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Choose d(O',p, - < 5P, 3) as in Assumptions 8 and 9. 14) Po [Ilog Vn(s, i') d< _Pot (i, ', d(I',p,i3)) < -n/ - +log m(p, ) . 15) p,/)) Y(xi,0',d(O', < inf exp -Pt- logm(p,') -n3 - logm(p,/) n p(t, 0', )) d(O', p, which is finite by Assumption 9. If inft for n > _logm(p, p(t, 0', d(O',p, 3)) is attained for t = to which is strictly positive by Lemma 1 ofe-t+ [4], we can choose, N(p, /3)d max (I - logm(p,/) ' tologm(p, log 2 ))<0. <-! 16) is bounded above by e-n(t-P) +pn for n > N(p, 0). 17) 1 n n log sup {IPol[Tn > t] : 0 Bo} 1log (e-n(t-) limsup n n = max[-(t - p), log p].

S. P0. lim inf N(e, s) > E--+o- loge J(O) 3. General Assumptions and a Useful Lemma Before giving further structural assumptions needed for Sections 4 and 5 we prove a simple general lemma already implicit in [4] stating a useful sufficient condition for a sequence {Tn} to be optimal. 1) 1 lim - log L,(s) = -J(0) n n 22 Bahadur and Bickel with Po probability 1 for all 0 E 8 - 00. 1) implies, (cf. s. s. P Lemma 2. s. Po, (i) lim inf n (ii) lim sup log (1 - G,(t)) d -t, n whereG,(t) = inf{Fn(t, Bo): 0oE o} and 0 ranges over E - 0o, then {Tn} is asymptoticallyoptimal.

22] SHAO, J. (1999). Mathematical Statistics. Springer, New York. [23] STIGLER,S. (1986). The History of Statistics. Harvard Univ. Press, Cambridge, MA. [24] TUKEY, J. W. (1961). Statistical and quantitative methodology. In Trends in Social Science (D. P. ). Philosophical Library, New York. (Reprinted in Vol. ) [25] TUKEY, J. W. (1962). The future of data analysis. Ann. Math. Statist. (Reprinted in Vol. ) [26] WALD,A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large.

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