Chibisov, D.M.

Zwet, W.R. (Willem) van

English

Zwet, Willem

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On efficiency of first and second order : (preprint)

CWI's Institutional Repository

stichting mathematisch centrum AFDELING MATHEMATISCH STATISTIEK SW 69/80 (DEPARTMENT OF MATHEMATICAL STATISTICS) P.J. BICKEL, D.M. CHIBISOV & W.R. VAN ZWET ON EFFICIENCY OF FIRST AND SECOND ORDER Preprint ~ MC NOVEMBER kruislaan 413 1098 SJ amsterdam PJunted a:t. .the Ma.the.ma.tic.al Centlie., 413 K!UJ.i..6la.a.n, Arrv.i.teA.dam. The Mathematic.al. Centlie , f,ounded .the 11-.th of, Febll..Ua/ty 1946, ,u., a. non-p.tr.o f,li .ln6.t,l,tu,.ti,o n aiming a:t. .the p.tr.omo.t.lo n of, pWLe. ma:t.he.matiC-6 and il-6 applic.a.t.lon6. 1.t ,u., J.ipon60ll.ed by .the Ne.the.tr.R..a.nd6 Gove.tr.nment .th.tr.ough .the Ne.the.Jli.and6 O.tr.gan.lza.t.lo n f, on .the Advanc.e.ment of, PUILe Ru eMc.h ( Z. W. 0. l • 1980 Mathematics subject classification: Primary 62F05, Secondary 62G20 On efficiency of first and second order*) **) ***) ****) P.J. Bickel , D.M. Chibisov & W.R. van Zwet ABSTRACT It has been noted by a number of authors that if two tests are asympto-tically efficient for the same testing problem, then typically their powers will not only agree to first but also to second order. A general result of this type was given by Pfanzagl (1979) in a paper entitled "First order ef-ficiency implies second order efficiency". Because of their technical nature, however, these contributions give little insight into the nature of this phenomenon. The purpose of the present paper is to provide an intuitive un-derstanding of the phenomenon by proving a simple theorem of this kind under mild assumptions. KEY WORDS & PHRASES: Asymptotic efficiency of tests, second order efficiency, deficiency **) ***) ****) This report will be submitted for publication elsewhere. Research supported by the U.S. Office of Naval Research, Contract N00014-80-C-0163, the National Science Foundation, Grant MC S76 10238 A01, and by the Netherlands' Organization for the Advancement of Pure Research. Research supported by the Netherlands' Organization for Pure Scienti-fic Research. Research supported by the U.S. Office of Naval Research, Contract N00014-80-C-0163. 1 I. INTRODUCTION For N = 1,2, .•. , we consider an experiment with outcome XN taking values in an arbitrary sample space. Let PN O and PN I be two possible ' , distributions of XN, with densities pN O and pN I with respect to some ' ' dominating measure µN. We shall write EN O and E~ 1 for expectations ' ' under PN,O ratio by and PN I respectively. Define the logarithm of the likelihood ' PN 0 , and/ or pN 1 • ' with the usual conventions for vanishing Now consider a sequence aN E(O,I) and let ¢N(AN,aN) denote the test function of the most powerful level-aN test for against PN l , thus ' 0 for A < cN(aN) ¢N(AN,aN) N = for A > cN(aN) N , with EN 0 ¢N(AN,aN) = aN ' ¢N(AN,aN) = * EN I nN(aN) ' , * maximum attainable where TIN(aN) is the power against p N, 1 at level aN For N = I , 2, ••• ' let ZN be a random variable depending only on the outcome ~ of function of the We have EN 0 ' EN I ' where TIN(aN) the N-th level-aN 0 ljJN(ZN,aN) ljJN(ZN,aN) experiment and let right-sided test based = = for ZN < dN(aN) for ZN> ~(aN) aN nN(aN) ' ljJN(ZN,aN) denote on the statistic is the power of this test against PN 1 ' the test ZN ' i.e. . For a sequence 'N E(0,1] , we shall say that the sequence of level-aN tests 1/JN(ZN,aN) is ,N-efficient if, for N-+ 00 , In a more usual terminology first and second order efficiency correspond to 'N-efficiency with T = 1 N and Finally, let us define for 0 -1 , = N 2 respectively. N N = 1,2, ••• , + ex, LiN = otherwise , and let us denote the indicator function of a set B by 1B. 2 Having established our notation we shall now give an informal description of the phenomenon we wish to study. Let us think of N as denoting sample size, i.e. N is the number of independent random variables involved in the N-th testing problem. We are interested primarily in