1512513 I .LEES? -‘ RY .' RflcfiEggm gt Unive /\ may \I This is to certify that the dissertation entitled '<‘ ‘ at ob; 1m VN W W presented by W.W.§oflmflg has been accepted towards fulfillment ofthe requirements for Dev D degree in thA/WMA \lobm W‘W Major professor DMMQLOMB .MSU is an Affirmative Acrionx Equal Opportunity Institution 0—12771 )V1531;1 RETURNING MATERIfigg: Place in book drop to LIBRARIES remove this checkout from JIIIKSIIIL. your record. FINES will be charged if book is returned after the date stamped below. ,": i.’ .3“ A 3‘. , ’ ' ‘ - '. , a - _ mg. 4 I}. J .P‘! 3‘ '*y m} r . . ‘ r - ' . ‘ 0N ADAPTIVE ESTIMATION By Anton Schick 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 l983 :41, W30 ABSTRACT ON ADAPTIVE ESTIMATION By Anton Schick A general method for the construction of locally asymptotically minimax (LAM) - adaptive estimates is given under conditions weaker than those in Bickel (l982). In particular, we show that Bickel's condition 5* is not necessary for LAM-adaptive estimation and replace it by a weaker condition. This new condition is found to be necessary and suf- ficient for a class of estimates to be regular, a property which implies LAM-adaptivity under Stein's (1956) necessary condition for LAM-adaptive estimation and which coincides with Bickel's notion of adaptivity. We demonstrate our method by constructing an LAM-adaptive estimate in a situation where condition 5* fails. To my parents, my wife Jeanette and my son Andreas ACKNOWLEDGEMENT I wish to express my sincere thanks to Professor Viclav Fabian for his guidance in the preparation of this dissertation. The advice and encouragement he gave are greatly appreciated. Also, I would like to thank Professor H. Koul for awakening my interest in adaptive estimation and for his review of my thesis and Professors J. Hannan and S. Axler for serving on my committee. Finally, I wish to thank my parents for their constant support throughout my studies and my wife Jeanette for her understanding and encouragement. 2. 3. 4. TABLE OF CONTENTS Page Introduction ....................... l A necessary and sufficient condition ........... 3 An example ........................ l4 An auxiliary result .................... 21 Bibliography ......................... 25 I. INTRODUCTION In a recent paper Fabian and Hannan (1982) use results on locally asymptotically minimax (LAM) estimates for locally asymptotically normal (LAN) families to reformulate Stein's (1956) heuristic arguments on adaptive estimation. The authors define LAM adaptivity of estimates, prove that a condition S, due to Stein, is necessary for the existence of such estimates, and give a sufficient property - regularity - for estimates to be LAM adaptive (see their Theorem 7.10). Bickel (1982) formulates a condttion 5*, stronger than S. He then constructs regular (and thus LAM adaptive) estimates under Condition 5* and when estimates of the nuisance parameter are available. He uses this result to obtain regular estimates in several important cases. Bickel states that 5* is “heuristically necessary" for the existence of LAM adaptive estimates (preceding Conditon C, following Condition H). We give a simple counterexample to the necessity of 5* and obtain re- sults on the existence of regular estimates without 5*. It is seen that if 5* does not hold regular estimates are more difficult to con- struct in that a certain rate of convergence for the estimate of the nuisance parameter is required. We consider estimates of a certain type and obtain a necessary and sufficient condition for such an estimate to be regular. The class of