Limits of experiments associated with sampling plans

Journal of Stztistic21 Plannimg and irifwence 19 (l988) l-29 North-Holland

thematik VII, Universitiit.Bayreuth,Postfach lOlZSl, 0-8.5l?OBayreuth, West Germany Received I3 October 1986; revised manuscript received 8 June 1987 Recommended by B. Torgersen

Abstract: Starting from Milbrodt (I985), the asymptotic behaviour of experiments associated with Poisson sampling, Rejective sampling and its Sampford-Durbin modiW&ion is investigated. As superpopulation modeis so-called p-generated regression parameter farri!W (1 s TI 2) are considered, allowing also the presence of nuisance parameters. Under some assumptions on the first order probabilities of inclusic~ it can be shown that the sampling experiments converge weakly if the underlying shift paameter families do so. In case of convergence the iimit of the sampling experiments is characterized in terms of its Hellinger transforms and its Levy-Khintchine representation, leading to criteria for the limit to be a pure Gaussian or 2 pure Poisson experiment =4ied to the situation of S~~,nlinag in the presence of rsss&Jrn respectively. These res-ultsarti1~+hnn Waiipyyy non-response, and to establish local asymptotic normality (LAN) under more restrictive co&tions. Applications also include asymptotic optimality properties of tests based on HorvitzThompson-type statistics, and LAM bounds and criteria for adaptivity, when testing or estimating a continuous linear functional in LAN situations. They especially cover the case of sampling from an unknown symmetric distribution, which has been subject to de*ti!ed investigations in the i.i.d. case. ~jK462 vau

e?- a b 5_3.k

O&UJCd

Ckzssification:Prima9 62D05; Secondary 62F99.

Key words and phrases: Poisson sampling; Rejective sampling; Sampford-Durbin

sampling; superpopulation models; L’-generated triangular arrays; nuisance parameters; weak convergence of experiments; local ,,ymptotic normality; LA bounds; adaptivity.

This paper contributes to t tion, when su the superpopulation parameter.

0378-3758/88/$3.50 0 1988,

olland)

find out about this amount, one first selects a random pendently determines a value of X for all indkiduals belon two-step procedure wi!lbe called a sa;lrpling iflcid!y introduced - a randomixa

rt or the whole of the population one chooses t re 5anzplingpkin means a some specified samph all (unordered distribution p ;(?2ihc; it is assumed that the characteristic f n chssicalsurvey without error. Fknce, the second step of th andomness; it reduces to recording the true characterisics Xj of sam ochran (I.939) points out that “the finite population itse!f le from some infinite popula?ion”. This at the charact~~s~~ andom vector, whose law , 8~ 63,the true parameter Mng unknown. Then the second step experiment is random, too. The family f ithin the superpopulation approach one may pursue two different aims: infer-

erg the o&r &nQ, parameter LB,the risk is

c0rqxdm? @fbot

given in S&~%3l _.C%@ _._

(4 - T& (x%~,i,.,))2P~ , some real function f of the su

tion

3

elusion probabilities could ective sampling, to iduak are included independentiy in

iIf ihb. ,zqzcied sampIe si is defined as

sfmplikvg r

r(s) = p(s 1size s = n),

pi is an integer, the corst%pO oisson sampling given that the sa se S.

Rejective sampling can also be realized as conditio W replacement and d.r?xmg probabilities

-fold drawing fro

with

H. Milbrodt / Experiments associated with sampling phzns

4

framework it is often possible to establish asymptotic formulae for the inclusion probabilities, for the variances of certain estimators and to prove their asymptotic eic (1p64), RO&I normality. Highlights in this development are the papers of e present work the first of which has also inspire (1972) and Iachan (198 and the monograph o .2. Experiments Recall that the mathematEicalmodel for a ‘r ’ random experiment is a statistical sisting of a sample space 52, a (ieqwi,zienr, i.e. a triplet E 8) of distributions on d which might ily {Ps 119 field cabs 2o of events an govern the process. Consider a sampling experiment with underlying sampling plan for the various individuals are independent. nd assume that the observations en, the superpopulation model is given by the product F= @ieI IF;:of experi)>, which describe the observations for the single ments Fi = (I?‘,$B(R’), { ;=~;A-Q!c of the sampling experiment are of the form i@A 7;wx-ii;-=_ iz I_ Sime Al outcome (s, (Xi)ias), where s is the sample obtained with probability p(s) in the first step and (Xi)icsare the characteristics of sampled individuals determined in the second step, the statistical experiment is pF:=

