : ² :, Ž /, is well defined and bijective PROOF Consider any P Ž and g From Ž 212 it follows, that Ž g Ž g Ž g Ž g Ž g Ž g and : ² : : ² : ² : Ž g Ž g Ž g Ž g On the other hand, by Theorem 22 and Proposition 23 of AP Ž 1993, it follows that P Ž if and only if and Ž g Ž g Ž g Ž g 1 1 : ² : : ² : Ž g Ž g, 1 for all and g Thus P if and only if : ² : M Ž ² : and P Ž for every Ž Now we discuss some invariance properties of P Ž, compare to AP Ž 1993, Section 24 Let M Ž denote the algebra M Ž M Ž, and L Ž the group of nonsingular elements in M Ž Note that L Ž,,, L Ž L Ž The following lemma generalizes Ž 219 in AP Ž 1993 LEMMA 22 Ž 214 The mapping M Ž M ² : M Ž, Ž, is well defined and bijective ž : / A A, A,

15 SYMMETRY AND LATTCE MODELS 539 PROOF The proof is similar to that of Theorem 21 Consider any A M Ž For Ž and g, it follows from Ž 212 that and : ² : 1 1 : g A g g A g 1 1 g A g g A g On the other hand, by Ž 219 in AP Ž 1993 together with the fact that Ž g M Ž, g, it follows that A M Ž if and only if ŽŽ g AŽ g 1 A and ŽŽ g AŽ g 1 A, for every Ž : : Thus A M Ž if and only if A M Ž ² : and A M Ž : for every Ž REMAR 21 Under the correspondence Ž 214, the subset L Ž, corresponds to the subset M ² : L Ž Ž The following lemma is a generalization of Lemma 24 in AP 1993 LEMMA 23 For any element ž / R, Ž M ² : P Ž, Ž : there exists a matrix A L such that for every,, R A A 1, A A : : ² : PROOF We shall use induction on q Ž f q 1 the assertion follows from the fact that L Ž acts transitively on P Ž Ž see Proposition A1 Next, assume that the theorem holds for every ring of subsets L Žof any index set where Ž L q Furthermore, let denote a maximal element in Ž, and define the set ŽL Ž L Finally, denote by L the sublattice LL 4 of Since is maximal, it follows that Ž L Ž 4, and hence, by assumption there exists a matrix B L Ž such that for every L Ž 4,, L R B B 1, B B L: L: ²L: L L L Furthermore, because L Ž acts transitively on P Ž, there exists a matrix A L Ž such that A A Since ² :, we can define A : R : B ² :, and it is straightforward to verify that A M Ž ² : : By Remark 21, the family ŽŽ B, B L Ž L together with Ž A, A L: L : uniquely determines a matrix A L Ž,, which satisfies the required conditions

16 540 S ANDERSSON AND J MADSEN The following theorem is a generalization of Theorem 23 of AP 1993 THEOREM 22 Ž 215 The action L, Ž P, Ž P, Ž, Ž A, A A, is well defined, transitive and proper PROOF Obviously, the action is well defined since L Ž acts on P Ž see AP Ž 1993, Theorem 23 and L Ž acts on P Ž Ž see Appendix A The one-to-one correspondence Ž 213 commutes with the action Ž 215 and with the action given by the restriction of the action Ž 226 in AP Ž 1993 to the subset L Ž ŽŽM Ž ² : P Ž Ž, Therefore, by Lemma 23, the action is transitive t is proper since L Ž and P Ž,, are closed subsets of LŽ and PŽ, respectively The model Ž 211 is invariant under the action of L Ž, on the observa- tion space given by the restriction of the action Ž 22 and the transitive action of L Ž on the parameter space P Ž,, 3 Likelihood inference Consider the S-LC model Ž 31 N P Ž Ž cf Section 23 Since P Ž P Ž, it follows that the likelihood function L: P 0, for the model Ž 31 has the following factorization: Ž 32,, LŽ, x det exp 2 trž xx Ł ž ž Ž 2 Ž : ² : ² : det exp tr x x ž // / Ž Ž Now consider the problem of existence and uniqueness of the maximum ˆ likelihood estimator x of P, based on an observation x Because of Ž 32 and the factorization of the parameter space P Ž, given in Theorem 21, it suffices for each Ž to consider the problem of maximizing ž / Ž 2 Ž : ² : Ž 33 det exp tr x R x Ž Ž ² : for R M and P The subspace : 4 Ž 34 L Ž x R x R M Ž ² : ² : : ² : : of is preserved by M Ž, that is, M Ž L Ž x L Ž x ² : ² : Thus, except for the parametrization of L Ž x ² : by the regression parameter R M Ž ² :, Ž 33 : is the likelihood function for a linear group symmetry Ž LS model determined by L Ž x and, as defined in ² :