estimates we consider includes estimates considered by Bickel, but it is a larger class. The results described above are derived in the case of i.i.d. observations under weaker conditons than Bickel's regularity conditions (see Remark 4.4). Some notation will be introduced next. If P,Q are probabilities on a o-algebra x, and 0+ is the absolutely continuous, with respect to P, part of Q, then any Radon-Nikodym derivative of 0+ with respect to P will be called a pseudodensity of Q with respect to P, and also a pseudodensity of J-dQ with respect to J-dP. He shall talk frequently about expectations using terms which make sense when applied to the probabilities. If E and F are expectations on a o-algebra 5, then dF/dE denotes the set of all pseudodensities of F with respect to E which are non-negative and finite valued. k I! denotes the k—dimensional Euclidean space._B_k the o-algebra of the Borel subsets of Rk, R = IR], g: g]. < > will be used to denote finite or infinite sequences, and, in particular, points in 19‘. In matrix calculations, points in IE‘ are columns. .1 denotes the identity matrix. The dimension is not displayed in '1' but will be clear from the context. If Hn IS an expectation on a o- algebra 1mg" an <§n,§K> k measurable transformation for each n = l,2,..., and if c E I2 , then (i) we write gn +'c 1n -prob. if an{u + O for every gn-cu > e} e > O and (ii) we say is bounded in -prob. if is tight, where Fn is the distribution of gn under Hn' 2. A NECESSARY AND SUFFICIENT CONDITION We begin by specifying the asymptotic estimation problem we shall consider throughout this paper. 1. Assumption. 9 is a non empty set satisfying 9 = e] x 92 with 91 an open subset of IE". 9 = is a point in o. 5. is a c-algebra and for every 6 E e, E is an expectation on 5, X],X2,... <5 is a sequence of s-dimensional random vectors which are independent and identically distributed with distribution F5 under Ea‘ for each 5 e 9. There exists a o-finite integral J such that each F6 has a density f3 with respect to J. For every 5 = E o the map u 6 G1 + f?” v is differentiable at t in L2(J) with derivative > h5 = h(-,t,v). Furthermore J heh; is nonsingular and the map u E o] + h(-,u,ez) is componentwise continuous at 91 in L2(J). 2. Notation. For every 6 E o and n = l,2,... we denote by 5n the o-algebra generated by X1,...,Xn and by End the restriction of E5 to fin. We set E = (Ené’é é e> and Ev = <5 v E oz. For convenience in notation we shall often write f(-,t,v) n’ t E 61> for instead of f and similarly for other functions gs, 5 E e. We also set for 5 E o and n 1,2,... . -T (l) u(s) 4 a héhé 3". 6 f6% {f5 > 0} and n (3) Yné (nM(6))'L2 Z By 5 we denote the set of all 5 = in <3 for which M(5) is nonsingular and u E 91 + h(-,u,v) is componentwise continuous at t in L2(J). 3. Remark. We are interested in estimating the first component 51 of an unknown point 5 in o. o] specifies our knowledge about this component while 92 summarizes our knowledge about the second component, the nuisance parameter. If 5 E 5 we obtain from Theorem 4.8 in Fabian >. 6 In this case we want our estimate to be LAMA(A5,5) where A5 is the and Hannah (1980) that E5 satisfies condition LAN<5],nM(5), yn 2 class of all subproblems which are LAN for some M(5) and En and satisfy Stein's necessary condition (N(5))12 = 0, see 6 Fabian and Hannan (1982), Section 7. By Theorem 7.10 in the same paper this can be done by constructing an estimate which is regular at 5, i.e. satisfies (1) (nM(5))%(Zn-51) - yné + o in - prob.. Since 5 is unknown, this suggests to construct an estimate which is (globally) regular, i.e. regular at each point in 5. In some cases, however, estimates which are regular at just one point, say a, are also of interest. For this reason we restrict ourselves to the construction of an estimate regular at 6. This will facilitate the treatment and the reader will have no difficulties to see under what conditions this estimate is globally regular. 