P(S) @ 4, SES

itzs

the mixture of product experiments @its F; belonging to the samples s, mixed by the sampling plan p: Its sample space is the free union of Euclidean spaces IRS (SE S), the o-field and the probability measures are defined by requiring that their trace on each IRSequal the Bore1 g-field &B(P) or p(s) &,,-_ ?9(8 t’ @I respe&e!y. TAaS.h:C-.‘,, - AUAAuvrrr ~-~dm-~~ wppriqwrst lUGllLll~llqj 5ji -a ycJia*I&,,

wit-h itc, __‘-~,*~rrrrH&.~wC1a ~qthprnatifial mnht . . ____ __L rravurr)

nJ7 will p’ . ..&a Opfa “I t4wmd -rL=c-w-

experiment generated by the sampling plan p and the superpopulation ere is a well-developed theory of statistical experiments, see EeCam Strasser (1985a) and BAlbrodt and Strasser (MS). Taking up a suggese discussion of his survey paper on “Comparison of Statistical , Torgersen (1982) applied this theo&ry to sampiing from a finite is results were non-asymptotic and referred to classical survey sampling. An asymptotic theory of sampling exbwrimcnts was iaitia!ed by ,Milbrodt

Ii. Milbrodt / Experiments associated with sampling plans

5

{I, ..*JVJ be a pop n of size NVE h\l, &, the set of samples from I,, and pv a issons&xpling plan 0 with probabilities of incllu‘EI,; in case its expected s le size rsD, = xie 1 pvi is an integer, rV ectke sampling defined by pv and rt its Sampiordcation. If Fyi is a statistical experiment for some parameter space 8 individual i~1,, (v E 94), we consi er the superpopulation models of the roN=wise products of the triangular array ptotic behaviour of sam VE hd, for (q,,)=(pvjo (qvj=(rvj and (q,,)=(I$), as NV+ models, we impose so-called L’-$en regression parameter arrays (allowing also nuisance parameters j: Let U# 0, ility measures MEW}beasetofpro on the line, QIU:= E~*Q~,,the transl QoUby TVlkr (UE Z;, a normed real vector space and w : B --) UN. Assume there is an rz 1 such that for every /3~ there is a C>O satisfying r every v E INlet I.

(d(m, . j denotes the Hellinger distance of probability measures,j Suppose also that the sequence ~(0) is constant, au assumption which is satisfied by all reasoaabk local reparametrizations w. Let 11’denote the univariate Lebesgue measure a&! -----:-L---..btlA rcIs.9 L’[O,1j the space of equivalence classes of r-times Lebesgu.=.= -I-. WVI’CI =- -.- &~.f4Phqn b lU,,W68 1 Urrrii-v’ird ‘;ii [0, 1j. Eve;;y a EL’[O, 1) generates “infinitesimal shifts’

denoted by q,,Pv8, a singleton), ~9will be dropped fro

6

H. Milbrodt / Experimentsassociated with samplingFlans

ties at the jumping points) to a sequence of k-sample location models with sampling proportions asymptotically equal to the lengths of the subintervals. Note that i these situations the regression constants I@,,~) tend to zero at rate NY’/‘, in i, as V-=L Thi takes care of the fact that in non-re ar cases, ifferent from fli are required to tain a n le (3.8) below). otic r’tssJlts of the fo ng chapters are given in t ~iilGX!lltS, ‘We fin recall some concepts and n from the theory of statistica! riments. Let T be a non-void pararreter set, a zt=l), GEA( }) is an experiment, let ?E cw))denote the (_r;lJ: -c,--+ [OS 8] OfE,, tte/d(T), as the function