17 SYMMETRY AND LATTCE MODELS 541 Section A8 with the sample space replaced by Let P MŽ be the unique orthogonal projection matrix of onto L Ž x ² : wrt all P Ž ; compare Lemma A5 Thus for arbitrary P Ž, Ž 33 is maximized for any element Rˆ M Ž ² : : that satisfies the equation Ž 35 Rˆ : x² : P x t follows from Ž 212 that 1 Ž S ² : Ý Ž g S g ² : ² : ² : g ² : Ž S ² :, 1 Ž S Ý Ž g S g g Ž S, 1 Ž S : Ý Ž Ž g S Ž g : ² : g and Ž S Ž S ² :, where S xx is the overall empirical covariance matrix Since 1 P Ž and Rˆ M Ž ² : :, it follows that for every R M Ž ² : and g, : 0 R x x P x : ² : R x x Rˆ x Ž : ² : ž : ² :/ ž ž ˆ / / : : ² : ² : ž ž ˆ : : : ² :// ž ž ˆ : : : ² :// ž ž ˆ // ² : : : : ² : ž ž ˆ // : : ² : : ² : ² : ² : tr R x R x x tr R S R S tr R g g S R S tr g R g S g R S tr R g S g R g S g Thus for every R M Ž ² : :, tr R S Rˆ S 0 ž ž // : : : ² : ² : ˆ Because Ž S R Ž S M Ž ² :, it then follows that : : ² : ² : Ž 36 Ž S : Rˆ Ž S : ² : ² : Conversely it follows that if Rˆ is a solution to Ž 36 :, then it will satisfy Ž 35 Now define Ž 37 Ž S Ž S Rˆ Ž S ² :

18 542 S ANDERSSON AND J MADSEN From Ž 35 and the fact that Rˆ M Ž ² : :, it follows that Ž Ž x P x Ž x P x ž ž /ž / / x Rˆ x x Rˆ x : ² : : ² : Ž S Rˆ : Ž S ² Ž S and thus Ž S does not depend on the solution Rˆ to Ž 36 : t now follows from Theorem A2 that the ML estimator ˆ for P Ž exists if and only if Ž S is nonsingular n this case it is unique and given by Ž 38 ˆ Ž S The maximum of the likelihood function 33 is then ž / 12 ˆ det exp We are now able to state the following theorem regarding ML estimation in the S-LC model given by and based on the observation x THEOREM 31 n the model Ž 211, the maximum likelihood estimator ˆ ˆ x of P, for the observation x exists if and only if the matrices Ž xx, Ž, all are positive definite n this case, there is a one-to-one correspondence between all families Ž Rˆ Ž : of solutions to the equations Ž 310 Ž xx : R : Ž xx ² :, where R M Ž ² :, Ž, and all ML estimators ˆ : for x, given by Ž 311 ˆ ˆ1 R ˆ, ˆ Ž xx, : ² : : Ž Ž cf Theorem 21 The maximum likelihood estimator ˆ is unique if and only if the equations R xx 0, : ² : where R M Ž ² :, Ž :, only have the solutions R : 0, Ž The proof follows from the above considerations and Remark 31 REMAR 31 For any Ž, the following statements are equivalent: Ž a The equation Ž S R Ž S : : ² : ² : has a unique solution in R : M Ž ² : Ž b The equation R Ž S 0in R M Ž ² : : ² : ² : : only has the solution R : 0 Ž c The parametrization mapping R R x of L Ž x : : ² : ² : by R : M Ž ² : is one-to-one