4. Remark. Bickel (1982) defines adaptivity at 6 for an estimate by s (1) For every sequence in 91 such that is bounded the distribution of (nM(e))E(Zn-tn) under E converges weakly n to the m-dimensional standard normal distribution. Condition (1) is equivalent to (3.1). This follows from Theorem 6.3 in Fabian and Hannan (1982), Theorem 6.1 in Bickel (1982) and the note thereafter. Hence an estimate which is adaptive at e in Bickel's sense is regular at e. Bickel claims that the existence of a regular estimate implies that each subproblem obeying his regularity condition R satisfies Stein's condition (M(d))12 = O, for every regular point d. But the proof of this claim is incorrect due to an inappropriate reference to Héjek (1972): Bickel considers only local alternatives for the paramenter of interest and not local alternatives of both the parameter of interest and the nuisance parameter as needed in Héjek's Theorem 4.2. Thus it remains an open question whether Bickel's claim is indeed true. Next we define a map 0 from G into I?" by Q(tsv) = J 2(°,t,V)f(’,t,62) if the integral is well defined and 0 otherwise. Bickel's condition 8* is (5*) Q = o The example below shows that (5*) is not necessary for the construction of regular estimates. Another example is given in Section 3. 5. Example. Let 01 = (O,m), oz = B? and let the Xi's be normal random variables with mean u and standard deviation 0 under E’ i.e. we want to estimate the standard deviation in the presence of an unknown mean. Easy calculations show that assumption 1 holds and that (1) z(°.0su) = -o" + o'3(--u>2 (2) 5 = o and (3) (No.11) = o‘3h-e2)? , for o s 0, M2 e IR Furthermore for every 5 E o, the full problem E satisfies condition LAN<5,nN(5),?n5>, where with 5 = ~ _2 2 o (4) M(6) = o o l and N _% n X-‘ll 2 -}' n X.-u (5) ind = <(2n) .2] ((-%;—i - 1). n 2 .2] -%;—-> J= J: and the estimate defined by ~ _ -1 " -— 22v — (6) Zn - <(n .; (xj-xn) )2 , Xn> 3-1 with 76 the sample average satisfies (7) (nfi(6))%(in~6) - t'ms + o in 56 - prob.. Verifications of the above are easy. We refer the reader to example 9.2.12 and Theorem 9.4.33 in Fabian and Hannan (1983).From (4) and (7) we obtain that Stein's necessary condition for LAM-adaptive estimation holds and that , with 2n the first component of in, is a globally regular estimate. 6. Remark. We now motivate a natural choice for an estimate regular at e and then give a necessary and sufficient conditon for this type of estimate to be regular at e. We begin with a definition. 7. Definition. We say is an auxiliary estimate at e if each . _ 35 _ . Un 1S a 91 valued in measurable random vector and -prob.. we say is a consistent estimate of the information matrix at e if the wn are positive definite matrix valued random vectors -% -% + - _ on 5n such that M(e) wnM(e) l_ in (Ens) prob.. We say is a local sequence if is a sequence in a] and is bounded. 