Equivalence and wmk convergence of experiments may be put in terms of equality and pointwise convergence of their ellinger transforms (LeCarn, 1986). E is pairwise imper$mt if no two of its u~&~ly%rgprobability measures are disjoint. When aSis a normed vector space, then E is continuous if its parametrization is HelXingesuicontinuity of a set of experiments is similarly defined, E is staM r> 0, if it is continuous and if for every n E hd resealing of E by 3 1/r same as taking azindependent observations according to E. Retail from Strasser (1985b) that E is stable with exponent r iff it is continuous and its ellinger transforms are homogeneous of degree r as functions of the paralmeters,

e begin with an eouic~ntinui~~ lemma which applies to R= -WV-*~ =5~*izil.__ -wu~?lsce ,” EZ.I~I_L~AJ v E IN. For fixed p^sB’ we consider the subexperi 1 acL’[O, 11) Of Fyi (idV), F&T) :=@JEiv the sequence of samp us on L'[O, 1)for every .

experiments (q,_al(

t ,- m c !/YI)is translation invariant and

l’r3

yp

?‘\

H. Milbrodt / Experiments associated with sampling plans

c 1

jii7/‘dA”,

VE

J0

IN, adL’10, 1).

ence the assertion follow he importance of Lemma 3.1 stems from the fact that in case of equicontinuous sequences of experi ter space E cowverge --“-_. ence on a dense subset of S. No;v, we pr be established by y statement of Theorem 3.3, which is the main result of this section. Let 2” be a vector space and S#& statistical experrment F= (X,x {Rt r I(&) E TXB}) is partially transkztion invizriant w.r. t. t, if for every s E T the iranslate (X,jic;{&+ t,5 f (t, kJ) E TX 2’)) is equivaknt to F. 1 (t, j9) E IR’ x Denote by Ey : = (lRv9 iB(R?v)9 { yq#(p)” product experiments generated by {QrU1 (t, a) E II?’ x U} results we need the following assumption:

), e

the localized For our convergence v E iN ,-

(1) Ev-E, weakly for some continuous an pairwise imperfect experiment E=(~,~{PI,BI(t,B)EIR1X (2) 14’can be decomposed as E= &r@t&, where 8r is stable with exponent r and the parametrization of & does not depend on the location parameter f E m’. These assumptions need some discussi . Of course, the .2)(2) need not be uniquely determined. is stable with exponent :. Note that - without any additional assumptions - stability of E automatically holds if v is of the form w(B), = u+ v+#?,

j?E

for some u E U. ( his follows along the lin however, we may have different rates of s bility -wiflnot be sati ssumptions (A. 1) and singleton, i.e. if no nuisance parameters are

dt / Bperiments associated with sampling,~km

8

to experiments associated with ilar to the parametrization of our regression par generally assume that robabilities of inclusion are generate by a score function: arrays ow, we turn especial

orel-measurable map lie: [0,I]-+

, converge to 1: 1cdill. 3) hold. Then the sequence of F’, v E N, converges weakly to an infinitely divisible exllinger tra cl)? (A.2) and

is always partially t~a~lation inrvariantw,r. t. a, and also stable with RTcponentr

ling cme’ of the classi3.3. is also new in the ‘non cal &al asymptotk theory (n= I), i.e. when pling the whole popu---:---*A /?f iaiimi, Mid %VSihO-Iii mmia~lbc enrnmntnm yaiarrhudi, IV @i-ho@. If {a, b) CL’ [0,1) and no u Usaa8.Y nuisance eters are present, then by (3.3.1), translation invariance and stability

