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Thelen has been accepted towards fulfillment of the requirements for Ph.D. degree in Statistics "==£Ziflgi‘2**fidalliitziz__‘_ # Major professor Date November 13, 1986 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 F '58?» \ MSU LIBRARIES —,_- RETURNING MATERIALS: ace in 00 rop to remove this checkout from your record. FINES will be charged if booE is returned after the date stamped below. FISHER INFORMATION AND DICHOTOMIES IN CONTIGUITY/ASYHPTOTIC SEPARATION by Brian J. Thelen A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1986 VJ 7/9/55 0 there exists h 6 L1(0.3.v) such that I (Ifal-h)+ dv < e for all a e I. where (x)+ = max(x.0) for all x G R. For more details regarding uniform integrability in this general framework. see Fabian/Hannan (1985). Section 4.8. or Bauer (1981). Section 2.12. A simple but useful result. which is proved in the appendix as Proposition A.1. is that since u is a-finite. (fa: a e I) C L1(v) is u.i. if and only if for every sequence (an) C I. there exists a subsequence (an.} such that {fa } is u.i.. n Let (On.$n.(Pn.Pn}) be a sequence of experiments. The sequence {PD} is contiguous to the sequence (Pu) (PD 4 P“) if for each sequence {Fn} where Fn 6 9n and Pn(Fn) 9 O as n 9'”. implies Pn(Fn) a 0 as n a m. The sequences {Pn} and (PD) are mutually contiguous (Pn 4) Pn) if (Pn) is contiguous to (Pu) and vice versa. The sequence (PD) is asymptotically separated from {Pn} (Pn A Pn) if there exists a subsequence {n'} and a corresponding subsequence of sets (Fn.) such that n. ~nI Fn. 6 3n.. P (Fn.) 9 1. and P (Fn.) 9 0. It is easy to 11 see in the special case of (On.3n) = (0.3). P = P. and Pu = P for all n. that Pn 4} PD (Pn A P“) if and only if P E P (P I P). and in this sense contiguity (asymptotic separation) is a generalization of equivalence (singularity). Let (0.3.(P9: 6 € 9}) be an experiment with 8 C Rd. The experiment is dominated if there exists a a-finite measure v on (0.3) such that Pe <( v for all 6 E 9. In the case of a dominated experiment there is a notion of differentiability which is defined as follows. Let 9 e are/av and h9 = félz The experiment is differentiable at 9 = 96 if the mapping f e L2(n.s.u) for all e e e. 6 E 9 4 h is differentiable at 8 = 8 as a mapping from 9 o B to the Hilbert space L2(O.$.v) i.e. there exists d 2 vh9 6 HL (0,3.v) such that 1 O T (1.1) lim "he-he —(e-ao) -vhe n/le-eol = o 6460 0 0 where the limit is through 8 € 9. (8-90)T is the transpose of (9-90). and the norm on Rd. IOI. is the usual one. Note that we do not assume that 9 is open. The experiment E is differentiable if it is differentiable at all points in 9 and E is regular if it is continuously differentiable. It is easy to see that differentiability is not dependent on the dominating measure. Throughout this thesis all parameter spaces are assumed to be in finite dimensional Euclidean space. and in any Euclidean space we use the usual norm which we denote by I'l. Suppose (0.3.{P.P)) is an experiment. The Hellinger distance. H(P.P). between P and P is defined by 2 ~ ~ (1.2) 2H (P.P) = ] (h-h) do where v = P+P. h E (dP/dv)1/2. and h 6 (dP/dv)1/2. It is easy to see that u could be replaced by any a-finite measure which dominates P and P (when of course h and h are replaced by the obvious functions). and that o g H(P.P) g 1 with H(P.P) = o (H(P.P) = 1) if and only if r = F (P 1 F). For the rest of this thesis we use the notation f anxi h for densities and square roots of densities respectively along with subscripts or superscripts (fa corresponds to P9. etc.) to indicate their associated probability measures without further comment. There are important relationships between the Hellinger metric and contiguity/asymptotic separation. The following equation is well known (cf. Strasser (1985). Lemma 2.15) and quite useful. (1.3) 2H2(P.P) g nr-Fu g 2H(P.P)(2-H2(P.P))l/2 where N°N is the total variation norm. Since Pn A Pn if and only if (1.4) lim sup urn-9“" = 2. nan (1.3) implies A P“ if and only if lim sup H(Pn.Pn) = 1. 11-)” (1.5) P For the infinite product situation. we can easily monitor the Hellinger distance by monitoring the Hellinger distances ~ P then ni between the components. If Pn = HPn and Pn = l i HfiS lO Q 2 n ~n ~ H (P .P ) _ 1 - q! hmhn1 doni (1.6) Q 2 ~ = 1 - ¥(1-H (Pni’Pni))’ ~ n ~n where uni = Pni+Pni' By (1.5) and (1.6). P A P if and only if either ”2 ~ lim sup 2H (P .P ) - “a“ 1 ni ni (1.7) or lim sup sup(H(Pn1.Pni)= i e N} = 1. 1'1"” For further details regarding contiguity. differentiability. and the Hellinger metric and their connection to asymptotic statistics. see Strasser (1985). Let (0.3.{Pei 8 € 8)) be an experiment. We now list some assumptions which will be invoked later in this thesis. (A1) P9 5 P9. for all 9. 9‘ € 9 (homogeneity). (A2) lim H(P9.P9.) = 0 for all 9 € 6 (continuity). Wherl 6'46 there is no possible confusion we write H(9.6') in place of H(P9.P9.). 11 (A3) lim H(8.8 ) = 1 for each 9 € 8 and sequence n900 n {9n} C 8 such that IGnI 9 m or 6n 9 t 6 ENG. Here 8 denotes the closure of 9 (asymptotic separation at the boundary). (A4) P9 f P9. for all 8 g 9' (identifiability). Since 9 c Rd. (Al). (A2), and (A3) imply 5X9 is closed. This is stated and proved in the following proposition. Proposition 1.1. Let E = (0.3.(P6: 6 € 9)) be an experiment which satisfies (A1). (A2). and (A3). Then 5N9 is closed. Proof: If BN9 = e. we are done. So suppose it is not null and let {t1} C BN6 be such that tJ 9 t E Rd. Let 80 € 8. By (A3) there exists {GJ} C 8 such that It -eJ| » o and H(9J.9°) » 1. Thus it t e 9. J H(GJ.t) 9 0 by (A2). This would then imply H(8°.t) = 1. a contradiction to (A1). since H(Go.t) 2 H(GJ.9°) - H(OJ.t) for all 1. Hence t 6 9 and this implies t 6 9\9. D Finally whenever there is an infinite product measure it is implicitly assumed the component probability measures 12 are compact. i.e. there exists a compact subclass (cf. Neveu (1965). pg. 26) in the component a-field such that the measure of any set is the supremum of the sets in the subclass (this is needed to ensure the existence of the infinite product measure). 13 CHAPTER II SUFFICIENT CONDITIONS FOR DICHOTOHY 2.1 Hoin ro_ultoo A corollary. of a more general result in Lipster. Pukel'sheim. and Shiryaev (1982). which gives necessary and sufficient conditions for contiguity in the infinite product situation is stated and proved below (this is actually a generalization of a result in Oosterhoff/Van Zwet (1979)). Proposition 2.1. Let Eni = (oni’gni'{Pni’Pni}) be an m N ”N experiment for all n.i e N. Pn = HP . and PD = HP . 1 ni 1 ni Then Pn 4» P“ if and only it (2.1) lingup 2H2(Pn1.5n1) < w and ”N ~ (2.2) ii: li:azup fpniuni > Kfni) = o and 14 Q (2.3) ii: lizqzup fpni(fni > Kfni) = 0. Proof: By the remark following corollary 1 in Lipster. Pukel'sheim. and Shiryaev (1982). PD 4 Pn if and only if (2.1) holds. (2.4) ii: lingup Pn(s:p(fnilfni) > K) = 0. and (2.5) ii: 11:6:up Pn(sup(fn1/fni) > K) = 0. But (2.4) is equivalent to m A! ~ (2.6) ii: ligazup [1-llI(1-Pn(fn1 > Kfni))] = 0. By taking the exponential of both sides. (2.6) is seen to be equivalent to (2.2). By symmetry. (2.3) is equivalent to (2.5). D. Before stating and proving our main results in this chapter. we need a technical lemma which shows that contiguity and asymptotic separation are not affected by measurable transformations which are one-to-one, onto. and have a measurable inverse. 