8. Remark. In Section 4 we prove that £(Xj,Un,ez) "1V3: ._l _ -l - Un + (nwn) (.1. is regular at e if is a discrete auxiliary estimate at e and is a consistent estimate of the information matrix at e. For a discussion and the use of discrete estimates we refer to Fabian and Hannan (1982) and Bickel (1982). The estimate in (1) is of limited practical value since it presupposes the knowledge of 92, but it suggests an obvious candidate for a regular estimate. Simply replace 82 by an estimate. A different method consists in estimating the score-function E(-,-,ez) directly as Bickel (1982) does. But the present method serves us better for the purpose of illustration. Substituting an estimate for 82 in (1) has to be done with some care. For technical reasons we adoot an idea Bickel (1982) uses, but modify it to obtain better estimates of the nuisance parameter. Recall that Bickel splits the sample in two unequal parts, estimates the nuisance parameter based on the observations in the smaller subsample and evaluates the scorefunction only at observa- tions of the larger part. We divide the sample in two equal parts, obtain an estimate of the nuisance parameter from each part and when evaluating the scorefunction with an observation of the first part we use the estimate of the nuisance parameter based on the second part and vice versa. Thus our estimates of the nuisance parameter are based on half the sample and not just on a small proportion of the sample. This improvement is vital, since it turns out that the estimate described above is regular at e if we can construct an estimate of the nuisance parameter with a certain rate of convergence. To have the above method well defined we make the following assumption. 9. Assumption. Assumption 1 holds. 62 is a topological space. The map 2(-,t,-) is measurable for each t E 01 and (I) Jué('9tav) " 2(°9t962)“2f('9tae "" 0 2) as in o converges to e. For every n = 1,2,... there is a measurable map hn from CR§)n into 92 such that Vn = hn(X],...,Xn) converges to 62 in Ee-prob.. is an auxiliary estimate at e and is a consistent estimate of the information matrix at e. 10. Remarks and Notation. Note that Assumption 9 implies (3.11) of condition H' in Bickel (1982) with Zn(-,.,x],...,xn) = é(.,-.vn). Indeed, we have (1) aué(-.tn.vk ) - é(-.tn.e2)uzi(-.tn.e2) + 0 T1 in Ee-prob. for every sequence in a] converging to 61 and every sequence of integers tending to infinity. Also observe that under Assumption 9 J E(-,t,v)f(-,t,62) is well defined for in a neighborhood of e. The estimate described in Remark 8 is formally defined by mn n (2) 2(U ) = U + (Mi )“(2 2(x.,U ,v )+ l I:(x.,U .v )) n n n n j=1 J n n2 j=mh+1 J n nl with a discrete auxiliary estimate at 9, mn the integer part of n/2 'and (3) an - hmn(X],. ,an) and Vn2 = h (an + 1,. .,Xn) 10 Typically will be a discretized version of . But we do not want the regularity at e of <2n(Uh)> to depend on the way we discretize. In other words, we want to be regular at e for each discrete auxiliary estimate at e. We now give a necessary and sufficient condition for this to happen. 11. Theorem. Suppose Assumption 9 holds. Then the following are equivalent. (1) <2n(Uh)> is regular at e for every discrete auxiliary estimate at B. (2) For every local sequence g . n Q(tn,Vn) + O in Ee-prob.. Proof: Note that (1) is equivalent to (3) is regular at e for every local sequence . Also recall (see Remark 8) that (4) is regular at e for every local sequence . We shall show thathor every local sequence A (5) n%(zn(tn) — Zn(tn,ez) - w"R (t )) + o in Ea prob., n n n n1))‘ = -1' - where Rn(t) n (an(t,Vnz) + (n mn)Q(t,V Combining the above shows that (1) is equivalent to ll (6) nENaiRn(tn) + O in Ee-prob. for every local sequence . By the consistency of and the independence of an and Vn2 (6) is equivalent to (2). Thus we are left to verify (5). Again by the consistency of , (5) is equivalent to m -g n n , (7) n (-E Tn2(xj’tn) + .2 Tn](Xj,tn))-+ O in Ee-prob., 3-1 