sequences of testing problems where aN ~ E and n;caN) ~ 1-E for some E > 0 and all N. Such sequences exist if it is impossible to discriminate perfectly between PN O and PN 1 ' ' even as N-+ 00 and this is true if PN O and PN 1 are contiguous. A sufficient ' ' condition for contiguity - and one which is often fulfilled in this case - is asymptotic normality of AN both under PN O and PN 1 • But if AN is ' ' asymptotically normal, it will usually also be possible to obtain an Edgeworth expansion for its distribution function under PN O and PN 1 and this will ' ' yield a similar expansion for the power of the test based on AN, viz. (I.I) Typically the remainder term on the right in (1.1) will be O(N- 1) • Suppose that the sequence of tests 1/JN(ZN,aN) is asymptotically efficient to first order, or I-efficient in our terminology. For most statistical problems such I-efficient tests abound. They are usually based on statistics ZN that closely resemble AN. Typically 6N = ZN - AN tends to zero in probability both under PN,O and PN,I and in the situation we have described so far, this suffices to ensure I-efficiency. Of course, these }-efficient tests can also be based on statistics ZN which don't resemble AN at all, because the test statistic associated with a test is by no means unique. However, we shall not be concerned with such alternative representations and suppose that 6 -+ 0 N in 3 PN,O - and PN,l - probability. Recalling that N is the sample size, we note that one often finds that a sequence of random variables tending to zero in probability does I -1 so at the rate of N 2 • Thus, for ]-efficient tests, N2 ~ will typically N be bounded in probability both under PN,O and PN 1 ., Hence, one may expect ' to be able to establish Edgeworth expansions for the distribution functions of ZN under PN 0 and PN I ' which differ from those for AN only in the N-! , term of order and in those of smaller order. This yields a similar expansion for the power of the test based on ZN, ( I. 2) where the remainder term on the right will typically be O(N- 1) or of slightly larger order. The fact that the leading terms in expansions (1.1) and (1.2) have the same value c0 reflects the }-efficiency of the sequence 1/JN(ZN,a.N) There would seem to be no reason a priori to expect that also _1 c 1 = cj , which would entail N 2 -efficiency or efficiency of second order. However, in those cases where expansions (1.1) and (1.2) were explicitly computed, one does indeed find that c = c' and hence that the sequence J } } 1/JN(ZN,a.N) 1.s N- 2-efficient. This phenomenon was noticed in Pfanzagl (1973), (1975) and Chibisov (1974) for a number of tests for the parametric one-sample problem and in Bickel and Van Zwet (1978) for rank tests for the nonparametric two-sample problem. Some tests for the one-sample problem for the case where nuisance parameters are present were considered in Chibisov (1973) and Pfanzagl -1 (1974) and also found to be N 2 -efficient. Finally, it was shown in Pfanzagl (1979) that first order efficiency forces seco~d order efficiency for a large class of one-sample tests in the presence of nuisance parameters. With an appropriate definition of efficiency a similar result was obtained for estimators. -1 In each of these contributions, N 2 -efficiency 1.s established by imposing the conditions needed to obtain expansions (1.1) and (1.2) and then checking that these expansions are in fact identical. This method of proof coupled with its extreme technicality makes an intuitive understanding of the phenomenon rather difficult. The purpose of the present paper is to provide such an intuitive understanding by proving a simple theorem of this kind under rather mild assumptions. Since our aim 1.s to provide insight rather than generality, we shall only be concerned with the simple hypothesis testing problem described 4 above and avoid the technicalities inherent in the treatment of nuisance para-·meters and estimation problems, although extension to these situations is certainly possible. In section 2 we present