Supplementary Materials: Proofs and Technical Details for Parsimonious Tensor Response Regression Lexin Li and Xin Zhang

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1 Suppleentary Materials: Proofs and Tecnical Details for Parsionious Tensor Response Regression Lexin Li and Xin Zang A Soe preliinary results We will apply te following two results repeatedly. For a positive definite atrix B, In addition (Kolda, 2006; Proposition 3.7), B = arg in A>0 {log A + tr(a B)}. A = B C 2 N C N A (n) = C n B (n) (C N C n+ C n C ) T. For te envelope paraeterization of B = Θ; Γ,..., Γ, I p, we ave B (+) = Θ (+) (Γ T Γ T ), and Θ (+) = B (+) (Γ Γ ). As for Y = Z; Γ,..., Γ R r r, we ave te following results by treating Y as (+)- t order tensor wit r + =, and treating vec T (Y) as te ode-( + ) atricization. vec T (Z) = vec T (Y Γ T 2 Γ T ) = vec T (Y)(Γ Γ ), vec(z) = (Γ T Γ T )vec(y). B Proof for Proposition and 2 We first prove Proposition. Ten Proposition 2 follows directly fro te results of Proposition and te definitions of reducing subspace and Tucer decoposition.

2 Under te tensor linear odel Y = B (+) X + ɛ, we can write Y Q = (B (+) X) Q + ɛ Q = (B Q ) (+) X + ɛ Q, wic iplies tat Y Q X Y Q is equivalent to B Q = 0. Siilarly, Y Q Y P X is equivalent to ɛ Q ɛ P, were te independence of two tensor-valued rando variable is defined as independence of teir vectorized fors: vec(ɛ Q ) vec(ɛ P ). Because of te tensor noral distribution, te independence is equivalent to cov{vec(ɛ Q ), vec(ɛ P )} = 0, and ence to Σ Σ + Q Σ P Σ Σ = 0. Terefore, we ave sown tat Y Q Y P X Q Σ P = 0 Σ = P Σ P + Q Σ Q. C Proof for Proposition 3 We first sow tat for subspaces E = E E and covariances Σ = Σ Σ, were E R r and Σ R r r, =,...,, te two conditions are equivalent: () E reduces Σ; (2) E reduces Σ for all =,...,. Te stateent (2) iplies () is straigtforward. We need to sow tat (), E = E E reduces Σ = Σ Σ, iplies (2), E reduces Σ for all =,...,. By definition, () iplies tat P E E (Σ Σ )Q E E = 0, were Q E E = I P E E = I P E P E. Expanding tis, we see tat P E Σ P E Σ P E Σ P E P E Σ P E = 0. If we rigt-ultiply P E P E2 I r to bot sides of te above equation, we obtain P E Σ P E P E2 Σ 2 P E2 (P E Σ P E Σ P E ) = 0, 2

3 wic iplies tat P E Σ P E Σ P E = 0. Since P E Σ = P E Σ (P E + Q E ), we conclude tat P E Σ Q E = 0 and ence E reduces Σ. Using te siilar arguent, we can see tat E reduces Σ, for all =,...,. We next sow tat E = E E contains span(b T ) iplies E (+) contains span(b () ). Let G R r q be sei-ortogonal atrix suc tat span(g ) = span(b () ), ten we ave Tucer decoposition of B as B = η; G,..., G, I p for soe η R q q p. Terefore we can write B T = (G (+) G )η T. Hence E = E (+) E contains span(b T ) (+) iplies tat E contains span(g ) = span(b () ). D Proof for Lea Here we ai to sow tat Σ (0), te Kronecer covariance estiator fro Manceur and Dutilleul (203), is a n-consistent estiator for Σ, for all, if ε i, i =,..., n, are i.i.d. fro tensor noral distribution wit ean 0 and covariances Σ,..., Σ. First we note tat te ode- atricization ε i() R r ( j r j ), i =,..., n, are i.i.d. fro a atrix noral distribution wit ean 0 and covariances Σ (te row covariance atrix) and Σ Σ + Σ Σ (te colun covariance atrix). Ten following Gupta and Nagar (999; Capter 2, Teore 2.3.5), we ave te following second-order oents: E(ε i() ε T i() ) = Σ tr( ), were tr( ) is te trace operator of atrices. In te iterative algorit of obtaining Σ (0), te starting value for Σ(0) is n j r j n e i() e T i(), wic is tus n-consistent for Σ up to a scalar difference due to tr( ). As we discussed in Section 4. regarding te identifiability of Σ s, te scalar difference can be resolved after noralizing Σ (0) = τσ (0) Σ (0) according to te scalar τ = (n j r j ) n vec(e i ){(Σ (0) ) (Σ (0) ) }vec T (e i ) at te end of eac iteration. 3