J-mfil where Tni(°’t) = £(°’t’vni) - £(-,t,92) - 0(t’vni) for i = 1,2 and t E 01 . Now fix a local sequence . Abbreviate E by En n and note that and are mutually contiguous. Next observe that for j = l,2,...,!"n (8) E‘(Tn2(xj,tn)lxmn+,,....xn) = o a.e. En n and thus m (9) Enuln'i ,2: Tnzlxj,tn)llzlxmn+1....,xn) -1 "I"-— 2 _. = n jg] EnlflTn2(Xj,tn)H Ixn%+1.....xn) a.e. En J “é(°atnavn2) ' é('3tn992)“2f('stnsez) a.e. E IA n by a property of conditional variances. Using the mutual contiguity of and and (10.1) we find that (10) converges to zero in -prob.. This shows that n Tn2(xj’tn) + 0 1n -prob.. _1 6 J (10) n urvj a 1 In the same way we obtain that 12 -g n + . ._ _ (11) n jZm +1 Tn1(xj’tn) 0 1n prob.. n Using the mutual contiguity of and we conclude \a from (10) and (11) that (7) holds. This completes the proof. 12. Remarks. Note that (11.2) is trivially satisfied if (3*) holds and in this case consistency of guarantees the existence of a estimate regular at a. This is Bickel's (1982) result. But if (5*) fails consistency of alone does not suffice to construct an estimate regular at e. In this sense LAM-adaptive estimation is more difficult in cases when (5*) fails. Assume for the moment that oz is an open subset of IE) for some positive integer p and that the whole problem E satisfies conditon LAN <5,nfi(5),?n5> for each 5 E 6. Also assume regularity conditions which allow the Taylor expansion _ T ~ Q(tav) ' Q(taez) ' (V'az) M]2(8) + 0(“t-81“) + 0(“V'92“2) as + e. In this case the necessary conditon for LAM-adaptive estimation N]2(e) = 0 implies that 2 (l) Q(t.v) = 0(Ht-61H) + 0(“v-9 ) ll 2“ Note that Q(t.62) = 0. Thus (1) shows that (11.2) is satisfied if % . (2) n (Vn-ez) + O in Ee-prob.. Obviously (2) is weaker than 13 (3) n%(Vn-ez) is bounded in Ee-prob., a condition which together with the existence of an auxiliary estimate at e and of a consistent estimate of the information matrix at e suffices to construct LAM-adaptive estimates if Stein's condition holds (c.f. Theorem 6.15 and Theorem 7.10 in Fabian and Hannan (1982)). We remind the reader that (1) is satisfied in example 5. 3. AN EXAMPLE 1. Description of the example We consider the regression model (1) Y. = a J 1 + 82(Tj) + e. j = 1,2,... J where T],T2,... are i.i.d. random variables with uniform distribution on [0,1], e1,e2,... are i.i.d. random variables with Lebesgue density 9 and independent of T],T2,...,e1 is a real number and 92 is a real valued absolutely continuous function on [0,1] with square integrable 1 derivative 65 and J 62(t)dt = 0. We suppose that the density 9 0 satisfies the following conditions (2) i x g(x)dx = 0 (3) 1x2 g(x)dx = «:2 < .. (4) g is absolutely continuous with derivative 9' and has finite Fisher information . 2 1(9) =lfl§§§§L dx (5) with L = - ”(iii—Hg > 0} we have (5a) J Jm (L(x + v(t)) - L(x))2 g(x)dxdt + 0 O 14 15 and l 1 (5b) ID I” L(x-v(t))g(x)dx dt = O (J0v2(t)dt) 1 l for J v(t)dt = O and J v2(t)dt +.0 . O 0 Note that (5) is satisfied if L is twice continuously differentiable with bounded derivatives L' and L". 2. Remark, The above regression model satisfies Assumption 2.1 with a] = 12,92 the family of all realvalued functions v on [0,1] which are absolutely continuous with square integrable derivative and satisfy ,J; v(t)dt = O, Xj = , J the integral induced by the Lebesgue measure on the Borel field :ofR x [0,1] and f5 and h5 defined by f(x,t,v) = g(xl-t-v(x2)) and h(x,t,v) = 3, L(xrt-vuz))gié(x]-t-v(x2)) with x = in II x [0,11 and 5 = in e. The differenti- ability in L2(J) follows from (1.4) and Lemma A.3 in Héjek (1972), while the required continuity of the derivative is a consequence of Theorem 9.5 in Rudin (1974). Note also that o = o by the translation invariance of the Lebesgue measure. Furthermore, if we endow 02 with the topology induced by the norm H-Hz defined by “VHS = [; v2(t)dt for v in 02, then (2.9.1) follows Y.