our result, discuss its meaning and explain why it is true. A formal proof of the theorem is given in section 3. Though this \ proof is straightforward, the non-mathematically inclined may wish to skip it. 2. DISCUSSION OF THE RESULT In the notation introduced in the previous section we have the following result. THEOREM Suppose that (2. I) lim inf aN > 0, N and that there exists A > 0 such that for every x0 E JR , every y > 0 and N ➔ oo , ! I (2.2) sup PN O(x-,; $ /1.N $ x) = 0 ( '[ J) x$xo ' (2.3) EN olt1NI 7 1 ( J l\N I ) = o ( ,N) ' • (y-r~,A) (2.4) PN O(llN 2'. A) = o(,N) ' , (2.5) PN l(l\N $-A)= O(,N) . ' Let us briefly discuss the conditions of this theorem. First of all, assumption (2.2) is clearly satisfied for any sequence 'N if the distributions of /1.N under PN,O possess uniformly bounded densities. More generally, will hold if the distribution functions FN ~f /1.N und_er P N, 0 can be approximated with a uniform error of order , 2 by distribution functions N I with uniformly bounded densities, i.e. if supJFN(x) - GN(x)J = O(,J) . X (2. 2) 5 This is certainly the case when FN has a normal approximation or an Edgeworth expansion with the required error. If T ➔ 0 N , assumption (2.2) clearly implies that the distributions of AN under PN O don't tend to a degenerate limit. In view of this, conditions , (2.3) - (2.4) serve to ensure that under PN,O , l6NI is small compared to AN. Note that these conditions refer only to values of l6NI the variation of which excede and that they are satisfied if the distribution of under I either assigns probability 1 to a set where l6NI = O(,J) , or has at most very small tails outside that range. Thus, roughly speaking, 6N is required I to be o(,J) under PN,O ; under PN,I condition (2.5) is even weaker, but this is to a certain extent artificial and is due to our efforts to replace conditions under PN I as much as possible by conditions under PN O which ' ' will usually be easier to verify. It follows that one can't hope to say more about the differences between the distribution functions of and under I PN O and PN I ' , than that they are * o(,J~ . One would therefore expect to be able to prove that TTN(aN) - TTN(aN) = o(,J) but, somewhat surprisingly, the conclusion of the theorem is the stronger l is O ( ,;) The condition that roughly 6N l improved. By taking 6N = 6,~ where 6 * statement that TIN (aN) - TTN(aN) under PN O cannot essentially be ' is independent of AN for every N, one easily constructs examples where * 1T (aN) - TT(a,N) is of exact order 'N. A formal proof of the theorem will be given in section 3. At this point we shall be content to provide an intuitive explanation of the result by sketching the proof for the special case where there exist numbers oN such that for N = 1,2, •.• , (2.6) By making this assumption we avoid certain technicalities and bring out the essential simplicity of the proof. Let us write cN and dN for cN(aN) and dN(aN) respectively. Since the tests tN(AN,aN) and ~N(ZN,aN) have the same level aN, (2.6) clearly implies that we may assume that lcN-dNI < oN. Invoking (2.6) once more, we see that if AN~ cN and ZN~ dN, then dN - oN ~AN~ dN + oN; the same conclusion holds if AN~ cN and ZN~ dN. It follows that on the set where tN(AN,aN) # ~N(ZN,aN) we have IAN-dN! ~ oN, and again because both tests have level aN we find with the aid of (2.2), 6 Note that we need to have dN bounded above, but this is an easy consequence of (2.1) and (2.6). The above sketch should make it clear that the essential thing which makes the theorem work is that not only do ~N(AN,aN) and ~N(ZN,aN) resemble each other closely, but that also AN is almost constant on the set where they differ. Let us finally discuss the relevance of the theorem to the problem of first and second order efficiency. As was pointed out in section I, first order ! efficiency of ~N(ZN,aN) will typically imply that N2~N is bounded in ! r EN olN2~N1 and ' r > I and this (or even uniform probability both under ! r EN,! JN 2~N1 will PN 0 ' usually be I and PN I • Also, , bounded