132

Th . Pfaff

4 . 13 . Definition. Т*^^^ [Т*^^^] denotes the subclass of all elements of T^^^ [T^^h satisfying relation (4.4).

4 . 14 . Theorem. Let (f), (jj), fjjjj be satisfied.

( i ) Then for every (T'^^h^^^ ^ T^^^ [T^^^^] and for every WeW^^^ [W^^^]

/ W (Tf""', в) dP^ = П-' /il (в) + n^l^ ß2 (0) + «-2 Мз (0) + о Гя-'Л (4.15)

locally uniformly in в. (ii) Letq € Q be defined as in Theorem 4.6. Then for (T*'^>)^^-^ e

7 * ( 2 ) [7*(3)j ^^^ sequence т'"") := T*'""' + ai"^ ф^^^ (T*Ml « G N, is in 7-*(2) [j''0)]and for every H^Ga/(l> [W^^)]

/ W(ff'^K S) dP^ = n"^ Ai (6) + я-^/2 Д2 (Ö) + n'^ Дз (<9) + о (n'^J, (4.16)

locally uniformly in в. We have ßi = Mi, Д2 = M2> ûfAzJ Д3 < Дз/ f/zw meaws

jW ( f<''> , e)dF^ ^iWfTf'^), e)dP^-^o(n'^). (4.17)

toca / / y uniformly in в, for every WeW^^^[{JJ^^^l In this sense, for every (Т^'^О^^^^ ^ T*^^^ [T*^^^] /Ле ctos ofestimator- seqUences (T*^^^ + /i~^ 0 f^*^^^^JnGN' ^ ^ ß' '^ asymptotically essentially complete of order о (n'^) in T*^^^ [Г^^)].

Note that for the version of Theorem 4.14 stated in square brackets rather weak assumptions are needed. Firstly, the sequences in T^^ are not assumed to have totically bounded moments (whereas the elements of T^у and of T^^ satisfy (3.2)). Secondly, the functions r-^W (г,в) have to be smooth only at one point (whereas the elements of L^ ^ are smooth at every point).

5 . Regularity Conditions

The conditions (j) — (jjj) refer to the family of probability measures under ation:

( j ) P^ I A, Ö G 0, are mutually absolutely continuous, (jij) Zr (Ö) is positive definite for Ö G 0.