> is an 1 J ),...> is a from (1.5). Also observe that the sample average satisfies % 2 . (1) n UVn - ezuz + O in Ee-prob.. We shall now construct such an estimate. 3. Construction of the estimate . We let denote a seouence of positive integers and set bn = a;]. For each n = 1,2,... we partition the unit interval [0,1] in an intervals Ini’ 1 = l,...,an of equal length bn' We let mni denote the midpoint of Ini and Xni the indicator of Ini' Furthermore we assume that the intervals Ini are numbered in such a way that m . < mnk for 1 5 J < k 5 an. Next we set "J (1) '1 i U = n _ Y. and -l " . (2) Yni - (nbn) jg] ijni(Tj) , 1 - 1,...,an and define Vn by / Ynl'Un 0 f t 5 mnl A t"mni (3) Vn(t) = [Yni-Un + —E;—— (Yni+1-Yni). mni f t < mni+1 Y -U m < i; <1 K nan n nan - - It is easily verified that Vn is a oz-valued random vector, e.g. 1 an (4) J Vn(t)dt = E b v . - U = 0 0 i=1 17 4. Lemma. If the sequence is chosen such that 4 2 n + O and nbn + w (l) nb then nEE “V -e U2 + O a n 2 2 Proof: For i = 1,2,...,an set 1 (2) C - = an J xni(u)62(u)du and note that (3) EeYni = 61 + Cni Easy calculations show that 2 -1 2 l 2 2 (4) E6(Yn1-61-Cnl) f 3(nbn) (9] + “92“2 + 0 ) and (5) 59(un-e,)2 5 n"(o2 + 16213) Next note that by the Schwarz inequality for O 5 U1 < u2 f l U 2"2. 2 2.2 (6) (e2(u2>-e2(u1)) = (j e2(x)dx) g (“2’”1) j (e,(x)) dx U] U] Using this and the Schwarz inequality we obtain (7) i(62(t)'cni)zxni(t)dt ilan l (e2zxn,+,(u)xn,> (jgl L(vj-un-vn2(ij))+ jgm +1 L(vj- -U vn,(r ))) n is a regular estimate, where is a discretized version of and mn, V and Vn2 are as described in Remark 2.10. n1 * Next observe that conditon (S ) does not hold for a proper choice 0f 9. 8.9- With g(x) = % e-|xl we obtain for v = 82 + r in oz by easy calculations 1 m (2) Q(t ,v)= I ! sign(x- r(u»g(x)dx du o—e . 1 = J sign(r(u eINu)I -l)du O 1 l sign(r defined by (5.1) is LAM-adaptive (Aa,e) where A6 is the class of all LAN We)12 = 0 (see Remark 2.3). We now describe a class of subproblems which belong to A6: Let r denote the family of all one to one maps y from an open neighborhood 5 around 0 in TE) into 92 for some positive integer p satisfying y(0) = 92 and 20 (l) “Yial'Y(0) - aTuHZ = 0(HaU) as a + 0 for some vector p = <5]....,¢p> such that wi is in oz for i = l,...,p and N = I] w(u)wT(u)du is nonsingular. For y in r weodefine the subproblem by 90 = o] x yLS] and o(t,y(a)) = for E I2 x S (for the defintion of sub- problem see Definition 7.3 in Fabian and Hannan (1982)). This subproblem satisfies condition LAN with (2) E = 1(9) 0 N and ~ ~ -g n (3) Yn = (0M) .2 L(Yj-81-92(Tj)) J l and hence satisfies “)2 = o. The above follows from Theorem 4.8 in Fabian and Hannah (1980), since the map E I? x S +-f%(-,t,y(a)) is differentiable in L2(J) at <6],0> with derivative A given by (4) 5(X) = 5(x1,61,62) for x = in tzx£0,1] This is easily verified using (1.4), the arguments in the proof of Lemma A.3 in deek (1972), Theorem 9.5 in Rudin (1974) and the properties of y. 4. AN AUXILIARY RESULT 1. Remark and Notation. In this section we shall prove that the estimate Zn(U n we abbreviate M(e) by M and y .62)> as given in (2.8.1) is regular at 6. To simplify notation ne by y". Also we shall use M(t) and y nt short for E E ,M(t,ez) and E a n t’ Ent’ , with t E o Yn 1' 2. Lemma. Suppose Assumption 2.1 holds, and are local sequences and g E dE /dE Then n nun ntn 109 gn-wl ;"tn + %“wn“2 + 0 in Eel~prob.. n = (nM)'35 X é(xj.t a w1th wn - (nM) (“n-tn) 60d n j=1 n’ 2)- V 'nt Proof: Let 5n = tn-e and Tn = {t E o]: t + 5n E 0]} and set for 1 c t E Tn, Hnt = Ent+sn’h nt Jf%( ,t+s n,62) and hnt = h(.,t+sn,e 2) . Using the fact that the map t 6 o] + f%(-,t,92) is continuously dif- ferentiable at 61 in L2(J) we obtain for every local sequence <5n> with 5n in Tn and for every e > 0 21 (1) n J(hn6n-hne]-(5 '91)Tfine])2 .. 