19 SYMMETRY AND LATTCE MODELS 543 REMAR 32 t follows from Ž 32, Ž 39 and Ž 25, that the maximum of the likelihood function is 12 ˆ det exp REMAR 33 The explicit expression for ˆ may be obtained from Ž 311 by means of the reconstruction algorithm given in AP Ž 1993, Section 27 COROLLARY 31 n the model Ž 211, the maximum likelihood estimator ˆ ˆ x of P, for the observation x exists and is unique if the matrices Ž xx, Ž, all are positive definite n this case, ˆ is determined by 1 1 ˆ ˆ : ² : ˆ : ² : Ž 313 Ž xx Ž xx, Ž xx, cf Theorem 21 PROOF Let Ž f Ž xx is positive definite, then Ž xx ² : is positive definite and thus the equation R Ž xx : ² : 0 implies R : 0 Furthermore Ž xx Ž xx is positive definite REMAR 34 Note that the condition for existence and uniqueness of the ML estimator in Corollary 31 is equivalent to the condition that the matrices Ž xx are nonsingular for all maximum elements in the partially ordered set Ž We shall now establish a result for S-LC models which is similar to the result in Proposition A2 for S models First we need to generalize the Lemmas A2 and A4 For PSŽ the subspace N x xx 04 of is called the null-space for Note that for any B LŽ, NBB 1 B N As in Section A2, let A Ž denote the subalgebra of MŽ generated by Ž LEMMA 31 Let N be a -subspace Then for x, A x N if and only if N Ž xx N PROOF Let x First assume that A x N and let z N Ž xx From the proof of Lemma A2 it then follows that zž g x 0 for all g This implies that z is orthogonal to A Ž x and therefore z N by the assumption On the other hand, assume that N Ž xx N and that A x N This implies that there exists a z Ž A x N But from the proof of Lemma A2 it then follows that z Ž xx z 0; that is, z N Ž xx, which is a contradiction

20 544 S ANDERSSON AND J MADSEN LEMMA 32 Let N be a -subspace and an M Ž -subspace, that is, Ž N N and M Ž N N, respectively Then we have the following a The orthogonal complement N to N Ž cf Section 21 is a -subspace and an M Ž -subspace b For any x, A Ž x N if and only if A Ž P N x N, where P M is the orthogonal projection matrix of onto N Ž cf Section 21 N The latter condition states that P x is a regular element in N wrt Ž N cf Section A2 Ž c The set x A Ž x N 4, N is open, and if it is nonempty, then the Lebesgue-measure is concentrated on ; that is, has Lebesgue-measure zero, N, N PROOF t is easy to see that N is both a -subspace and an M Ž -subspace To show Ž b, observe that P M Ž N and that PN commutes with all matrices in M Ž From the bicommutant theorem cf, eg, Bourbaki Ž 1958, Section 4, Number 2, Corollary 1 it then follows that P A Ž N Now, if A Ž x N then A Ž P x P A Ž x P N N, since P M Ž N N N N On the other hand, if A Ž P x N, then A Ž x A Ž P N N x N, since PN A Ž Ž 1 To show c, note that P Ž, N N N, where N is the set of regular points in N wrt The assertion now follows from Lemma A4 with and replaced by and N, respectively N Next, for, let A Ž denote the subalgebra of MŽ generated by Ž When is subsumed, we denote A Ž by A For Ž we define and 4 ² N z : R M ² : : R z 0, ² : ² : : : ² : Ž 314, ² : x² : ² : A² : x² : E, ² : 4, where E, ² : N, ² : is the orthogonal complement to N, ² : in ² : When is subsumed we denote N, ² :,, ² : and E, ² : by N ² :, ² : and E ² :, respectively t is easily verified that N² : is a -subspace and an Ž² : ² : M -subspace of Thus by Lemma 32, E² : is also a -subspace Ž² : ² : and an M -subspace of ² : LEMMA 33 Let For any x² :, the parametrization mapping R R x of L Ž x by R M Ž ² : : : ² : ² : : is one-toone if and only if x² : ² : PROOF Let x² : ² : and assume that R : x² : 0 Then R : A² : x² : A R : x² : 0 4