15 Lemma 2.1. Let En = (0n.$n.{Pn.Pn}) be a sequence of experiments and let In be a measurable transformation from (On.9n) onto itself which is one-to-one. onto. and has a measurable inverse. Let Qn = PDOIn and 6n = PDOIn. Then P“ 4 PD (3“ A P“) implies 5“ 4 on (5“ A on). Proof: Let {Pu} be such that Qn(Fn) a o. This is equivalent to Pn(I;1(Fn)) 9 O which implies Pn(I;1(Fn)) 9 0 since Pn 4 Pn. But this is equivalent to 6n(Fn) 9 O and hence 6n 4 Qn' The proof of the asymptotic separation result is similar. 0. Remark. Let Eni = (0.3.(Pn1.Pn1}) be an experiment for Q and Pn = HPn . Let Qn and Q“ Q all n.i e m, Pn = HP ni i 1 1 be obtained by a common Juxtaposition of the component probability measures of PD and PD respectively (Juxtaposition can be different for each n). By Lemma 2.1 P“ 4 Pn (Pn A P“) if and only if an 4 Qn (an A On). This fact will be useful for simplifying some arguments and calculations. Based on the previous results we now state and prove a technical theorem from which the main two results of this chapter follow. 16 Theorem 2.1. Let E = (0.3.{Pei 9 € 9}) be an experiment Q Q satisfying (A1) through (A4). Pn = HP9 . and PD = HP; 1 ni 1 ni where (6 = n.i e N} C 9 . 9 a compact subset of 9. ni c c Also assume that if v is a dominating a-finite measure and 8; is any compact subset of 9. then 2 | : 2 . . . (2.7) {(he-he.) /|e-e e e BC. 9 e BC} is u.i. and (2.8) lim inf{H(6.9')/I6-9°I: 9 6 ac. le—e'l < p} > 0. p90 n N Then Pn 4» P“ or P A Pn with the former occuring if and only if (2.9) lim inf inf{dist(9 5\9): i e m} > 0 new ni' and Q ~ 2 (2.10) lingup flen1-9n1| = M < a where 9 is the closure of 9 and dist(9ni.¢) = 1 by convention (e is the empty set). 17 Proof: First suppose (2.11) lim sup sup{|9n i E N} = w. I: 11'9“o i By the remark following Lemma 2.1. without loss of generality there exist subsequences {en'l} and {9n.1} | a m and e such that I9 9 90 € 9. Then n'l n'l H(Go.9n.1) 9 1 by (A3) and H(9°.9n.1) 9 O by (A2). But by the triangle inequality and algebra. ~ H(en.l,en.1) 2 H(9°.6n.1)-H(9°.9n.1) for all n' and on taking the limit infimum of both sides. ~ n ~n H(9n.1.9n.1) 9 1. Thus by (1.7). P A P . Now suppose (2.9) is false. By the above argument if (2.11) if true. then Pn A P“. so assume that (2.11) is also false. By a previous remark. without loss of generality there exists subsequences {en'l} and (9n.1} and a subset {tn.} C 9N9. such that 9n.1 9 t 6 Rd. 9 9 60 € 9. and I9 tn" 9 0. By the triangle n'l n'l- inequality tn. 9 t and hence t E 9 by Proposition 1.1. Thus Pn A P“ by (A2). (A3). (1.7). and an argument similar to the one used in the previous paragraph. Now suppose (2.10) is false. and we want to show Pn A Pn. By previous arguments it suffices to prove Pn A Pn under the additional assumptions that (2.11) is 18 false and (2.9) is true. By these additional assumptions (note we are not using that (2.8) is false in this statement) there exists an N e N and a compact set a; c e such that (Sui: n 2 N. i e u) c 9;. Let p0 > O and do > 0 be such that (2.12) H(6.9')/I9—6'I > a0 for 8 € 9c. 9' 6 9;. and IB-O'I ( p0. Also (H(9.9')/I6-9'|: e e BC. 9' e 9;. le-e'l 2 p0} is bounded away from 0. To show this. assume it is not bounded away from O and choose sequences {OJ} C 90 and (83} C 9c such that eJ » 90 e o, 93 e a; e o. IOJ-93I 2 p0 for all J. and H(9J.63) 9 0. By (A2). H(9°.9;) = 0. which contradicts the identifiability assumption (A4). Hence combining this with (2.12). (H(9.0')/I9-6°|: a e 90,9' 6 9;} is bounded away from O. and by (2.7). it is bounded above. Thus there exists a.B € (0.") such that (2.13) aIG-G'I2 g H2(9.9') g ale—e'l2 for 6 6 9c. 9' € Bé. Since (2.10) is false, (2.13) and (1.7) imply Pn A PD. For the final case suppose (2.9) and (2.10) are true with H as the constant given in (2.10). By the remark following Lemma 2.1. without loss of generality 19 (2.14) leni-enil 2 |9n.1+1-9n.1+1| for all i,n e m. As in the previous paragraph. since (2.10) implies (2.11) is false. there exists a compact set 9; C 9 and an No 6 m such that (9n n 2 No. i e u) c 9;. Since (2 13) is only 1: dependent on the hypothesis given in the statement of theorem. it holds in this case as well. This combined with (2.10) implies (2.1) is true. We now want to verify that (2.2) and (2.3) hold. Let 0 e. e > 0. Then let {91} C 9c and {93} C 90 be sequences such that 9J ¢ 93 for all j and [91-93] 9 0. Then there exists subsequences {91.} and {93.} such that 93. 9 9°. OJ. 9 9°. hej' 9 h9° a.e.-v. and h93‘ 9 h9 O a.e.-v. This implies 2 2 2 lim P ((h —h . ) 2 e h }/H (9 .,9°.) 1'9” 91' 91' 91' ° 91' J 5 '9m {he -he. ) 2 eohe ) 1' ' J. I J. . 2 2 . _ . 2 —1 I93. 93" ) dv . (60H (91..9J.)/|91. 91.| ) = 0 by (2.13). (2.7). and (A1). since (A1) implies that the integrand is converging to 0 a.e.-v (cf. Fabian/Hannan 20 (1985). Lemma 4.8.3). By an exactly analogous argument 2 2 6°h2 6j.}/112(9J..9'.) e o. 1im P . ((h -h . ) 91' 9 9J' J j.“ J. Since the original sequences were arbitrary we have actually shown 1:: sup{P9((he-he.)2 2 eoh§)/H2(9.9°): |9—9'| g p} = o p and lim sup(Pe.((h9-h do .)2 2 eohg.)/H2(9.9'): |9-9'| g p} = o p 9 where both limits are over 9 € 9c and 9' e 9; with 9 fl 9'. Thus there exists p > 0 such that (2.15) P6{(he-he.)2 2 eoh§)/92(9.9') < e and 2 2 2 . (2.16) P9.{(h9-he.) 2 eoh9.}/H (9.9 ) < e for 9 9 9c and 9' e 9; such that |9—9'| < p. To show Pn 4) P“. it suffices to prove that any two subsequences (Pn } and (PD ) have mutually contiguous subsubsequences. Hence it suffices to prove Pn 4} PD for two subsequences 21 2 2 for which 9n. 9 91 6 9. 9 9 9 € 9. f 9 f i n'i i a.e.-v and f~ 9 f~ a.e.—v for all i 6 m, since 9 . 9 n i 1 (A2) is true and since 9c and 9; are both compact. Let C = max{H/p2.1}. By (Al) there exists K ) (1+eo)2 such that (2.17) :im P (i6 > Kf” ) = P9 (P6 > Kigi) < e/C and (2.18) lim P~ (i; > Kfe ) = P31(f3 > Kfei) < e/C for i S C. There exists N e N such that n 2 N and i > 0 implies |9n -9n1| < p by (2 10) and (2.14). By 1 combining (2.17) and (2.18) with (2.13) and (2.16). (2.19) lim sup 2P9 (fe > ng ) < e(1+MB) n'9loo 1 n'i n'i n'i and Q (2.20) lim sup 2P3 (f3 ) Kf6 ) < e(1+MB). n'9'IID 1 n'i n’i n'i Since e was arbitrary. this implies Pn 4) Pn by Proposition 2.1. D. 22 Remark. Let E = (9.3.(P9: 9 € 9}) be a dominated experiment and let u 92 be two dominating a-finite 1 I measures. Suppose E is differentiable at 9 = 90 with differential vhJ for measure n J = 1.2. Then it is an 90 J’ easy exersise to show that [ vh; ~(vh; )T dol = I vhg ~(vhg ) dv . 0 0 O O In particular I vb; °(vh; )T dv1 is non-singular if and o 0 only if I vb: -(vh§ )T dv2 is non-singular i.e. O O non-singularity is independent of the dominating measure. Theorem 2.2. Let E (0.3.