4 Terefore, after te first iteration, we ave obtained n-consistent estiators Σ (0) for Σ. In te iterations tat follow, te updating equations of Σ (0) for =,...,, are obtained by axiizing te tensor noral lieliood function. Hence it is guaranteed tat te final estiators of Σ (0) fro te algorit are also n-consistent for Σ. E Proof for Teore E. Consistency of te one-step estiator Fro Coo and Zang (206; Proposition 6) we now tat if M and Û are n-consistent estiators for M > 0 and U 0, ten te D algorit for iniizing J n (G) = log G T MG + log G T ( M + Û) G produce n-consistent estiator for te projection onto te envelope E M (U). We use tis result to prove te n-consistency of P os Γ, noting tat f (0) and Algorit 2 are special instances of J n and te D algorit. First Σ (0), te Kronecer covariance estiator fro Manceur and Dutilleul (203), is n- consistent estiator for Σ. Next, by coparing f (0) (G ) to J n (G) we see tat N (0) = (n j r j ) n Y i() Σ (0) YT i() is analogous to ( M + Û) in J n(g), were we ave defined Σ (0) ((Σ(0) ) (Σ (0) + ) (Σ (0) ) (Σ (0) ) ). Terefore we focus on N (0) Σ (0), wic is analogous to Û in J n(g). Recall tat Y i = B (0) (+) X T i +e i and tat Σ (0) is obtained based on e i as Σ (0) = (n j r j ) n e i() Σ (0) et i(). Hence, N (0) Σ (0) = (n j r j ) n (Y i e i ) () Σ (0) (Y i e i ) T (), were we recognize Y i e i = B (0) (+) X T i. Define te following scaled regression coefficient tensor B {} = B; Σ /2,..., Σ /2, I r, Σ /2 +,..., Σ /2, Σ /2 X. Ten we see tat N (0) Σ (0) = ( j r j ) B {} ( B {} ) T for B {} = B; (Σ (0) ) /2,..., (Σ (0) ) /2, I r, (Σ (0) + ) /2,..., (Σ (0) ) /2 /2, Σ X, 4

5 were Σ X = n n X i X T i. We clai tat N(0) Σ (0) is a n-consistent estiator for B {} () (B{} () )T up to an upfront scaling constant, ( j r j ), based on te fact tat te saple atrices Σ (0) j and Σ X are asyptotically independent of te OLS estiator B (0). By Coo and Zang (206; Proposition 6) we now see tat P os Γ is a n-consistent estiator for te projection onto te envelope E Σ (B {} () (B{} () )T ) = E Σ (B {} ). By definition of () B{} and te property of Tucer operator, we ave B {} = B () () (Σ /2 Σ /2 Σ /2 + Σ /2 Σ /2 X ), wic iplies span(b {} () ) = span(b ()), and ence E Σ (B {} () ) = E Σ (B () ). So far we ave sown tat P os Γ is a n-consistent estiator for te projection onto te envelope E Σ (B () ). Te second part of te proposition is based on te n-consistency of B (0) = B OLS and P os Γ, =,...,, and te definition of B os : B os = B (0) ; P os Γ,..., P os Γ, I p. E.2 Consistency of te lieliood-based estiator Te n-consistency of te lieliood-based estiator fro iniizing l(b, Σ) relies on Sapiro s (986) results on te asyptotics of over-paraeterized structural odels. Te proof is parallel to te proof of Proposition 4 in Coo and Zang (205) and is tus oitted. F Proof for Teore 2 (including derivations for te updating equations in Section 4.) Following te discussion in Section 5.2 of te paper, te iterative estiator is not guaranteed to be necessarily te axiu lieliood estiator (MLE), due to te existence of ultiple local inia. However, it is asyptotically equivalent to te MLE. Tis is because, under te tensor noral distribution, te initialization of Algorit is built upon n-consistent estiators, wile eac paraeter in Algorit is iteratively obtained along te partial derivative of 5