21 SYMMETRY AND LATTCE MODELS ² : n particular R E 0 Thus for any z, : ² : ² : 0 R P z R z R P z R z, : E ² : : ² : : N ² : : ² : ² : ² : and it follows that R : 0 Here PN and PE denote the orthogonal ² : ² : ² : projection matrices of onto N and E, respectively Ž cf Section 21 ² : ² : On the other hand, assume that x² : ² :, that is, A² : x² : E² : E Then U Ž A x E 0 4 ² : ² : ² : ² : ² : Furthermore, U² : is a -sub- space orthogonal to N and x U Let y U 0 4 ² : ² : ² : ² : ² : Since y N, there exists R M Ž ² : such that R ² : ² : : : y² : 0 Now let R be the matrix for the linear mapping ² : defined by : R : z² : R : z² : if z² : U² : and R : z² : 0if z² : U ² : Then R M Ž ² : :, R : 0 and R : x² : 0 This shows that the parametrization mapping is not one-to-one Now for define, ² : ² : ² : N x,0 x N, and, x A x E 4, where E, N, is the orthogonal complement to N, in When is subsumed, we denote N,,, and E, by N, and E, respectively LEMMA 34 Let and x Then x if and only if Ž x x is nonsingular and the equation Ž 316 R x x : ² : Ž ² : ² : 0, Ž ² : only has the solution R 0 for R M : : PROOF Define the matrix ž / 1² : 0 B, Rˆ 1 where Rˆ is any solution in M Ž ² : to the equation Ž x x R Ž x x Then B M Ž, B is nonsingular and : : : : ² : ² : ² : Ž 317 B Ž x x B Ž x x, where / ž Ž x x ² : ² : ² : 0 Ž x x 0 Ž x x

22 546 S ANDERSSON AND J MADSEN We now have that x N N N Ž x x Ž x x N Ž x x is nonsingular and N Ž x x N ² : ² : ² : ² : Ž x x is nonsingular and A x E ² : ² : ² : Ž x x is nonsingular and Ž 316 only has the solution R 0, : where the first biimplication follows from Lemma 31, the second from Ž 317 and from the fact that Ž B 1 N N, the third from Ž 315, the fourth from Lemma 31 and the fifth from Lemma 33 and from Remark 31 Now define 4 Ž 318 x Ž : A x E PROPOSTON 31 Let x Then x if and only if the maximum likelihood estimator ˆŽ x of P Ž in the model Ž 211, exists and is unique The proof follows from Lemma 34 and Theorem 31 LEMMA 35 The set is an open subset of in f, then the Lebesgue measure on is concentrated on, that is, has Lebesgue measure zero PROOF By 318, Ž 1 p We can assume that t then follows from Ž c in Lemma 32 that 1 p is open and that the Lebesgue measure on is concentrated on 1 p Ž for all Ž Hence the same holds for PROPOSTON 32 The maximum likelihood estimator ˆ of P Ž, in the model Ž 211 exists and is unique with probability one wrt all NŽ, P Ž, if and only if, The proof follows from Lemma 35 and Proposition 31 REMAR 35 Note that Proposition 31 and Proposition 32 imply that either the maximum likelihood estimator ˆŽ x exists and is unique for almost all x or else for any x it will not exist or it will not be unique The model Ž 211 is called regular if the equivalent conditions in Proposition 32 hold

23 SYMMETRY AND LATTCE MODELS 547 REMAR 36 Consider the special case where the interplay between the representation of on and the lattice yields that N 04 ² : for all Ž Then x A x 4; that is, is the set of regular elements in wrt, Ž Ž Ḳ cf Section A2 From Lemma A2 and Proposition 31 it then follows that in this case the condition in Corollary 31 is also necessary ŽŽ Let p, d, n be the structure constants for the representation of on, Ž cf Section A3 Let, with Since Ž Ž u u are -subspaces, it follows that p p,, with equalities for all if and only if Furthermore, for ŽŽ, let p, d, n be the structure constants for the repre- sentation of on Let Since u Ž is the direct sum of Ž the -subspaces u,,, it follows that p Ž Ž Ý p,, n particular p Ý p Ž, ŽŽŽ We shall call the family p Ž, d, n the structure constants given by and for the model Ž 31 PROPOSTON 33 The S-LC model Ž 31 is regular if and only if Ž 319 Ž : p 0 n p and PROOF For, let T ² : ² : T be the unique decompositions of ² : and ² : into the orthogonal sums of their isotypic components wrt Ž cf Section A3 Since N² : and E² : are Ž² : ² : -subspaces and M -subspaces of, they are both orthogonal sums ² : of some of the isotypic components T,, cf, eg, Bourbaki Ž 1958, Section 3, Number 4, Proposition 11 t is easy to see that for, T ² : N² : if and only if T ² : and T are disjoint, or equivalently, if ² : T 0orT 0 Ž cf Section A3 From this it follows that ² : Ž 320 N² : Ž T, p 0 and ² : E² : T, p 0 From the definition of N and E it then follows that and Ž N T, p 0 Ž 321 E T, p 0,