0 (2) allfinGIIIZXW-‘nelu > "%€“ne]} -> o and (3) M(tn) + M From (3) we obtain that M(tn) is invertible for all n 3 no, for some integer no. We now obtain by Theorem 4.5 in Fabian and Hannan (l980) that the family (Hnt’t E T", n 3 n0> satisfies condition LAN . This shows that n T ~ ~ 2 . (4) log gn-wn Yntn + lawn“ + 0 1n (”ne:-pr°b°’ with Eh = (nM(tn))%(un-tn). Now note that T ~ ~T (5) w v =w 1' n ntn n ntn and that by (3) <6) :1:an - 111nm? -» o The desired result follows now from (4), (5) and (6) and the mutual cont1gu1ty of and . 3. Proposition. Suppose Assumption 2.l holds, is a discrete auxiliary estimate at e and is a consistent estimate of the information matrix at 6. Then 23 is regular at 8. Proof: We have to show that (l) (nM)%(Zn-e]) - yn + 0 in E61-prob.. By the discreteness of and the consistency of it suffices to show that k _. ~ . (2) (nM) (tn 8]) + Yntn - yn + 0 1n Eel-prob., for every local sequence . Let be a local sequence and set un = tn + (nM)‘%u for u e lRm. With 9n 6 den“ /dEn we obtain from Lemma 4.3 in Fabian and t n n Hannan (1982) and the mutual contiguity of and ntn n61 (3) lo - uT( -t ) + %”uV2 + 0 in E - rob 9 9n Yn n h 1 6 P ., l with En = (nM)%(tn-e]). On the other hand Lemma 2 shows that , ~ 1 12 . (4) log gn - u Yntn + adud + 0 1n Ee1-prob.. Combining (3) and (4) shows that (5) uT(t + ~ - ) + 0 in E -prob n Yntn Yn a] ' for every u E film. From this (2) follows which concludes the proof. 4. Remark. Bickel (l982) constructs an estimate as in Proposition 3 under stronger conditions than ours. It is easily checked that his re- gularity conditions R(i), R(ii) and UR(iii) imply continuous dif- ferentiability at a] in L2(J). Also note that we can choose wn to 24 be M(Un) if the latter is nonsingular and .1 otherwise. However, estimates which are regular at e and do require the knowledge of 62 can be constructed under weaker conditions than ours; see Theorem 6.l5 in Fabian and Hannah (1982). The estimates constructed there are based on difference quotients rather than on the "derivative" z. Since the use of £ facilitates our treatment we have chosen to work with Assumption 2.l. BIBLIOGRAPHY BIBLIOGRAPHY Bickel, P.J. (1982). On adaptive estimation. Ann. Statist. 19, 647-671. Fabian, V. and Hannah, J. (1980). Sufficient conditions for local asymptotic normality. RM-403. Department of Statistics and Probability, Michigan State University. Fabian, V. and Hannan, J. (1982). On estimation and adaptive estimation for locally asymptotically normal families. 1: Nahrschein- lichkeitsthenrie verw. Gebiete 53, 459-478. Fabian, V. and Hannan, J. (1983). Introduction to Probability and Mathemathical Statistics. (Forthcoming book) Hajek, J. (1972). Local asymptotic minimax and admissibility in estimation. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. L, 175-194. Univ. Calif. Press (197?). Rudin, w. (1974). Real nd complex analysis, (2nd ed.). McGraw Hill, New York. Stein, C. (1956). Efficient nonparametric testing and estimation. Proc. Third Berkeley Synpos. Math. Statist. agg_Probab. L, 1874195. Univ. Calif. Press. 25 "'ll'flllfitllflljllfllflaflligfllfllfllfillllfl