6 te log-lieliood. Fro te classical teory of point estiation, we now tat one Newton- Rapson step fro te starting value provides an estiator tat is asyptotically equivalent to te MLE even in te presence of ultiple local inia (Leann and Casella, 998, p. 454). Consequently, to prove Teore 2, we only focus on te asyptotic properties of te teoretical MLE. Specifically, we need to derive all te objective functions and updating equations in Section 4. fro te negative noral log-lieliood function: l(b, Σ) = log Σ + n n {vec(y i ) B T (+) X i}σ {vec(y i ) B T (+) X i}, wic is to be optiized over Σ = Σ({Γ, Ω, Ω 0 } = ) and B = B({Γ } =, Θ) for seiortogonal Γ R r u, positive definite and syetric Ω S u and Ω0 S r u, and Θ R u u p. Since it is ipossible to obtain explicit fors of MLEs, we sow tat te series of equations used in Algorit coe fro partially iniizing l wit specified paraeters fixed. We suarize our findings in te following stateents and give detailed derivations iediately after. Note tat te updating equations in Section 4. is obtained fro te following equations by superscripting (t) on te left and sides and superscripting (t+) on te rigt and sides of equations. Under te tensor noral assuption, te MLEs satisfy te following equations, Θ = Z (+) {(XX T ) X}, B = Y P Γ 2 P Γ (+) {(XX T ) X}, = B OLS ; P Γ,, P Γ, I p Ω = n s i() {Ω Ω + Ω Ω }s T i() n j r j Ω 0 = n n j r j Γ T 0Y i() {Σ + Σ Σ }Y T i() Γ 0, were te data tensors Z i and Z are defined according to Z = Y; Γ T,..., Γ T, and te residual tensor s i = Z i Θ (+) X i. Under te tensor noral assuption, te MLE of {Γ } = can be 6

7 obtained as iniizer of te following objective function were M, N are defined as M = (n j N = (n j f (Γ ) = log Γ T M Γ + log Γ T N Γ, r j ) n δ i() (Σ + Σ Σ )δ T i(), r j ) n Y i() (Σ + Σ Σ )Y T i(), were δ i() is te -t atricization of te residual δ i, δ i = Y i Y P Γ 2 ( ) P Γ (+) P Γ+ +2 P Γ (+) X T i{(xx T ) X} = Y i B OLS ; P Γ,..., P Γ, I r, P Γ+,..., P Γ, I p (+) X i. F. Estiation of oter paraeters given {Γ } = We first decopose te log Σ ter as log Σ = log Σ Σ = = {( r ) log Σ } = j = {( r )(log Ω + log Ω 0 )}, wic is essentially 2 additive ters of log Ω and log Ω 0. Te conditional log-lieliood can be separated into two independent parts regarding P(Y) X and Q(Y) Q(Y) X. Studying te regression of P(Y) on X is essentially studying tat of Z = Y; Γ T,..., Γ T on X. Following te discussion in te paper, we ave te MLEs for Θ, B and {Ω } = given {Γ } =. We next derive te MLE equations for {Ω 0 } = wit given {Γ } =. Witout loss of generality, we write down our derivations wit respect to Ω 0. We decopose Σ = + 0, according to Ω and Ω 0 in te decoposition of Σ = Γ Ω Γ T + Γ 0 Ω 0Γ T 0. = Σ 2 (Γ Ω Γ T ) = (Γ Ω Γ T + Γ 0 Ω 0Γ T 0) (Γ 2 Ω 2 Γ T 2 + Γ 02 Ω 02Γ T 02) (Γ Ω Γ T ), 0 = Σ 2 (Γ 0 Ω 0Γ T 0) = (Γ Ω Γ T + Γ 0 Ω 0Γ T 0) (Γ 2 Ω 2 Γ T 2 + Γ 02 Ω 02Γ T 02) (Γ 0 Ω 0Γ T 0). j 7