((jIbsor n(a- b)‘dA’)“‘,O) @

0, (jfpcbl n(b -

a)‘dl’)“‘}

l

0 vycuI..II A+AmA frnm f z zz1, this can also bW -___ Jpt~n~swn(19851 \-- -_,: Theorem 5. tnploying stability, (3.3.1) can be rewritten:

it s

S

lbrodt / Experiments associated with sampIing plans

H.

the set of interv

eas

e ste

If 01,E IR’,0, E !R’(VE !M&then II,- O,signifies that li 1,Pd, ...9 (a,, &J) C 2-x jumping points 0 x/+1--x/, /=o, . ..) aj(.Xl), ajvi

- Ntl’r

[N&l] + 1
ai(xr-1

i=[N,q]+l,

1=

In view of Amunptions (

n-1

rNw+ll

I=0

i=[N,xj]+2

= n-l + I

Pv,[N”x,]+l( Ul(X/- 1)(&X/ . .. ,

[N&r]) +U~(x/)([~xr] f 1-N&h%-I”, w(81)Nv9

(x/v I)(N,xI - [N,x/]) + uin(x/)CWvx/l+ 1-

NaWi-I”,

'dh~)N,,

ccasionally, we write Sm instead of S{r,_..,m). Uptder (A.2) the mqw

.

1

!i

1, l *,iP,) +@ n log ((P

IrBP l **’

0

*) are continuous on L’[O, l)* for every

(t?ZEiN,&&Jk

n:

p, !] --) [O,

11.

pparently, it suffices LOconsider the case that E=t”r is stable with expoet woE hJ, (/I,, . . . , rS,) E B” and 3:E S”. S&e E is continuous, infinitely divisible and pairwise imperfect, the map

f:(t*,~~~,t*;Y1,=**,~*)W1og ued and continuous on fR”X verges to (al’), . . . , tz$f’). Then

4tmd(z) ppose (@I”) . . e4a$&

A’fo(*~‘,...,~~‘;~*,...,IP,)~A*fv(~~o’,-;~,~~~’;~,,**~,~*)

E I?loC O*

con-

(A’)*

oreover, since E is stable with exponent r, the sequence is uniformly r-times indenote the maximum of 1 f1on {(t 19l*-9t*; SrBr, l a*,S*&) 1 lsii 5 l, !til S 1, lki=m}. Then

which clearly is uniformly integrabie. ence, the asserted continuity follows from the generalized dominated convergence theorem (cf. Chung (1974), Theorem 5.4). El 3.3. Let B. be a finite subset B, equipped with the discrete a 3.1, 3.5 and Lemma (3.6) in lbrodt (1985) the sampling experiments (p,F,), EN convergence weakly on L’[O,I] x to a continuous experiellinger transforms on T are given by formula xtends to the whole of continuity lemma this relat artial translation invariance w.r.t. itability of F foilow immediately fro

is a covarimce

of the biting

According to this theorem, a covariance and the from the covariance or Levy measures respective fuctions pointwise as location parameters and It is sometimes hard to obtain a is more explicit that in Theorems 3 likelihood ratios of Fare derived in S (2.9) and (3.8)). In Poisson or ‘mixed involved. Although desirable, a sati Poisson part of F is not yet availabl is sometimes possible to describe the of 0. 3.8. Let fY= (Q 03) and defiotc by Qou the

parameter u > 0:

Taking

:=lR’ and

it is not hard to verify that =

ifor

3

ZjV V

s

k?. Milbrodt / Experiments associated with sampling plans

12

and

s 00

=

l+T/v

ence,

Let &I = t@@denote the image of tire translation parameter experiment of one-sided exponential distributions with fixed standard deviation l/u under the projection ussian shift experiment (t, p) ++f, and &‘2=&‘p)the image of the one-dimensional riance (8, y) ++~-~/3y under the projection (t, /3j - /K Then