{Pez 9 € 9)) be an experiment differentiable at 9 9 and assume E satisfies (A1) 0 through (A4). Let U be a dominating measure. vh9 be 0 the differential corresponding to v. and assume that Q I vhe -vhg do is non-singular. If Pn = HP9 and o o 1 0 ~ m ~ ~ Pn = HP; , then Pn 4» PD or Pn A Pn with the former 1 hi being true if and only if (2.21) lim inf inf{dist(9n .9\9): i e m} > 0 n9" 1 and 23 Q (2.22) lim sup 2|90-9 n90 1 2 MI < 9. Proof: Let 9 = {9 } and note that c 0 lim inf{H(9°.9°)/I9°—9'I= lea-9'l < p) p90 = lim inf{fl(9°-9')T-vh9 n/|90-9'|: loo-9'I < p} p90 0 2 ini{tTo([ vh8 -vh: du)-t: t 6 Rd. |t| = 1} O 0 > 0. Thus (2.8) holds. Let 9; be a compact subset of 9. In order to verify (2.9) it suffices. by Proposition A.1. to prove that if {91) is a convergent sequence in 9é\(9°} satisfying that (93-9°)/I9J-9°I converges to t 6 Rd. then ((hGJ-he )2/I91-9OI2: 3 e u) is u.i.. To show this we 0 consider two cases. First suppose 9J 9 90. Then (h -h )/I9 -9 I 9 tT°vh in L2(v) and hence 9J 90 j 0 90 (h -h )2/I9 -9 |2 e (tT-vh )2 in L1(v) which implies DJ 90 j o 60 u.i.. For the second case suppose 9J 9 9 ¢ 90. Then by 2 (12). (heJ-h9°)/|93-9o| 9 (he-heo)/I9-9°I in L (D) and 24 2 2 2 2 1 hence (heJ-heo) Ilej-eol e (he-heo) /|9-9o| in L (D) which again implies u.i.. Hence we have verified (2.9). so by Theorem 2.1. the result follows. El. Theorem 2.3. Let E = (0.3.(P9: 9 E 9}) be a regular experiment which satisfies (A1) through (A4). Let u be a dominating a-finite measure and assume that I vh9°vhg do is non-singular for all 9 € 9. Furthermore assume that 9 is locally convex and (Gui: i,n 6 N} C 9c C 9 where 9c Q Q is compact. If Pn = HPe and Pn = 9P5 , then PD 4» PD 1 ni 1 ni or Pn A Pn with the former being true if and only if (2.9) and (2.10) are true. Proof: Let 9; be a compact subset of 9. ‘To show (2.7) and (2.8) are true. and hence obtain the desired result. it suffices to prove that if {OJ} C 90 and {93} C 9;, there exists subsequences {GJ'} and {93.} such that ("hej._hej.fl/I9J.-93.I} is bounded away from o and 2 ((h -h . ) /|9 .-9°. OJ. OJ. J j subsequence notation for convenience. it suffices to prove I2) is u.i.. Thus dropping the {uh9 -h9.u/|9 -9'|) is bounded away from o and J J J J ((he -h9.)2/I9 -9'I2) is u.i. under the additional assumptions that 9J 9 90 € 9. 95 9 9; e 9. and 2 (9 -9j)/|9J—93| 9 t c n . J 25 If 90 = 9;. then by the local convexity of 9. continuity of the differential. and a standard differential calculus result for normed linear spaces (cf. Loomis/Sternberg (1968). pg. 149). (2.23) (h -h .)/|9 -9'| 4 tT°vh in L2(v) 9 9 j 1 9 J J 0 2 . 2. which implies ((h -h .) /I9 -9 | . j e N} is u.i.. Also GJ 6J J J (2.23) implies (2.24) lim inf H(9 .9')/|9 -9°| > 0 14¢ J J J J T since I vh9 °vh6 do is non-singular. O O In the case 90 ¢ 9;. (2.25) (had-h93)/|91-9J| 9 (hGo-h6;)/I90-9;I in L2(v) by (A2). Hence we again have {(119 ~he.)2/I9 -9'I2= j G N) is u.i.. Also in this case J J J J (2.24) again holds by (2.25) and (A4). The result now follows. 0. 2.2 Application to Gaussian Seguences. In this section we use our results to prove the contiguity/asymptotic 26 separation dichotomy for sequences of Gaussian processes with arbitrary index sets. This is a generalization of Corollary 4 in Eagleson (1981). which dealt only with triangular arrays. First we prove the dichotomy for the case of countable product measures in Corollary 2.1. Secondly in Corollary 2.2 we prove this generalizes the triangular array result. Finally in Corollary 2.3 we use these two results to obtain the general Gaussian contiguity/asymptotic separation dichotomy. k. c 6 PD}) Corollary 2.1. Let E = (Rk.9(Rk).{PuC: p e R where PD is the set of all positive definite k by k matrices and PuC is multivariate normal with mean u and Q Q covariance matrix C. Let Pn = HP C and Pn = HP~ E . 1 ”0 0 1 p'ni ni ~ Then Pn 4} PD or Pn A Pn with the former being true if and only if (2.26) 1im inf inf(det(Cn1)= i e w) > 0 new and ” ~ 2 ~ 2 (2.27) lim sup 2(Iun1-uol +|cni-c°| } < m. n9no 1 where det(Cni) is the determinant of Cni and the norm on k2 matrices is as elements of R 27 Proof: For the sake of brevity and clarity we only prove this for the case k = 1. since conceptually the proof in the multivariate case is the same. Let A. Lebesgue measure. be the dominating measure. By examples 3.1 and 3.2 on pages 47-49 in Roussas (1972). the mapping u E R 9 h“C is differentiable for all C > O and the mapping C is differentiable for all u e R with the L2—derivatives coinciding with the Lz-equivalence classes C > O 9 h u containing the pointwise partial derivatives ((-1/4C) + ((x-u)2/2C2)) —(1/4) 9/90 huc(x) (21C) exp(—(x-u)2/4C) and )-(1/4)exp(-(x-u)2/4C)- 9/9u huC(x) ((x-u)/C)(2wC It is easy to verify that the above are continuous as mappings from 9 = RXR+ to L2(A). Thus by a standard result in differential calculus for normed linear spaces (e.g. Loomis/Sternberg (1968). Theorem 3.9.3). the experiment is differentiable. The remaining assumptions in the hypothesis of Theorem 2.2 are easily verified and are left to the reader. 0. 28 Remark. Corollary 2.1 is subsumed by an upcoming example in Section 2.3. Specifically we show in the example that under fairly general conditions. exponential families generate regular experiments and hence Theorem 2.3 is applicable. However in the Gaussian example. since one can always translate and rescale. we only needed Theorem 2.2 to prove the dichotomy. Corollary 2.2. Let E be as in Corollary 2.1 and let 11 ~ n ~ Pn = HP" c and Pn = HP; 5 . Then Pn 4» Pn or 1 o o 1 ni ni ~ Pn A Pn with the former occuring if and only if (2.28) lim inf inf{det(C )= 1 g i S n} > O ni 11-)” and (2 29) lim sup §{|; -u |2+|E —c I2) < 9 ° ni 0 ni 0 ' n9‘0 1 Q ~ ~ Q Proof: Let Qn = an n P and on = anIH P where P is n+1 n+1 multivariate normal with mean 0 and identity covariance matrix. By Proposition A.2 in the appendix. Qn 4» Qn (on A on) if and only if Pn 4» Pn (Pn A P“). Thus by Corollary 2.1. the result follows. 0. 29 In the appendix we prove a proposition (Proposition A.3) which essentially says contiguity and asymptotic separability can be monitored on fields which generate the a-fields. Using this result and Corollary 2.2 we have the following corollary which gives the dichotomy for Gaussian processes with arbitrary index sets. Corollary 2.3. Let E = (93.9(RS),(P“,PD)) be an experiment where S is an arbitrary index set and PD and ~ Pn are Gaussian probability measures. Then PD 4) "d? 01' P11 A P”. Proof= By Propositions A.2 and A.3. it suffices to prove the dichotomy for the sequence of experiments En = (Rn.9(Rn).(Pn.Pn)) where Pn and P“ are Gaussian probability measures. If the cardinality of the set (n e N: PD l P“) is infinite then clearly Pn A Pn. If it is finite we can assume without loss of generality that Pn and Pn are non-degenerate. By translating the components and invoking Lemma 2.1 we can assume without loss of generality that Pn has mean 0. Next by diagonalizing the covariance matrix of Pn. rescaling the components of PD. and invoking Lemma 2.1. we can assume without loss of generality that Pn has an identity covariance matrix. Finally we diagonalize the covariance matrix of Pn (this doesn't affect the covariance matrix of PD since it is the 30 identity) and again invoking Lemma 2.1. we can assume without loss of generality that PD is a product of one-dimensional normal distributions. Thus we can assume that PD is an n-fold product of N(O.1) and Pn is an n-fold product of (N(uni.a§1)= 1 S i S n}. By Corollary 2.2. the result now follows. 