8 Hence, we can write te negative partial log-lieliood for solving Ω 0 as l(ω 0 {Γ } = ) ( r j ) log Ω 0 + n n {vec(y i ) B T (+) X i} T 0 {vec(y i ) B T (+) X i}. j> Recall tat B (+) = η (+) (Γ T Γ T ), ence 0 B T (+) = 0 as a result of Γ 0Ω 0Γ T 0 Γ = 0. Terefore, l(ω 0 {Γ } = ) ( r j ) log Ω 0 + n n vec T (Y i ) 0 vec(y i ). j> Te quadratic for vec T (Y i ) 0 vec(y i ) equals te squared nor of vec(y i Ω /2 0 Γ T 0 2 Σ /2 2 Σ /2 ) vec(v i ). By definition of tensor nor, V i 2 = vec(v i ) 2 = V i() 2 F = tr(v T i() V i()), were V i() is te ode- atricization of V i : Hence V i() = (Ω /2 0 Γ T 0)Y i() (Σ /2 /2 2 ). tr(v T i() V i()) = tr(v i() V T i() ) = tr {Ω 0Γ T 0Y i() (Σ 2 ) Y T i() Γ 0}. Te partial conditional log-lieliood becoes l(ω 0 {Γ } = ) ( r j ) log Ω 0 + n n tr(v T i() V i()) j> = ( j> r j ) log Ω 0 + n tr [Ω 0 n wic lead to te following equations for iteratively solving Ω 0. Ω 0 = (n j> {Γ T 0Y i() (Σ 2 ) Y T i() Γ 0}], r j ) n Γ T 0Y i() (Σ 2 )Y T i() Γ 0, were Σ = Γ Ω Γ T + Γ 0 Ω 0 Γ T 0 for =,...,. It is ten easy to obtain te following result, for any, Ω 0 = (n j r j ) n Γ T 0Y i() (Σ + Σ Σ )Y T i() Γ 0. 8

9 F.2 Estiation of Γ given {Γ, Ω, Ω 0 } =2 Treating te oter paraeters {Γ, Ω, Ω 0 } =2 as fixed constants, we write B = B(Γ ) and Σ = Σ(Γ, Ω (Γ ), Ω 0 (Γ )) as functions of Γ. We ten plug te into te lieliood l(b, Σ) to partially optiize over Ω and Ω 0 analytically, and ten te objective function for optiizing over Γ R r u as follows. First, ignoring all te fixed constants, te log-deterinant ter in l(b, Σ) becoes log Σ ( r j ){log Ω (Γ ) + log Ω 0 (Γ ) }. j=2 Siilar to te previous section, we can decopose Σ = Σ Σ into two parts according to te decoposition Σ = Γ Ω Γ T + Γ 0 Ω 0Γ T 0. Ten we can write = n {vec(y i ) B T (+) X i}σ {vec(y i ) B T (+) X i} n tr {Ω 0Γ T 0Y i() (Σ 2 )Y T i() Γ 0} + n tr {Ω Γ T e i() (Σ 2 )e T i() Γ }, were e i e i (Γ ) = Y i B(Γ ) (+) X T i and B(Γ ) = Y P Γ 2 P Γ (+) {(XX T ) X}. Ten, Γ T e i() = (Y i Γ T Y Γ T P Γ 2 P Γ (+) X T i{(xx T ) X}) () = (Y i Γ T Y Γ T 2 P Γ (+) X T i{(xx T ) X}) () = {(Y i Y I r 2 P Γ (+) X T i{(xx T ) X}) Γ T } () = Γ T δ i(), were δ i = (Y i Y I r 2 P Γ (+) X T i {(XXT ) X}) does not involve Γ. Te partially axiized negative log-lieliood now becoes 9

10 l(b, Σ) = log Σ + n n {vec(y i ) B T (+) X i}σ {vec(y i ) B T (+) X i} ( r j ){log Ω (Γ ) + log Ω 0 (Γ ) } j=2 + n n tr {Ω 0Γ T 0Y i() (Σ 2 )Y T i() Γ 0} + n n tr {Ω Γ T δ i() (Σ 2 )δ T i()γ }, wic leads to partial MLE of Ω (Γ ) and Ω 0 (Γ ) as Ω (Γ ) = (n j=2 Ω 0 (Γ ) = (n j=2 r j ) n Γ T δ i() (Σ 2 )δ T i()γ r j ) n Γ T 0Y i() (Σ 2 )Y T i() Γ 0. Ten, substitute tese bac to l(b, Σ) to get te lieliood-based objective function for Γ : F n (Γ ) = log (n j=2 + log (n j=2 r j ) n Γ T δ i() (Σ 2 )δ T i()γ r j ) n Γ T 0Y i() (Σ 2 )Y T i() Γ 0 log Γ T M Γ + log Γ T N Γ, were M = (n j=2 r j ) n δ i() (Σ Σ 2 )δ T i() and N = (n j=2 r j ) n Y i() (Σ Σ 2 )Y T i(). Te last step of te above equations coe fro te fact tat log ΓT 0AΓ 0 log Γ A Γ for any positive definite syetric atrix A. G Proof for Teore 3 Fro te proof of Teore 2, we ave seen tat B Γ = B OLS ; P Γ,..., P Γ, I p, were P Γ, =,...,, are all true projections onto te envelopes (i.e. population values). Ten vec( B Γ ) = (I p P Γ P Γ )vec( B OLS ) = P t vec( B OLS ). Te OLS estiator vec( B OLS ) is n-consistent and asyptotically noral wit ean zero covariance equals to U OLS = 0