(tT$)(zj= exp 8, is stable with exponent 1, the parametrization of &$ does not depend on the bcation parameter a&

se that the regression is liuear

the nuisance-parameter the

The proof of the Theorem 3.7 is prepare Let p>O and normed real vector space 5 with Gaussian (R2y 1 &ZZj). Let K denoti jaEAtE,the system of L&y measures of Then: is stable with exponent p iff are both stable with exponent p. case of stability, is continuous and homogeneous of degree reover, denoting .

the map

is continuous and homogeneous of degree p fs,* e”:r:j $ : Sm ==+R’ continuous. Since (1) is almost obvious, we restrict ourselves to the and homogeneity of follow from the fact that

ssion (2.9) and reca

the ‘partial homogeneity’ of weakly. air (

n

-+

0

such

that f

- s;.’

is

measures of the factors respectively, it suffices to consider the case that E= 81 ic ctakla w;tk nvnnn6nt p U&W”A’CI Vl_LU \v’AYCrrrr--V -By Lemma 3.9@j, & is :y&d&Ilcd; obvinusly, it is a covariance oh G. In or&z is t system of Levy measures, we first prove that for every to show that - . - -6) as &f&a 1~ Lc;ii,sia J ~, ~~ 8 kernel from QF?’x 3)” 2 Q ic to SEe By definirnEN; AI &, &); 0) is a measure on (S,). To prove measuraas a function of the first arguments9 let be a compact subset of , a.O, Mn}. Choose conthlusus maps fn:8"--I, [Q, =),nEIN, decreasing tc the indicator function of A and such that fivanishes locally at (ldm, . =. , Mn). @@(* ;f,) ismeasurable by Lemma 3.9(2). Standard fll’(* ;A)=inf,:EN the Bore1 measur~bihty of M@@(.; A) fol every yStHIldgUIllC:ntS eEd the system ,J is well-defined. laji. 0~) fix a={(a,,&), *. ,(a,,&)} eA(O and denote ar=l+maxt,j,,n ecause of the continuity of the map have to show that $, is M,Jntegrable. @)(q; s,$) (Lemma 3.9(2)),

Together with the homogeneity part of Lemma 3.9(‘1)this yields

Thus,

R is a Levy measure. Let

aking use of Theorem 3.3, it is mere routine to check that (K,,M,,) Levy-Khintchine representation of &:

Suppose that (

gives the

= {u} is a singlet0 heorem 3.7 a% Strasser (l985b),

(ru--sI’+ lb-sl’ussi

la-bl’)n

hp. Mb&t

/ l3perinznt.s associated with sampling plans

As an easy application of gnd strasser (1985) we &tain the of experiments associated with Poisson sampling to purely Gausskn or

15

pur&y

3,XL Sg4ppse ?!zcati$~l)~ (A.2) and (A.3) are satisfied. Then: (‘I) The limit exb *imentF of (pVFV),. N is a Gaussian experiment iff for every *E R’, 0, y)&,

(2) F is a P&NMZ experkent iJffor every t E I?‘, (fl, y) EB2,

2 dQv-11~u/(p) " 11 dQO,,(,), = 0. v-w' &10 \iI~Q,-"~~.w(B)Y/~Qo.w(~)Va ~I<4I dQo,;w. I lim lim SIRS17’

ive

s

Our goal in this ~ctic~~ is to show that under mild additional assumptions on G-E asymptotic b&&our e
This condition was introduced (1987a) for the approximation o we need:

16

H. Milbrodt / &periments associated with samplingplans

ecah that (Eyi) is (d,)-infirritesimal for some sequence Ed&> 0, v E IN, iff for every a 63&E), ZE Sar

meter space 3.

lim dy max (i -

= Oe

V”Qo iel, 4,

t

EI Q) are weakly asympto V

Tvi: s w l,(i) denote th

icator of inclusion of unit i ective sampling rV, v E IN. ective sampling equal

ility of inclusion un ratios of Sampford-

(3) and ZE Sa. Then

exp

9

s emma (2.1) in O
[

j

U%&(z) -

l)rYj 1(I-

ilbrodt (1985). Now, employing (a,)-infinitesimality, choose

lim c,b, rnEy( V+oO

id(z)) = O

V

, p. 1503) constructed n plans which ejective sampling. Copying the estimates in the rodt (1985), p. 146 we obtain

(I- I,,)d(r, -p,) 5

(Isize-n,l ~x,B,)

VI

.