0. 2.3 Examples. We now give two examples where the dichotomy results in Theorems 2.2 and 2.3 apply. The second subsumes the first. but the first example is presented because of its simplicity and transparency relative to the previous theory. (1) Nultinomial. Let n = (1.....d) and. E = (0.2”, (Pa: 9 e 9)) where 9 = (9 9 m3: 9(3) 2 o for all d d j. 29(3) = 1}. and dPeldv = 29(J)1{ where v is the counting measure on (0.20). For 1 € 9. let eJ e R0 be defined by 0 H J' #J 81(1')= 1 If 1' =j 9 Then for 9 € R+. ((e+eeJ)1’2-el’2) = {c(eii)+e)1’2-(eii))"2llell{J} 9 (1/2)(9()))'1/21{j} as e 9 o. 31 where the last convergence is in L2(v). Thus the mapping 9 e R? 9 91/2 is partially differentiable and it is easy to see that the partial derivatives are continuous as functions 0 from 3+ to L2(v). Thus by a standard differential calculus for normed linear spaces (e.g. Loomis/Sternbert (1968). Theorem 3.9.3). and since 9 € 9 9 h9 is Just a restriction of the above mapping. the mapping 9 € 9 9 h9 is continuously differentiable with -1/2 -1/2 T 2vh = 9 1 1 ...., 9 d 1 . 9 [1 ( )) {1) ((n (9)3 Q ~ Q Hence E is regular. Let Pn = HP and Pn = HP~ 9 9 1 hi 1 ni and assume there exists a p > 0 such that p ( Gni(J) < l-p for all n.i e N and J e Q. ~ ~ Then by Theorem 2.3. PD 4) Pn or Pn A Pn. with the former being true if and only if lim inf inf{I9ni(j)I.Il-9ni(j)|: i e m. 3 e n) > 0 new and and ~ 2 11:22“? f f|9n1(J)-9n1(i)l < 9- 32 (2) Exponential fopily. Let E = (0.3.{P63 9 € 9)) be an experiment where 9 is an open subset of Rd. Assume that there exists a a-finite measure U which dominates E and there exists random variables {T 1 S j S n} such that J! n c(9)exp(293TJ) € dPeldv for all 9 € 9. 1 This is the exponential family with the natural parameterization. and we further assume that the random variables (T1) are affinely linearly independent. i.e. ZtJTJ = O a.e.-v implies tJ = O for all j. and there does not exist (t1) and c 6 R such that EtJTJ = c a.e.-v. Then (A1) holds since P9 5 v for all 9 € 9. Also P = P implies 9 = 9' by the affine linear independence 9 9' of {TJ) and hence (A4) holds. By Theorem 78.2 in Strasser (1985). E is differentiable with vh - [(T -P T )h (T —P T )h ]T 9" 1919""'t19t19 where PGTJ = ] TJ dP9 for all 1. Also the mapping 9 € 9 9 PGTJ is continuous by Lemma 3.5.5 of Fabian/Hannan (1985). By using this continuity and applying the lemma again. the mapping 9 € 9 9 vh9 is continuous i.e. E is regular. 33 Q Let Pn = HP and Pn = HP~ with (9 : n.i e m) 9 9 ni 1 ni 1 ni restricted to some compact subset of 9. If we assume (A3) is true then we have a contiguity/asymptotic separation dichotomy by Theorem 2.3. If we do not assume (A3) is true but instead only assume {9nit n.i 6 M} is also contained in some compact subset of 9 we again get a contiguity/asymptotic separation dichotomy by Theorem 2.3. This last statement follows by the fact that we could without loss of generality assume 9 is compact by taking it to be the closure of {eni}U{9ni}' and hence (A3) is satisfied trivially. 34 CHAPTER III NECESSARY AND SUFFICIENT CONDITIONS FOR DICHOTOMY 3.1 Preliminaries and Auxiliary Results. In this chapter. we prove a converse of the sufficiency result in Chapter 2 for a specific experiment E = (Rk.9(Rk).{PtR: t 6 Rk. R 6 9)). This experiment E is based on an underlying probability measure P on (Rk,9(Rk)) and rigid motion perturbations of P. Thus PtR = PORTt where Tt is the translation operator by the vector t and R is in 9. the set of all orthogonal transformations on Rk. Note that all rigid motions can uniquely be expressed as RTt for some R 6 9' and t 6 RR. For this experiment E. the parameter space 9 = ka9 C Rk(k+l) when k 2 2 and 9 = R when k = 1. We also prove a partial converse for an experiment E based on an underlying probability measure P on (Bk.9(Rk)) and invertable affine perturbations of P. Thus P(t.A) = POATt where A E d. the set of all invertable linear transformations from Rk to RR. For this experiment E. the parameter space 9 = kad C Rk(k+l). In this context for k = 1. Shepp (1965) proved a very Q Q interesting result. Specifically he showed that UP 1 HPt 1 1 i for all (t1) C 22. Even more interesting is that he showed 35 Q P E HPt for all (t1) 6 82 if and only if P E A (where 1 i A is Lebesque measure) and there exists f e dP/dA which v9=38 is locally absolutely continuous and such that . 2 (3.1) f ((f ) /r) dA < m i.e. f has finite Fisher information. This result was generalized to a more abstract setting by LeCam (1970), Proposition 2. This result included as a corollary a generalization of Shepp’s translation result to the multivariate setting. Steele (1986) generalized both of the above multivariate results by including all rigid motions. First Steele proved that HP 1 HPO for all (91) E £2 such that 1 1 i 91 9 O as i 9 w. Secondly and more importantly he showed that HP 9 for all {91} € £2 such that 91 9 0 if 1 1 i and only if P E A and for all one-parameter groups m m "U {p(s)= s G R) in the space of rigid motions. there exists a number K such that (3.2) II h(x) [d/ds(w(p(s)x))ls=ol dA(x) s Kuwu2 for all e e c:(sk). Here h e (dP/dA)1/2 and c:(mk) is the set of all infinitely differentiable functions with compact support. Steele defines finite Fisher information 36 by this last condition. We now show that if E is dominated by A and is differentiable at (0.1) (I is identity operator on Bk). then (3.2) is satisfied. Proposition 3.1. Let E be an experiment as given above and suppose E is dominated by A and is differentiable at 9 = (0.1). Then (3.2) is satisfied. Proof: Let h e dP/dA. and o e c:(nk). Then I! h(X)[d/ds(¢(p(s)X))ls=o] dA(X)| 11m I hix)[w(p(e)x)—w(x)1(e'1) dA(X)| e90 11m I [hipi—e)x)—h(x)lie“)w(xl dA(X)| 590 If ‘Vp10)'Vh(o.I)’ t d“ IA "Vp(o)°vh(o.1)fl2fl¢fl 2. In the above. we have used the Lebesque dominated convergence theorem. a change of variables. and the chain rule in differential calculus for normed linear spaces. Also we have used that one-parameter groups in a Lie group are differentiable (cf. Warner (1983). pages 102-103). D. 37 By Proposition 3.1. (3.2) appears to be a weaker condition than differentiability. However we will prove that the hypothesis of P E A and E being differentiable are necessary for an 82 dichotomy. Thus we will actually show that if P E A and satisfies (3.2). then E is differentiable. In proving the main theorem we need two results due to LeCam (1970) (Theorem 1 and Proposition 2) which we state as propositions for ease of reference. and a technical lemma. which is stated and proved. In the rest of this chapter A will always denote Lebesgue measure on Rk (or sometimes Rd) where we have suppressed the index k (or sometimes d) for notational convenience. Proposition 3.2. Let J: U c Rd 4 H where m is a Hilbert space and U is a Borel subset. Suppose (3.3) lim sup "w(u')-w(u)fl/lu'-ul < a u'9u for A-a.e. u 6 U. Then w is Frechet differentiable at A-a.e. u E U. Proposition 3.3. Let E = (9.9.