11 Σ X Σ. Since P t = I p P Γ P Γ is fixed as population trut, we ave proven tat te envelope estiator B Γ is n-consistent and asyptotically noral wit ean zero covariance U Γ = P t U OLS P t = Σ X P Γ Σ P Γ P Γ Σ P Γ. H Proof for Teore 4 Recall tat te paraeter vectors involved in tis Teore are: = ( 2 ) = ( vec(b) vec(σ) ), φ = φ φ 2 φ + = vec(b) vec(σ ) vec(σ ), ξ = ξ ξ 3+ were ξ = vec(θ), {ξ j } + j=2 = {vec(γ )} =, {ξ j} 2+ j=+2 = {vec(ω )} =, {ξ j} 3+ j=2+2 = {vec(ω 0 )} =. Te lengt of vectors are onotonically decreasing fro to φ and ten to ξ, because tey corresponding to tree nested odel assuptions: (M) corresponding to te unrestricted, vectorized linear regression odel, or te OLS odel; (M2) φ is fro te Kronecer covariance assuption, so letting = (φ) is essentially iposing te Kronecer atrix structure tat Σ = Σ Σ ; (M3) ξ is based on envelope assuptions of E Σ (B () ), =,...,. Fro Teore 2, we now tat ( ) = (φ ( ) ) = (ξ ( ) ) is te MLE under te envelope assuption, i.e. odel assuption (M3). Also, (0) = (φ (0) ) contains te OLS estiator and te Kronecer covariance estiator fro Manceur and Dutilleul (203). Hence it is te MLE under tensor noral assuption, i.e. odel assuption (M2). Since = (φ) = (ξ) is overparaeterized, fro Sapiro s (986) we see te following results: n( (0) True ) N(0, V 0 ) and n( ( ) True ) N(0, V ), were V 0 = H(H T J H) H T, V = K(K T J K) K T, and te gradient atrices H = (φ)/ φ and K = (ξ)/ ξ. Moreover, te Fiser inforation atrix J as te sae for as in te usual vector-response linear regression odel, J = ( Σ X Σ ET d (Σ Σ )E d ),

12 were te expansion atrix E d R d2 d(d+)/2 as te corresponding diension d = = r. Ten, H(H T J H) H T = J /2 J /2 H(HT J /2 J/2 H) H T J /2 J /2 = J /2 and siilarly, K(K T J K) K T = J /2 P /2 J KJ /2. By cain rule, we can write K = (ξ)/ ξ = (φ)/ φ φ(ξ)/ ξ = H φ(ξ)/ ξ. P /2 J HJ /2, Terefore span(k) span(h) and span(j /2 K) span(j/2 H), wic iplies tat P J /2 P /2 J HP = P J /2 K J /2 KP J /2 H. Finally, we ave arrived at V 0 V = H(H T J H) H T K(K T J K) K T = J /2 P /2 J HJ /2 J /2 P /2 J KJ /2 = J /2 (P P /2 J H J /2 = J /2 (P P /2 J H J /2 HP J /2 = J /2 P /2 J HQ J /2 KJ /2 0. K) J /2 J /2 K) K = So far, we ave proved te ain part of te Proposition 6. We next provide details about te gradient atrices H and K. First of all, since = φ = vec(b), we ave H = ( I p = r 0 0 vec(σ) 0 vec(σ ) vec(σ) vec(σ ) ). (H) For a syetric atrix A R a a, vec(a) = C a vec(a) and vec(a) = E a vec(a), were C a R a(a+)/2 a2 is te contraction atrix and E a R a2 a(a+)/2 is te extraction atrix. Terefore, vec(σ) vec(σ ) = C vec(σ) j= r j vec(σ ) E r, =,...,. We ten use te Kronecer structure Σ = Σ Σ to calculate vec(σ) vec(σ ). For = and =, we can use te forulas fro Facler (2005) for te derivatives of vec(a B) over vec(a) and over vec(b). For 2, tere is no copact way of writing down te atrix for but eleentwise derivatives, wic is straigtforward but non-trivial. 2