.s

/ Experiments associated with sampling plans

to the situation of

from the equicontinuity of the sa proper modifications of the restr vergence theorem 3.3 by literally (5. to

irst we note that cover the case of k

V-+00

KA Bo9

XBO

regression para

17

H. Milbrodt / Experiments associated with sampling plans

18

e conclude this section continu sidered already in ilbrodt (1985)

the example of random non-response con-

Let q& be a sampling design an t S;s responds e subp&lation he statistician uses (s, )q,,(ds) the samp ing design resulting to II+&.Follewing ab (1979), we conon design on s with probabilities of inclusider the case that every &SE gV, v E IN, independent of s. y Cywe denote the sion PinVir with probabilities of inclusion mvipvi, i E I,, v E N. If (&,&I,, YEh\lis a (a,)-infinitesimal triangular array of experiments with only pairwise imperfect accumulation Points9 b?v Qj 21”&I) and CCV @ie*” i) are weakly asymptotically equivalent under each of the following conditions: (1) @J = (M (2) (C&J= (PV)and (3) (qJ=(rz), (A. (1) is obvious, (2) ilbrodt (l9g5), Example (X4), and (3) follows by similar arguments using Theorem 4.1 above. .3) are satisfied. We study the asymptotic In the following assume that ( behaviour of the regression parameter sampling experiments (with side parameters) in case each population nsists of k subpopillations with constant response probabilities ml, . . . I mke ecisely: Let k~ N and {N-J) 11 skk} be eych that $, N,,(I) = NV; we assume that I mvi

N,(j)
= ml,

j=l

Nv(jL

VE N.

j=l

dditionally, we suppose that the following limits exist: 1

lim JNV j=!

NV(j) := cl, Orlsk.

V+Qo

lusion probabilities ives the asymptotic converges to the

hr.

Ibrodt / Experiments associated with sampling pIans

pt.&ally normal ( j2* I!?*which we assume turns r= 2, we shall also assume:

In this section we consider locally as

fix an inner product , stronger than E,, 3 E

weakly,

where E is the Gaussian shift experiment on IF?’ x

with covariance

First, we show how the results obtained so far can be zations of the LAN Theorems method of proof here is somew is based on the construction of a a central limit theorem for Poisson sampling. Assume that

.I) holds with r= 2 an

satisfied. Th& the sampling experiments (p,, with covariance

s 1

:(Lp,tT)w

0

and centered loglikelihood processes

H. Milbrodt / Experimentsmsociated with sampIingplans

20

e&n’s general asymptotic As is well known, results of this type may decision theory - be applied to problems of testing and estimation.

a score function n which is Sounded away fro denote the expectation functional for t inst {f #O} for some lev

v :

69(Xihes),

(These estimators have been termed n estimators since in case of standard Gaussian isance parameters of renormahzed v-1/2 ,vco,).

If no nuisance parameters are present, an asymptotically equivalent form of ( which reduces exactly to the classical orvitz-Thompson estimators in the standard normal case is given in iscussion (2.11) of ilbrodt (1985).) Note that by the results of Strasser (1985b, Theorem 4.1), (a,O;b,O)=8c ‘n(x)a(x)b(x)dx,

(a,b}CL2[0,1).