24 548 S ANDERSSON AND J MADSEN where T is the unique decomposition of into the orthogonal sum of its isotypic components wrt By Lemma 32, if and only if the set of regular elements in E wrt is nonempty The proposition now follows from Ž A3 and Ž 321 REMAR 37 t follows from Ž 320 that N² : 0 if and only if for all, p ² : 0 when p 0 Thus in the special case where N² : 0 for all Ž Ž cf Remark 36 and Corollary 31, Ž 319 reduces to the condition that n p, for all and REMAR 38 For the model H S in Example 11, the family of structure constants becomes Ž p, d, n Ž p, d, n Ž 3, 1, , and the family of Ž L structure constants for the model H then simply becomes p, d, n S-LC j j j Ž 1, 1, 1, L a1, a2 4, b1, b2 4, c1, c2 4, j 1, 2 This family is essentially also the family of structure constants for the model H S-LC in Example 12, but since Ž Ž, the regularity conditions become different For the model H S in Example 13, the family of structure constants become Ž p, d, n Ž p p p,1,1, Ž p, d, n Ž p,1,n 1, a b c a a a a a Ž p, d, n Ž p,1,n 1, and Ž p, d, n Ž p,1,n 1, respec- b b b b b c c c c c tively The family of structure constants for the model H by p a p, p b p, p c 0 a 0 b 0 p c and ½ S-LC is then given m p l, l m, p l 0, l m, for l, m a, b, c Similarly this family is essentially also the family of structure constants for the model H S-LC in Example 14, but in this case too, the regularity conditions become different ² : LEMMA 36 Let, x, and let Žl Ž x ² : ² : be the M Ž -dimension of the M Ž -subspace 4 L x R x R M ² : ² : : ² : : Ž cf 34 and Remark A6 f x, then ² : ² : l Ž x ½ p ² :, p 0, 0, p 0, ² : PROOF The vector spaces M Ž ² : and L Ž x ² : are both M Ž -modules under multiplication with the matrices in M Ž to the

25 SYMMETRY AND LATTCE MODELS 549 left, and the parametrization mapping M ² : L Ž x ² :, Ž 322 R : R : x ² :, commutes with the module structures, that is, Ž A R : x² : A Ž R : x² : for all A M Ž and R M Ž ² : : f x² : ² :, then it follows from Lemma 33 that Ž 322 is an isomorphism between the two M Ž -modules, and hence they have the same M Ž -dimension n particular, l l Ž x ² : does not depend on x² : ² :, Let x and define * p 04 ² : ² : t is easy to see that l 0 when p 0,, and thus we have that *: n p l : n p l The latter condition is by Remark A7 equivalent to the condition that there exists x such that Ž 33 with x Ž x, x ² : has a unique maximum for Ž R, M Ž ² : P Ž By Lemma 34 and Theorem 31 this is equivalent to the condition that there exists x such that x Ž x, x t then follows from Proposition 33 that ² : *: n p l *: n p, and since p p ² : p, the lemma follows REMAR 39 Assume that the model Ž 31 is regular and let Ž Then x is regular with probability one Žcf Ž 314 and Lemma 32 ² : ² : The function Ž 33 is the likelihood function for the conditional model of x given x Except for the parametrization of L Ž x ² : ² : by the regression parameter R M Ž ² : :, this is the likelihood function for a LSmodel given by L Ž x² : and Ž cf Section A8 By Ž A13 and Lemma 36, it then follows that the distribution of detž ˆ detž given x² : is the same as a product of independent variables ž / dn Ł Łž j / X j 1,, p, 2 Ž where X follows a distribution with d f j 1 j degrees of freedom, scale Ž d n 1 and f n p ² : Thus the distribution is independent of x² : ² : with probability 1 and is therefore also the distribution of detž ˆ detž 4 Testing problems n this section we consider the problem of testing additional conditional independence andor symmetry restrictions