13 Next we write K as K = H φ(ξ) ξ = HR, R = ( ( vec(b) ξ ) T ( vec(σ ) ξ ) T ( vec(σ) ξ ) T ) T, (H2) were we calculate eac blocs of R in te following. Recall tat ξ = (ξ T,..., ξ T 3+) T and ξ = vec(θ), {ξ j } + j=2 = {vec(γ )} =, {ξ j} 2+ j=+2 = {vec(ω )} =, {ξ j} 3+ j=2+2 = {vec(ω 0)} =. Te first bloc in R is vec(b) ξ = ( vec(b) vec(θ) vec(b) vec(γ ) vec(b) vec(γ ) 0 0 ), (H3) were te zeros are because of B does not depend on Ω or Ω 0. We re-write vec(b) as vec(b) = vec(b T (+) ) = vec ((Γ Γ )Θ T (+) ) = (I p Γ Γ )vec(θ T (+)) = (I p Γ Γ )vec(θ). Tus, vec(b) vec(θ) = (I p Γ Γ ). We next introduce te notation of re-arranging te vectorizations of a ode-n tensor T R d d N : squared constant atrixπ T n satisfies: vec(t) = Π T n vec(t (n) ). Terefore, vec(b) = Π B vec(b () ) and ence vec(b) = Π B vec (Γ Θ () (Γ T Γ T + Γ T Γ T )) = Π B ((Γ Γ + Γ Γ )Θ T () I r ) vec(γ ), (H4) vec(b) vec(γ ) = ΠB ((Γ Γ + Γ Γ )Θ T () I r ). Finally for Σ = Γ Ω Γ T + Γ 0 Ω 0 Γ T 0, we ave (H5) vec(σ ) ξ = ( 0 0 vec(σ ) vec(γ ) 0 0 vec(σ ) vec(ω ) 0 0 vec(σ ) vec(ω 0 ) 0 0 ), (H6) 3

14 were te tree nonzero eleents are, (analogous to Coo et al. (200)), vec(σ ) vec(γ ) vec(σ ) vec(ω ) vec(σ ) vec(ω 0 ) = 2C r (Γ Ω I r Γ r Γ 0 Ω 0 Γ T 0), = C r (Γ Γ )E u, = C r (Γ 0 Γ 0 )E r u. Finally te explicit gradient atrices H is obtained by plugging eac blocs (H3) (H6) into (H); siilarly, te explicit gradient atrices K is obtained by plugging eac blocs (H3) (H6) into (H2). We ave tus copleted te proof of tis teore. Additional References Coo, R. D., Li, B. and Ciaroonte, F. (200), Envelope odels for parsionious and efficient ultivariate linear regression, Statistica Sinica, 20(3), Coo, R. D. and Zang, X. (205), Siultaneous envelope for ultivariate linear regression, Tecnoetrics, 57(), 25. Coo, R. D. and Zang, X. (206), Algorits for envelope estiation, Journal of Coputational and Grapical Statistics, In press. Facler, P. L. (2005). Notes on atrix calculus. ttp://www4.ncsu.edu/~pfacler/ MatCalc.pdf, Tecnical Report. Gupta, A. and Nagar, D. (999), Matrix variate distributions, CRC Press. Kolda, T. G. (2006). Multilinear operators for iger-order decopositions. United States, Departent of Energy, Tecnical Report. Leann, E. L. and Casella, G. (998). Business Media. Teory of point estiation, Springer Science & 4

15 Manceur, A. M. and Dutilleul, P. (203), Maxiu lieliood estiation for te tensor noral distribution: Algorit, iniu saple size, and epirical bias and dispersion, Journal of Coputational and Applied Mateatics, 239, Sapiro, A. (986), Asyptotic teory of overparaeterized structural odels, Journal of te Aerican Statistical Association, 8,