(5.2.1)

s0 Wow, we assume in addition that

an orthogonality relation which will aiso play a role later in connection with adaptive estimation of f. (1/8c~,0; 0,8) = 0, and by (5.2.1) and linearity

e have

H. Milbrodt / Experiments associated with sampling plans roof

cf

Corollary (

elo - an =

0.

i

erefore, the sequence of tests with critical regions

optimal of level a for {f = 0) agai izes the asymptot sts which are as .14). Under the S) respectively, the same holds for rbin sampling.

itional assumptions

The application of results to the asymptotically efficient estimation off if no nuisance 5.1 the generalization to the

to the situation of

SGl :&all fro

and that the array is bounded iff 2

(

W

OQ,

Kvk~2;

Gaussianiff for every e>O, (c,(p) E 22,

e

VE

K

H. Milbrodt / Experimentsassociated with samplingplans

22

he proof follows the pattern of the proof of eCam (1969), I, Proposition 1 p. 47.49), which has een elaborated in ecker (1983) (Sect. 2.1).

for (qJ

= (p,),

respective con

(d&J

the theorem. Since,

aussian!), Lemma (5.3) yields

;

is Gaussian with note that any inner p Strasser (1985

to prove vi9VWV,

n case of

)-

0. (5.1.3) i

schebyschev’s inequality,

I)

#rymQm.w(b)J> 8

H

lbrodt / Experiments associated with sampling plans

= 1 an

Since lim, + tr& m

respective con the theorem. D =

(O}, i.e. if no nuisance p

. Sbippose that with r= 2 and that LAN on L2[Q91) wr’ppz covwiance

:

(a, b) - 8c

’

s0

a(x)

23

H.

24

Milbrodt / ??xperimentsassociated with samplingplans

are 3nkformly equicontinuous. ad?@ 1). ue to Theorem 5.11the processes totically linear random functions of a. Since th extends to dl of L2[0, 1). 0

.I) for every non-positive ))a&),l)r

‘)!E MS are

are b!so linear, relation (

cons&r the roblem of esti sting an arbitrary li which is continuous w.r.t. the product f assume Ph. 4 w: hxs is of the form Now

..

let

‘JS

lx

ejective

g or

lbrodt / &kperiments associated with satnpkng plans _

tie

its

25

ectiveiy.

mediate consequence of

is applicable since f is contin

erall the topology generated tion IC1 const. >

Then et (nk) E thl be any subsequence. Choose a furt O,cxcl such that

ce

~lbertia~ by (I).

is coarser e

H. Mllmdt / Experimentsassociated with sampling plans

26

_Yd

I ~~I~lM99

ll&>cdx,

J

where a := sup(lg(@i 1 llsl~

(a,O;a,O)s1}. slation invariant, this bou efelmeyer (1982, pp. ity as a local asymptotic at f is adaptively ne knowing /Y,i.e. if l~f~KK= lgil(.

Grossmann (1987

Suppose that the assymptio tively estimable, ifs IR’and

,

Apparently, always Ii assume that IR”and

of Theorem 5.5 are orthogonal under K.

Then for every (a, j3) E 0 0; a, O), showing that ~gll& if lK,. This

Conversely, suppose llglln (Theorem 5.5(1 0 for every aE

,O)=l and K,(a,,O;a,O)=

ssume there is a BE e show that a& Then F can be chosen such that *-2Kniag, 0; 0, p) giving

>

satisfying K,(a,, 0; 0,/J) *

io,8;0,/3)~-4-2

0 *=&(a,, j!3;as, j3)c 1.

at for every

sho

is

i

.

lbrodt / Experiments associated with sampling plans

H.

Q024:= u&P, UE u. reparametrizelocally around (Q,g).

e fixagEe/an

:= {jkL2(A’)

1 j?

symmetric about 0, plg”2}

(viewed as a close subspace of L2(A1)), f v(U(j9);‘2-g”Q e :=

cy :

e any lrmapsuch that

--Ir 8,

-(g 1’2)*and

is dense in

f /YE9, we define

and (A.1) obviously holds. ( ther choices sue w(j?), :=

((1-+~j?2dls~‘2g1’2+-+)f,

therwise, we c are also possi quence n,P 00 satisfying

aking /3,,:= yn, an

verify

is c

VEN,

27

28

H. Milbrodt / Experiments associated with sampling plans

e referees for valuable suggestions concerni bility of the paper.

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