{Pez 9 € 9}) be an Q Q experiment satisfying (A1) and 90 € 9. If HP9 E HP 1 0 1 for all (91) such that ((90-91)) 6 22. then 91 38 (3.4) lizesup H(9°.9)/I9°-9I ( m 0 Lemma 3.1. Let h e L2(Rk). If d is the set of all invertable linear transformations from RR to Rk. then (3.5) lim IlhOATt—hll2 = 0 (t.A)9(0.I) where the limit is over t 6 R1“ and A e d. Proof: Let 5 > 0 and g be a continuous function from RR to R with compact support such that Ilg-hll2 < 5. Then (3.6) HhOATt—hflz g NhOATt—gOATtNZ + HgOATt-gflz + "g-hflz. But NhOATt-gOATtNZ e flg-hN2 as (t.A) » (0.1). by a change of variables and the invariance of A under translation. Also since g has compact support. IlgOATt-gll2 9 0 as (t.A) 9 (0.1) by the Lebesgue dominated convergence theorem. Thus by taking the limit supremum in (3.6). lim sup llhOATt-hll2 S 25. (t.A)-’(O-I) Since e was arbitrary. the result follows. 0. 39 We now set out some notation before going to the main results of this chapter. For P. a probability measure on (nk.s(nk)). let E: = (Rk.9(Rk).{P9: 9 e mk)). a; = (Rk.9(Rk).{P9: 9 e 9 = ka9}) (k 2 2). and E; = (Rk.9(Rk).{Pe: 9 e 9 = kad}). where E: corresponds to the translation experiment. E; is the rigid motion experiment. and E; is the invertable affine transformation experiment. We sometimes suppress the superscript P for notational convenience when the underlying probability measure is clear. P 2’ at 9). Thus if E is dominated. £(E) represents the If E 6 (BE. E 9;). let e(E) = (9 e 9: (3.4) holds points 9 E 9 at which the mapping 9 G 9 9 h9 is Lipschitz. Let 9 € 9. As given in the above. 9 represents an element in some Euclidean space. However we will sometimes find it convenient to let 9 also represent the transformation i.e. in El' 9 = T9. etc. It will always be clear from the context whether 9 represents a transformation or an element of the parameter space. and hence we will not overtly distinguish between the two interpretations of 9 in the rest of this thesis. 4O 3.2 Main Results. The main result of this section is necessary and sufficient conditions for the contiguity/asymptotic separation dichotomy in the rigid motion experiment. We also prove a partial result in this direction for the invertable affine transformation experiment. These are given in Theorems 3.2 and 3.3. Before proving them we need some further results. related to E1. E2. and E3. which are interesting in their own right. First in Lemma 3.2. we show that in all 3 cases if P (S A and if (3.4) is satisfied at one point then (3.4) is satisfied at all points. This is equivalent to saying that if P << A and the mapping 9 € 9 9 h9 is Lipschitz at one point. then it is Lipschitz on all of 9. The usefulness of this result is derived from Proposition 3.2. and is given in Theorem 3.1. Specifically we show that if P << A and (3.4) holds at one point. then E is differentiable for I i = 1,2,3. Lomma 3.2. Let P << A. E 6 (El. E2. E3}. and suppose £(E) ¢ ¢- Then £(E) = 9. Proof: Let E = E1. 90 6 9(2). and 91 e 9 = Rk. Then 41 lim sup "h -h "/|9—9 | 999 9 91 1 lim sup "hoe—hoelfl/IG-B I 999 1 1 lim sup "h099; -1 9 -ho9 n/I99 9 -9 | < w. 9 o o 1 o 0 since 90 € £(E1). Note that the second equality follows by the invariance of Lebesgue measure under translation. and the invariance of the absolute value norm on Rk under translation. Thus 91 € £(El). k Let E = E3. 90 € £(E3). and 91 E 9 = R xd. Now temporarily substitute (to.Ao) for 90. (t1.A1) for 9 and (t.A) for 9. Then G II AT T A- t -t1 1 0 to A T 1 -1 . 0 (A0 A1(t-tl)+to) AA1 Thus -1 2 -1 . #1 2 (3.7) I99l 90-9ol |(Ao Al(t-t1).(AA1 -I)AO)I 2 I/\ -1 2 2 -1 2 IAo All It-tll + |Al Aol IA—All IA 2 2 x (A°.A1)I9-91I 42 where K(AO.A1) = max{IA;lA IAEIAOI). Note that the first 1|- inequality follows since the norm of matrix as elements of k2 R is greater than or equal to the norm of a matrix as a linear operator from Rk to Rk. By (3.7). there exists K < a such that (3.8) "he-helfllle-Gll -1 1/2 -1 g [uhee-l-leo-heou-IA1 AOI /|991 90—9ol] -1 -[|991 90-9°|/|9-91|] -1 g Knheei190-heon/I99l 90-9°| for all 9 € 9. This implies 91 € £(E3). The case of E = E2 follows by a similar and slightly easier argument than that Just given for E = E3. 0. Remark. Notice in (3.8). if we restrict ourselves to Rxdc where “c is a compact subset of 1. then we can find a K for which (3.8) holds for all 9 € Rxdc. In particular 1 this is true when dc = 9. This will be used later on. In Theorem 3.1. by using Lemma 3.2 and Proposition 3.1. we prove that a sufficient condition for differentiability of E1 is that £(E1) f e. for i = 1.2.3. The result for i = 1 was previously known (c.f. the remark following Proposition 2 in LeCam (1970)). The purpose of proving it 43 again in the case i = 1. is that it is useful in understanding the proofs for the cases i = 2.3. Before stating and proving the first theorem. we outline some of the key ideas of the proof. The essential idea behind the proofs of all 3 cases is to use Proposition 3.2 and Lemma 3.2 to show that E is differentiable except on a null set. and then to use the differentiability of E on a dense set in 9 to get differentiability on all of 9. The case E = E1 is proven easily this way since the differentials are all translates of each other. In the case B = E2. there are two main difficulties. First 9 is already a null set so Proposition 3.2 doesn’t even give differentiability at any points. To overcome this difficulty. we locally transform E2 into a new experiment E; with a new parameter space 9* which is not null. and then we invoke Proposition 3.2 for this new experiment. We then transform back to the original experiment E2 to get differentiability on a dense set in 9. The second difficulty is that differentiability at a point 90 is not directly transferable to other points in 9. However locally and asymptotically it is transferable. This last difficulty is also inherent in the case E = E 3. Theorem 3.1. Let (9.3.P) be an experiment dominated by P 2. (a) E is differentiable in the case i = 2 and in A, E e (E5. E E3) and suppose £(E) f e. Then 44 this case there exists a family of differentials (vh 9 € 9) which is uniformly bounded over compact 9: subsets of 9. (b) E is regular in the case i = 1. 3. Proof: Let E = El. Then 9(91) = 9 = Rk By Proposition 3.2. there exists 90 C 9 such that E1 is by Lemma 3.2. differentiable at 9 = 9 for all 9 € 9 and o o o’ A(9\9°) = 0. Let 90 € 9°. 91 € 9. Then for 9 E 9 T -1 "he-hel-(O-Gl) vheooe 91" T = uh ~he -(9-9l) ~vhe n O O O -1 991 9 and -1 |9-91| = |99l 90-9ol. Hence on letting 9 9 91. we see E1 is differentiable at 09:19 . By Lemma 3.1. the map 9 '1 th Vh = Vh 1 1 9 9 1 o 9 € 9 9 vh is continuous and hence E is regular. Let E = E Then exactly as in the previous case. 2. 