for some integrability of JN2~N1 that assumptions (2.3) -under PN,O and PN,l) (2.5) are satisfied for is amply sufficient to ensure -1 TN= N 2 . But then the theorem ensures that ~N(ZN,aN) is efficient to second order under the very _! mild conditions (2.1) and (2.2) for T = N 2 N An examination of our proof shows that if we replace (2.6) by (2.7) for M < oo and N = 1,2, ... , then we obtain the conclusion (2.8) Taking -1 T = N N we conclude that if is bounded and if e.g. the distribution of * TTN(aN) - TTN(aN) = AN under PN O tends to normality at the rate of O(N- 1) • Thi~ means that the tests based on ZN _! N 2 , then have a finite deficiency in the sense of Hodges and Lehmann (1970). That is, if we let aN = a E(0,1) for N = 1,2, ... * for which TTN 1 (a) ~ TTN(a) then and define N' lim supN(N'-N) to be the smallest integer < 00 7 _! Asumption (2.7) is of course not necessary to obtain (2.8) and N 2 -efficiency is frequently coupled with finite deficiency. However, the coupling is not inevitable. An example - in a nuisance parameter context - is provided by the normal scores test studied in Bickel and Van Zwet (1978). Note that this bears out what we have said about the order of the rem~inder terms in (1.1) and ( 1. 2). 3. PROOF OF THE THEOREM Take a sequence aN E(O,1) satisfying (2.1) and write cN = cN(aN) and dN = dN(aN) . If dN = - 00 for some N then I - 'ITN(aN) s PN, 1 (ZN = -oo) s p N 1 (.I\N = -oo) + p N, 1 (liN = -oo) ' = PN 1 (liN = -oo) = O(,N) ' because of (2.5), and 1/JN(ZN,aN) is clearly 'N-efficient. Take A as in (2.3) - (2.5). Then aN s PN,O(ZN~dN) s PN,0(.1\N~dN-A) + PN,O(liN~A) -dN+A -dN+A s e PN,l(.I\N~dN-A) + o(,N) s e + o(,N) because of (2.4). In view of (2.1) it is therefore no loss of generality to assume that for some D < 00 and all N, (3. 1) -oo < d s D. N Define -liN = 0 otherwise. Obviously, (3. 2) 8 + where x =xv O and x = (-x) v O denote the positive and negative parts of a number x. since {~N(AN,a~) - ~N(ZN,aN)} is nonnegative (nonpositive) if (AN-dN) is positive (negative) and equals zero if ~N = 0 and (AN-dN) IO. A similar argument yields and hence (3. 3) By (2.3) there exists a sequence yN + 0 such that -~ EN OJ e N_ I J 7 l ( J ~NI ) = o ( TN) ' (yNT~,A) for N + 00 • In view of (3.2) this implies --~ EN ale N-11 7 l cJ~NI) 7[0,A/l~NI) = O(TN). ' (yNTJ,ooJ (3. 4) Also (3.1) and (2.2) with x0 = D + I yield (3.5) If bN ~ 0 then bN ~ 0 and because of (2.4) -b (3.6) EN,ole N-1 I J[A,ooJ(bN) ~ PN,O(~~A) = O(TN) . If bN ~ 0 then ~N ~ 0 and (2.5) ensures that (3. 7) dN -bN. e EN,ole -11 7[-00,-AJ(~N) 9 Together (3.1) and (3.3) - (3.7) imply that ~;(aN) - ~N(aN) = o(TN) and the proof is complete. 10 REFERENCES [l] BICKEL, P.J. and VAN ZWET, W.R. (1978). Asymptotic expansions for the power of distributionfree tests in the two-sample problem. Ann. Statist. i, 937-1004. [2] CHIBISOV, D.M. (1973). Asymptotic expansions for Neyman's C(a) tests. Proc. Second Japan - USSR Symp. on Probability Theory, G. Maruyama, Yu. V. Prokhorov editors. Lecture Notes in Mathematics 330, Springer, Berlin, 16-45. [3] CHIBISOV, D.M. (1974). Asymptotic expansions for some asymptotically optimal tests. Proc. Prague Symp. on Asymptotic Statistics, J. Hajek editor, Vol.2, 37-68. [4] HODGES, J.L. Jr. and LEHMANN, E.L. (1970). Deficiency. Ann. Math. Statist. ~. 783-801. [5] PFANZAGL, J. (1973). Asymptotic expansions related to minimum contrast estimators. Ann. Statist . ..!_, 993-1026. [6] PFANZAGL, J. (1974). Asymptotically optimum estimation and test procedures. Proc. Prague Symp. on Asymptotic Statistics, J. Hajek editor, Vol.1, 201-272. [7] PFANZAGL, J. (1975). On asymptotically complete classes. Statistical Inference and Related Topics, M.L. Puri editor, Academic Press, New York, Vol.2, 1-43. [8] PFANZAGL, J. (1979). First order efficiency implies second order efficiency. Contributions to Statistics [J. Hajek Memorial Volume], J. Jureckova editor, Academia, Prague, 167-196. 

On efficiency of first and second order : (preprint)

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