26 550 S ANDERSSON AND J MADSEN Let M denote a subring of and H a subgroup of such that for all h H, Ž h M Ž Since P Ž P Ž M, H, M, the problem of testing the S-LC model Ž 41 Ž N P, Ž versus the S-LC model Ž 42 N P Ž, is well defined compare AP Ž 1995a n the usual statistical language, this is the problem of testing the hypothesis H, : P, Ž versus H H, M : PH, M Ž, based on a random observation x from a normal distribution NŽ, where P Ž H, M ² : REMAR 41 Quantities such as and depend not only on the subset of but also on the lattice of which is considered a member To alleviate this difficulty, the letter shall denote a subset of that is to be considered as a member of, while M shall denote a subset of that is to be considered as a member of M PROPOSTON 41 Let, and H, M denote the set of regular elements for the S-LC models Ž 41 and Ž 42, respectively cf Ž 318 Then H, M n particular if the S-LC model Ž 42, is regular, then the S-LC model Ž 41 is regular PROOF Let x For M Ž M we have M ŽM ² M : H, M M ŽM ² M : H and thus N, ² M : NH, ² M : which implies that E, M E Since M is a sublattice of, it follows see Andersson Ž 1990 H, M, Proposition 33 that there exists a surjective mapping : Ž Ž M such that Ž for all Ž and H, M Ž 43 M Ž Ž, Ž M, for all M Ž M, respectively Let Ž and let M Ž Moreover M let p : denote the coordinate projection, that is, p Ž x, M, M M x t follows easily that 4 N, y A MŽ : Ay 0, where M Ž, is the vector space of all matrices that commute with Analogously 4 M N, M y A MŽ M M : Ay 0 Since p, M commutes with the group representations on M and, it then follows that p Ž N N, and thus p Ž E E Since, M, M,, M, M,

27 SYMMETRY AND LATTCE MODELS 551 A Ž H A Ž M Ž M Ž, it follows that for any A A Ž H M, A x Ž Ax Ž Ax Ž A x M M M These considerations and the fact that A Ž H A Ž, shows that and hence x A Ž x A Ž H x A Ž H x, M M E E E H, M, M, Suppose that the normal model Ž 42 is regular t then follows from Proposition 41 and Ž 312 that the LR test statistic Q for testing H, against H exists with probability 1 and is given by H, M ž / 12 det Ž x Ž 44 QŽ x, det ˆ Ž x where x and is the MLE for P Ž in the model Ž 42 H, M Next we shall find the moments of the LR test statistic Q First note that L Ž L Ž, H, M The testing problem is invariant under the action of the group L, on the sample space given by the restriction of the action Ž 22, and the action of L Ž on the parameter space P Ž, H, M given by the restriction of the action of L Ž on P Ž Ž H, M H, M cf Theorem 22 with replaced by M and replaced by H Let : L, Ž denote the orbit projection Ž maximal invariant of the action of L Ž, on The LR test statistic Q is invariant under this action and thus QŽ x only depends on x through Ž x The central distribution of Q is then readily obtained from this fact and the theorem below THEOREM 41 Suppose that the normal model Ž 42 is regular Under H the orbit projection and the ML estimators ˆ of P Ž,, Ž, are mutually independent For the proof, see Appendix B t follows from Ž 44, Ž 28 and Theorem 41 that for every P Ž, and 0,, Hence ˆ ˆ Ež det / Ež det Q / Ež det / EŽ Q 2 ž Ž / E det Ž 45 EŽ Q 2 E detž ˆ ž / Thus it suffices to determine the moments EŽdetŽ ˆ 2, where ˆ is the MLE of P Ž in the S-LC model Ž 41 Furthermore it follows from Ž 28,