9(2 = 9 = ka9. Now let L = k(k-1)/2. By the general 2) implicit function theorem (e.g. Auslander/HacKenzie (1963)). for each R G 9. there exists neighborhoods V of R E 9 and u of o 8 RL. and w e Cl(U.V). such that 9(0) = R and w is a homeomorphism with the differential of w at u G U. dwu. having rank L for all u E U. Also there 45 2 exist a neighborhood W of R 6 Rk . where V = W09. and n e Cl(W.U) such that n is onto. and now is the identity mapping on U. Without loss of generality. we can assume U is convex. U is compact, and the above is true on a neighborhood of U with the same n and w. Now fix an R0 6 9 and let n. w. W. U. and V be as above. By the compactness of U. sup{fldwu-dwu.fl= u.u' e U} = B < n. Thus using a standard differential calculus result (e.g. Loomis/Sternberg (1968). Theorem 3.7.4). (3.9) Iw(u')-W(u)l S BIu'-u| for all u.u' e U. Thus (I¢(u')-w(u)I/Iu'-ul= u.u' e U} is bounded above. We now want to show it is bounded away from 0. Let {“3} and (us) be arbitrary sequences in U such that uJ i “J for all 1. It suffices to prove that (IW(u3)-w(uJ)I/Iu3-ujl} is bounded away from 0. Since this is true if and only if every subsequence has a subsubsequence which is bounded away from 0. it suffices to prove that the above sequence is bounded away from 0 under the additional assumptions that u 9 u 6 U and u' 9 u‘ e U. If u g u'. then J J {IW(u3)-W(UJ)I/Iu3-ujl) is bounded away from 0 by the 46 continuity and injectivity of w. If u'. then I: II (3.10) lim I¢(u3)-w(uJ)-dwu(u3-u1)I/Iuj-ujl = o j-W by the continuous differentiability of w. the convexity of U. and standard differential calculus. But dwu has rank L so (3.10) implies (3.11) 1im inf IW(u&)-W(uJ)I/Iu3-ujl ) 0. 190 Thus combining this with (3.9). there exists positive constants a.B such that (3 12) alu'-u| s lw(u')-w(u)| s BIu'-u| for all u.u' e U. Now consider a new experiment E; = (Rk.B(Rk). {P;*: 9* € 9*} where 9* = kaU. P(t.u) = P(t.¢(u))’ and n he! 6 (dP;*/dA)1/2. We define a new map WI: kaU 9 9 by ¢l(t.u) = (t.W(u)). By (3.12). (3.13) IwI(e*)-wlfe:)l s (1+91Ie*-etl 1)! N N for all 90. 9 6 9 . If 9: is a compact subset of 9*. then there exists K < m such that for all 9: 6 9c 47 xu/I9”-9:| < K(1+B) 0 (3.14) lim sup "hefl-h 9 x x 9 990 by dividing and multiplying by Iw1(9*)-¢1(9:)I. invoking (3.13). and recalling the remark following Lemma 3.2. Thus ((E;) = 9*. By Proposition 3.2. there exists 9: C 9* such that E; is differentiable on 9: and h(9*\9:) = 0. Now define a map n1: kaW 9 kaU by n1(t.w) = (t.n(w)). Note that n1 6 C1(kaW) and nlowl is the identity map on kaU. This last fact also implies wlonl is the identity map on kaV. Thus by the chain rule u E is differentiable on 90 = ¢1(9°) with x (3.15) vh6 = V"19th1(9) for 9 e 90. k Let to 6 R be fixed. Then there exists a sequence {(tj'uj)} € 9: such that (tj'uj) 9 (to.0). Letting 91 = w1(tj.uj). we see that 9J 9 90 = (t°.R°). Now for a fixed J and letting 9 e kaV. we have T -1 (3.16) "he-he -(9-9°) -vh9 o9 9 H o J 1 ° nho99;19J-h091—(9-90)T-vh9 n J g uhoeezlej—ho91-(99:193-9J)Tovheju + "(99:191-91-9+90)T-vh93u. 48 If we let 9J = (tJ.RJ) 99:39j as an operator. -1 (3.17) 990 9J — RTtT = RR-1 0 and 9 = (t.R). and if we view we get jT(R31Ro(t-to)+tj)° Thus corresponding to the first term in (3.16). -1 (3.18) I99o 9 J'BJ' For the second term in -1 -1 |(RJ R°(t-to).RR° RJ’lel I(t-to.RR;l-I)I I(t-t°.R-R°)I |9-9°|. (3.16). by (3.17) and the Cauchy—Schwarz inequality. -1 T T 2 (3.19) "(990 91-91-9+9°) ovhejfl -1 2 T 1/2 2 3 I990 91-91-9+9°| "(vhBJ-vhej) u 49 |((R31R.-I)(t-to).RR;1RJ-Rj-R+Ro)|2n(vhgjovh 1/2"2 ) 9: 1/2"2 9 ) I((R —R )(t-t ),(l-RR'1)(R -R ))I2fl(th -vh o J o o 0 j 9j j 1/2"2 l/\ IRO-RJ|2|(t-to.R-Ro)|2u(vhg-vhej) where in the last inequality. we have used that the norm of Ro-RJ as a linear operator is less than or equal to IRo-le. and we have also used that the inverse of an element in 9 is equal to its adjoint. On dividing the quantity in (3.16) by I9-9ol. and letting 9 9 90. we see that the first term goes to 0 by the differentiability of E2 at 9 and by (3.18). Thus by (3.17) and (3.19). if J j _ -1 g - vhe 09j 90. J (3.20) lim sup "he-he -(9-9°)T°gJ"/l9-9°I 999 o 0 g IR -R |u(vhT ovh )1/2u. 3 ° 6J 91 it Since vh9 = vnlejvhnl(ej). J (3.21) "(vhg ovhe )1/2u J J xT T * 1/2 = "(vh vn vn vh ) H. n1(91) 193 19J n1(91) 50 But vnla 9 vale by continuity of the differential and J o ("(vh*T °vh* )l/zfl) is uniformly bounded by (3.14). 91(9J) 91(91) Thus the quantity in (3.21) is uniformly bounded in j, and hence by (3.20). (3.22) lim lim sup "he-he -(9-9°)T-an/|9-9°| = 0. 19¢ 999° 0 k(k+1). Then H is an (L+k) Let u = ka(dwo(RL)) c n dimensional subspace since dwo has rank L. Now let 9: = (t°.O). Then by the differentiability of 91. (3.23) 1im lwl(9*)—wl(9:)-dwlex(9”-9:)|/|9*-9:| = 0. i N O 9 990 By dividing and multiplying the expression in (3.23) by le(9*)-wl(9:)l. noting that *1 is a homeomorphism. and invoking (3.11), (3.24) $13 I9-9°-Prn(9-9°)I/I9-9°|= 0. 0 where Pr" is the projection operator onto H. * Also if y 6 H and Iyl = 1. there exists 9: E 9 such that dwle*(9:-9:) = soy. where so > 0. Then 0 51 (3.25) ::3 Iw1(9:+s(9:-9:))-wl(9:)-sdwle:(9:-9:)I/sl9:-9:I :0, where the limit is over 8 € R\{O}. Hence if we again divide and multiply by Iw1(9:+s(9:-9:))-¢1(9:)I and invoke (3.11). we obtain (3.26) lim inf |((9-9°)/|9-9°|)-yl s 0 O for all y e n such that |y| = 1. Let 5 > 0. Then by (3.22) there exists J 5 R such that J 2 J implies (3.27) lim sup "he—he -(9-9°)T°gJfl/I9-9°I < e. 9960 0 Let y e u be such that |y| = 1, and let {9n} c 9 be a sequence such that 9n 9 90 and (3.28) ii: I((9n-9°)/|9n-9°|)-yl = 0. Then for J.J° 2 J and n 6 R. (3.29) uy-(gJ-gj')u g fly°gJ-((h9 -h9 )/|9n-9°|)n + n o "((hen-heo)/|9n-9°|)-yogj u 52 Now triangulate the first term on the right hand side of T. J (3.29) using the term ((9n-90) g )/I9n—9°I. and triangulate the second using the term T. J' ((9n-9o) g )/|9n-9°|. Then invoke (3.28) and (3.27). and apply the Lebesgue dominated convergence theorem as n 9 w to get (3.30) uy-(gJ-gj')u < 28. Since this is true for any J. J' 2 J. we have Just shown (y°gJ) is a Cauchy sequence in L2(A). -Since y was arbitrary. this shows (PrngJ} is a Cauchy sequence in k(k+1) 2 n L (A). Let g be the limit of (Prugj} in 1 k(k+1) 2 u L (A). By (3.24) and (3.30). 1 lim sup "he-he -(9-9°)T°g"/I9-9°I 3 2c 9990 c after triangulating the numerater on (9-9°)T°gJ and Prn(9-9°)'gJ for large J. Thus E2 is differentiable at 90. Since 90 was arbitrary. E2 is differentiable. Also by proJecting the differentials onto the appropriate subspaces (as was done previously with the subspace M), invoking the remark following Lemma 3.2. and invoking (3.26). we also obtain a family of differentials 53 {vh 9 E 9} which are uniformly bounded over compact e: subsets of 9. Let E E . By Lemma 3.2. 9 = £(E3). and by 3 Proposition 3.2. there exists a measurable subset 3 is differentiable on 90 and A(9\9°) = 0. Again let 9o 6 9 and {GJ} C 90 be such 90 C 9 such that E that 9 9 90. A similar argument as done in the case J E = B . shows that if g1 = vh o9'19 . then 2 9J J o (3.31) lim lim sup "he-he -(9-9°)T°gJ" = 0. J90 9990 o The only difference is that it is easier since one can carry out the argument directly instead of transforming the experiment. Since 9 is open it follows that {g3} is 2 k+k 2 J Cauchy in H L (D). By (3.31) the limit g of (g } is 1 seen to be the differential of h9 at 9 = 90. Thus E is differentiable at 9 = 9 . Also by Holder's inequality. 0 eagleju + Ilvh9 09:19j-vh9 n. O flvh -vh H S flvh -vh OJ 90 BJ 90 By a change of variables and Lemma 3.1. vhe 9 vh9 . Hence 3 o by an easy argument. this implies the map 9 € 9 9 vh9 is continuous and E is regular. 54 Remark. Let E 6 (Bi. ES. Eg). We will sometimes write EP and drop the subscript i in order to conveniently keep track of several experiments generated by different perturbations of the same probability measure. By Theorem 3.1. we now have a useful sufficient condition for checking when EP is differentiable. Namely we need only check whether £(EP) g e. Most often this will be done by checking if £(EP) contains the identity. We apply this in the next result. Proposition 3.3. to show that if P°.P1 are two probability measures with EPO being differentiable. then EP0*P1 is differentiable. Pr ti 3.3. Let i 9 (1,2,3) and (Rk.9(Rk).{P°.P1}) be an experiment such that Po << A and BIO is P *P differentiable. Then E10 1 is differentiable. Proof. Let P = POEP and note that by Fubini. P (S A. 1 Let 9 be the appropriate parameter space depending on whether i = 1.2. or 3. and let h9 € (dPeldA)1/2 for 9 6 9. and f G dPoldA. Then 0 (3 32) h9(y) = (I :.(ey-x) dP1(X))1/2(K(9)) for all y e Rk\B where A(B) = 0. K(G) 1/2 IAI 1 for i = 1 and 2. and K(9) = for i 3 where 9 = (t.A). 55 Let 9 € 9 be fixed. For 9' € 9\{9) 2 2 2 (3.33) (h9.—h9) = h9.-2h9.h9+h9 by algebra. But by (3.32). if ho € (dPO/dA)1/2. h9u(Y)h9(Y) = (I f°(9'y-X) dP1(X) I fo(9y-X) dP1(x))l/2(K(9)K(9')) 2 (I h.(6'y-x)h.(ey-x) dP.(x))(K(9)K(e°)) for a.e.-A y by Holder’s inequality. Combining this with (3.33). (3.34) (h..(y)-h9(y))2 s I (K(9')h.(9'y-X)-K(9)h.(9y-X))2 dP1(X) for a.e.-A y. Thus letting 9' 9 9. we obtain that 9 9 ((EE) by (3.34) and Fubini. By Theorem 3.1, B: is differentiable. n. We are now ready to state and prove the two main theorems of this chapter, which give a partial converse to Theorem 2.2 in the case E e (E1. E2) and a partial converse in the case E = E3. 56 Theorem 3.2. Let (Rk.9(Rk).P) be an experiment with E 6 {E5. E3). and in the case of E = E assume E 2’ satisfies (A4). Then the following are true: n ~n ~ ~ (a) P = "PG 1) P = HPG for all {Gni}' {Gui} 1 ni 1 ni such that Q ~ 2 (3.35) lim sup 2'9 -9 I < w “a” 1 ni ni if and only if P E A and E is differentiable. 0 N o N (b) Pn = UP A Pn = HP" for all {9 ). {9 ) 1 9ni 1 9111 ni ni such that Q ~ 2 (3.36) 11:4:up slant-9n1I = a. Proof: By Lemma 2.1. we can without loss of generality assume 9111 = O for E = E1 and {eni} C (O)x9 for E = E2. We will prove both cases simultaneously since many of the arguments for both cases are the same. The maJor differences will be pointed out when they occur. 57 (a) HP ' PEXHP for all 9 € 6 and hence (A1) is l 2 true. This implies that P E A (cf. Steele (1986). Lemma 4.1) and henceforth let A be the dominating measure. By Proposition 3.3. £(E) fl 9 and by Theorem 3.1, E is differentiable. For the converse clearly (Al) and (A3) are true. and by Lemma 3.1. (A2) holds. Also (A4) holds for E = El by a straightforward argument and for E = E2 by hypothesis. Thus by Theorem 2.1. it suffices to prove (3.37) lim inf {flh9.-hefl/I9'—9I: e e 9 . |9'-e| < p} > o 940 c where 6c = {0) for E = E and 9c = {0)XQ for E = E l 2' We first prove (3.37) for the case E = E1. Note that "h9.-hefl = "h9._e-hofl by the translation invariance of Lebesgue measure. Thus it suffices to show (3.38) lim "he-hofl/IGI > o. 940 Note that in (3.38). we can use 1im instead of limit infimum because E is differentiable. Suppose (3.38) is false. Then 58 n (3.39) uh —h u g zuh 1 o 1 (k/n)‘h((k-1)/n)" n Eflh 1 (1/n)-h0II = Hh(1/n)-hOH/(1/n) where the first equality is by the translation invariance of Lebesgue measure. On letting n 4 fl in (3.39) and invoking the falsity of (3.38). "bl-ho" = 0. This implies P0 = P1 a contradiction to (A4). We now prove (3.37) for the case E = E2. We first embed Rk into Rk+1 by the mapping (x1.....xk) € Bk a (x1.....xk.1) € Rk+1. On this embedded space in Rk+l. all rigid motions can be represented by a set of linear transformations forming a matrix Lie group (of. Auslander (1967). Theorem 1.6.6). In matrix form the element in the matrix Lie group representing RTt is denoted by G(R.t) and is given by R t G(R.t) = O l k+1 Also letting Io be the identity on R . note that (3.40) IG(R.t) - I°| = |(t.R) - (o.I)|. 59 This matrix Lie group has for each x. a tangent space 2 T(x) C R(k+l) which is a (k+L) dimensional subspace with L = k(k-l)/2. By the theory of matrix Lie groups there exists a ball of radius r about 0. Br(0), in the tangent space and neighborhood of (0.1) in 9 such that the map S e Br(0) » exp(S) is a continuously differentiable homeomorphism (cf. Warner (1983). Theorem 3.3.1 and Definition 3.8). The exponential, exp(S), is defined as N exp(S) = 2(Sn/n!) 0 where we remind the reader that S is in the tangent space T(0,I). In order to prove (3.37). it suffices to prove (3.41 11 i f "h -h "I 9- 0.1 > 0 ) 94:0,?) 9 (O.I) I ( )l by the remark following Lemma 3.2. So suppose (3.41) is false and there exists OJ 6 (0,1) such that 3.42 "h -h H/ 9 - 0.1 a 0. ( ) 91 (0,1, I J ( )I For large J. OJ = exp(aJSJ) where Sj € T(Io). [81' = 1, and aJ € (0,r). By choosing a convergent subsequence. we can without loss of generality assume SJ 9 So 6 T(Io). 60 Then a n (3 43) I(Io) exp(ajsjllllajl - lf(aJsJ) l/lajl m n—1 n s flajl ISJI ___ elaJI since ISJI = 1 for all 1. Thus (3.44) lim sup I(0,I)-GJI/Iajl g 1 j-m by (3.43), since IaJI » 0. Also (3.45) lim flhOexp(aJSJ)-h°exp(aJS°)fl/IaJI = o j-W by continuous differentiability of the map S e Br(0) é hOexp(S) (which we get by the chain rule) and standard differential calculus. Thus since flhOexp(aJS°)-hfl/Iajl S {"haexp(aJSo)-h0exp(aJSJ)fl + "hoexp(aJSJ)-hH}/Iajl. 61 we have (3.46) lim flhOexp(a So)-h"/Ia | = o H, J J by (3.42), (3.44). and (3.45). But the map a 6 (-r.r) 4 hoexp(aS°) is differentiable by the chain rule. Hence by (3.46) (3.47) lim "hoexp(aS°)-h"/Ial = o. ado Let a e (0,r). Then 0 N (3.48) HhOexp(a°S°)-hfl g f"h°exp(na°S°/N) - hOexp((n-1)aoS°/N)fl = flhOexp(aoSo/N)-hH/(1/N) for all N € N where the inequality is by Minkowski’s inequality. and the equality is from the invariance of A under rigid motions. Thus II 0 "hoexp(a°S°)-hfl by (3.47), and letting N 4 m in (3.48). Hence if 90 = exp(aOSO), P = P6 . a contradiction to the O 62 identifiability assumption in (A4). Thus (3.37) is true in the case E = E2 and we have proven the converse portion of part (a). (b) First we convolute P with N(O.I). Then note that by characteristic functions. Proposition A.4. and the invariance of N(O.I) under rotations. (A4) is satisfied prx(o.l). for the experiment Also by Proposition 3.3 EP*N(O’I) is differentiable. Hence by Proposition A.4 in the appendix. it suffices to prove the result under the additional assumptions that P E A and E is differentiable. By the proof of (a). (3.37) holds. Hence by (1.7) the result follows. 0. Ihegrem 3.4. Let (Bk.$(kk).P) be an experiment and E = E3 . If E satisfies (A4) then the following are true: ” N m R! (a) P“ = are 4» P“ = up; for all {9n1}' {Gui} l ni 1 ni such that {9 } C 8 C 9. where 9 is compact, ni c c Q " 2 (3.49) lim sup BIG -9 I < w. n 1 ni ni and 63 (3.50) lim inf inf{det(A )t i e M) > 0. ni 11-” ~ ~ N where (tni'Ani) = Gni' if and only if P h and E is differentiable. (b) If P E h and E is differentiable. then 0 Q n ~n ~ ~ P = HP9 A P = HPG for all {eni}’ {eni} such that 1 ni l ni {eni} C 9c C 8. where 9c is compact. and @ ~ 2 (3.51) lim sup 2'9 -9 I = w new 1 ni ni or (3.52) lim inf inf{det(A )= i e N) = 0. ni new Proof: (a) Suppose (3.49) and (3.50) both hold. Then without loss of generality. if eni = (tni'Ani)’ we can assume tn1 = O for all n.i 6 fl and ( Sui} C 6; C 9. where 9; is compact. Exactly analogous to the proof of (a) in Theorem 3.3. P E A and E is differentiable. For the converse. (Al) and (A3) are true trivially while (A2) follows by Lemma 3.1. Since invertable affine transformations again form a Lie group. an exactly analogous argument as in Theorem 3.2 shows that (3.37) holds in this case also. Thus by Theorem 2.1. the converse follows. 64 (b) By the proof of (a). (3.37) holds. The result now follows by (1.7) 0. Remark. We conjecture that a more complete converse result also holds in the case i = 3. Namely we conjecture that if P is any measure on (Bk.$(kk)). and E = E3. and if {eni} C kadc. where dc is a compact subset of d. then u m N UP A HP~ for all {9 } satisfying (3.51) or (3.52). 9 6 ni 1 ni 1 ni The main difficulty is that the technique used in the case 3 i = 2 does not work here. Namely we are unable to prove (or disprove) that if EP satisfies (A4). then E§*N(o-I) also satisfies (A4). 65 CHAPTER IV STATISTICAL APPLICATIONS 4.1 Asymptotic Normality of Likelihood gotio for Triangular Arrays Part of the proofs in both Theorems 2.2 and 2.3 can be combined with a result of Oosterhoff/Van Zwet (1979) (Theorem 2. pg 162) to prove asymptotic normality of the likelihood ratio of a triangular array with asymptotically negligible components from a differentiable experiment. More specifically under the above assumptions. the likelihood ratio converges weakly to W(-02/2.02). The statistical importance of this is well known (cf. Hajek/Sidak (1967). pages 208-210). For completeness and ease of reference. the necessary result of Oosterhoff/Van Zwet is stated below as a proposition. 0 . = n o 0 Proposition 4 1 Let En1 (0n1 ’ni {Pn1 Pni}) be an experiment for n 6 N and 1 S i S n. P = HP . P = HP .. 1 ni 1 n1 11 ~ and Ln = flog(fn1/fn1). Then (4.1) 2(LnIP“) 3 N(-02/2.02) and 66 (4.2) :1: inf{Pn1(Ilog(fn1/fni)| 2 e): 1 2 i 2 n) = o for all e > 0 if and only if n 2 ~ 2 (4.3) lim 2H (P ,P ) = a /4 ndfl 1 ni ni and n ~ 2 (4.4) lim 2 1 _~ (h -h ) dv = 0 new 1 I {Ifn1 fni|)6fni} ni ni ni for all e > 0 where v = P +3 ni ni ni' Theorem 4.1. Let E = (0.3.{P9= 9 € 9}) be an experiment satisfying (A1) through (A4) and which is differentiable at °th dv being 6 = 60 6 9 with the matrix I vhe 9 n non-singular. Let Pn = HPe . and L be as in the l o n previous proposition. and assume (4.5) lim sup{|9°-9 |: 1 g 1 g n} = o ni new and “ ~ T 2 2 (4.6) 1im 2u(6 -6 ) ovh u = a /4 > o. ni o 9 11-300 1 o 67 Then (4.7) 2(Lnlpn) 3 x(—a2/2.02). Proof: By (4.5). (4.6). and the non-singularity of T I (vheovhe) do. “I ~ T 2 2 ~ I lim sup 2 "(9 -9 ) °vh H -H (9 .6 ) n 1 ni 0 9° 0 ni “ ~ 2 ~ T 2 2 ~ = Ii: sup ileni-eol |u(en1-e°) .vheofl -H (6°.6n1)ll " 2 l9ni-eo| and thus (4.3) is true. Also by (4.5). the proof of Theorem 2.2. and (2.15). (4.4) is true. Thus Proposition 4.1 implies the desired result. 0. Remark. If the hypothesis in Theorem 4.1 is strengthened to include regularity of E and local convexity of 9. then 90 can be replaced by {eni} where {eni} C 9c C 9. with 90 being compact. to get a stronger result implying a kind of uniform asymptotic normality. The proof is exactly analogous to the one Just given except for the use of the proof of Theorem 2.3 in place of the proof of Theorem 2.2. 68 4.2 Necesoary Conditions for Consistency By using Theorem 3.2 (or Theorem 3.3) we can prove a necessary condition for consistency of estimation in a model which generalizes that of the nonlinear least squares given in Wu (1981) which was concerned only with translation. The result is stated precisely below but the proof is omitted since it is similar to that given in Wu. Theorem 4.2 Let (0.3.P) be a probability measure space. 9 be an arbitrary parameter space. and 61.62... be i.i.d. k-dimensional random vectors. Assume that Y1.72.... are k-dimensional random vectors given by Y1 = f1(9)e1 for i G N where f1=6 4 {Rth t 6 Rk. R 6 3). Suppose the experiment E = (Bk,$(Rk),{P(t R): (t.R) e kaa)) satisfies (A4). P E A. and E is differentiable at (0.1). If there exists an estimate 9n(Y1.....Yn) such that 9n(Yl.....Yn) 4 9 in Pe-probability for 8 € 9. then °° 2 2 (4.8) 2|c,(e)-t.(e')l +IR,(e)-R.(e')l = w 1 for all 9' fl 9 where f1(9) = R1(6)T and c,(e) {1(9') = Ri(9')Tti(e.). 69 Remark. Theorem 4.2 could be extended to allowing f1: 9 4 {ATt: A E d. t E Bk) again assuming the corresponding experiment satisfies the conditions stated above and {det(A1(6))= i e N) is bounded away from O and is bounded above. In this case the necessary condition for consistency is that for all 9' t 9. an analogous condition to (4.8) holds. 70 APPENDIX Proposition A.1. Let (0.3.v) be a a-finite measure space and let {fat a G I) C L1(v). Then (fa: a 6 I) is u.i. if and only if for all sequences {an} C I there exists a subsequence {an.} such that {fa ) is u.i.. n Proof: The "if" part is immediate by the definition of uniform integrability. For the "only if" part suppose (fa: a e I) is not u.i.. Let g G L1(v) be such that g > O everywhere. The existence of such a g is guaranteed by the a-finiteness of v. Now there exists 6 > 0 such that for each n 6 N there is an an e I such that (A.1) ] (Int-ng)+ dv > e. Let h e Ll(v). Then by the Lebesgue dominated convergence theorem. f (h-ng)+ dv 4 0. On combining this with (A1). (A.2) lim inf f (If I-h) dv > e. n + 11-)” Since h was arbitrary. this implies there does not exists a subsequence an. such that (fa } is u.i.. D. n O 71 Proposition A.2. Let E = (a .3 .(P".P“)) and n n n n = (0A.3£.Qn) be two sequences of experiments. Then Pn 4 Pn (Pn A P“) if and only if anQn 4 anQn (anQn A anQn). Proof: Let Pn(Fn) » 0. Then anQn(anD£) » 0 which in ~n n . _ ~n turn implies P xQ (anfln) - P (Fn) 4 O. For the converse let anQn(A ) 4 0 where A € 3 x3'. n n n n Then let Anlw. = (o e on: (o.o ) 6 An} and by Fubini n n n n . P xQ (An) _ I P (Anlw.) dQ (o ). Thus Pn(AnI.) a o in Qn-probability. Let 6 > 0. Then there exists p > 0 and N 6 N such that Pn(Fn) < p and n 2 N implies Pn(Fn) < 6. Thus liquup Pn xQn (An ) S lim sup I1{Pn(An | )2P}Pn (An I. ) dQn + lim sup I1{Pn(An I )