IEEE". UNIY-NILUAUKEE J SEDER 25 APR 85 AFOSR-TR i RFOSR UNCLASSIFIED F/G 12/1 NL

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1 D- Ai5S 762 SIEVES ND FILTER FOR AUSSI N PROCESSES (U WISCONSIN 1/1 UNIY-NILUAUKEE J SEDER 25 APR 85 AFOSR-TR i I IEEE". RFOSR UNCLASSIFIED F/G 12/1 NL

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3 AFOSii Retearch Progress and Forecast Report: April 25, 1985 Sieves and a Filter for Gaussian Processes: AFOSP Results on Conjecture 1. We use the definitions and notation of [1]. Let H be the subspace of 2, L.,,p spanned b 'X, tej, and let H = H(R,T', be the corresponding reproducing kernel Hilbert space. Assume that H has countable dimension, with CON basis :Ui. op. Fix a = {ak}, and consider a sample of size n. Proposition 1. Under P a' the Uk are independent, and Uk has the n(a k, 1/n) distribution. L" We consider the sieve estimator defined in [I] p. 7. Then for each m and for a sample of size n, the sieve estimator of the mean g(t) is (1) - I,' g(t) = Z kg k ( t ) l where,gki is the CON basis of H corresponding to {Ukl. We wish to pick m as a function of n so that Ili-gll -* 0 (norm in H). But ig= -a 2 (norm in 2 ) = a a H) Bt ksm fk-ak k>m ak = X n m k' ' say. The non-stochastic tail vanishes as-n if we require m By Proposition 1, nx is x 2 (m), and the following weak result describes C) nm the limitations we will need to impose on m = mn. Proposition 2. If m/n - 6, then X - 0 in P -probability iff B=O. * I.. ni a C.3 The Borel-Cantelli Lemma gives a sufficient condition for 9 a (X-0) Xnm - = 1: Proposition 3. Let {m n } be chosen so that m - nn sequence {cn} defined by and such that the D T IC.-1 u ELECTER% c,, = n 1 u-m) e - u du AU G Lq is summable for every c > 0. Then P a (Xnm4O) :1I. G as If m is chosen so that it is always even, then we may rewrite c n n cn e e,_,(an), f o.,. o" , " , , - ', ' "* =-.', * 5. "- ". -. "-""- ""S.

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5 -2- n k where a - c/ 2, n - 29, and en (x)- E X Invert -t(n), say n -(l). n kno I-01; Now we must choose A(Z) so that at least X/1 - and so that ee (ca) is. summable for every m > 0. Write el e (cax) + A Using results of Buckholtz (2], we may prove the following: Theorem 1. Let X = X(9.) be such that lk/ -k as Then for each aci(o,-) we have A.(a) ( 9) t From this and Stirling's formula we have Corollary 1. If X/ - as. -, then for each cic(o,) the sequence e a e 2 (cx) is summable. Translating back, we get the following consistency result. Theorem 2. If m = 2.(n) = o(n) and m - -, then a - a a.s. At first this holds a.s. P 8. but since the measures in P are equivalent, we may assert a.s. convergence without qualification. In any case, this settles Conjecture 1. In particular, the condition m = o(n), which was sufficient for weak convergence, also gives strong convergence, and since m/n - 0 > 0 does not even give weak convergence, our condition is in some sense best possible. Moreover, a result similar to Theorem 1 allows us to see just what does hold almost surely if we only require m/n 0 > 0:

6 ". -3- Theorem 3. If m/n - > 0, then for all c sufficiently large (depending only on 0) we have Pa (Xnm C i.o.) = 0 The distribution of the estimator g given by (1) is easy to describe. Consider g(t) as a stochastic process. From Proposition 1 we have the following. Theorem 4. Under P, the process {i(t), tct} is Gaussian with mean m m function I ag(t) and covariance function - E g.(s)g.(t). i=laigi g=11 i t For comparison we note that the reproducing kernel R(s,t) of H may be written R(s,t) - gi(s)gt(t) (as long as {gi } is CON in H), and that under Pa the true mean function is E aigt(t). What needs to a i=l be investigated now is the use of Theorem 4 to make confidence statements and to test hypotheses. Results on Conjecture 2. The set-up here is given in [l], pp Let us * fix a countably infinite orthonormal set {gkl (not necessarily complete) in H, and consider the subset PO c P consisting of measures corresponding to covariances of from S(s,t) = R(st) + Zlkgk(s)gk(t), where i = {l is in the subset e 2 of 93. Now the likelihood function depends only on Let UkcH correspond to 9k as usual, and let A = {wcfl: Uk(cO)to for every k. Certainly A is an event in A. Theorem 5. Q(A) = 1 for every QcP, and for each outcome wn the corresponding likelihood is unbounded over P 0. Thus Conjecture 2 is established.

7 References [1] Jay H. Seder. Sieves and a filter for Gaussian processes, AFOSR grant proposal. [2) J. D. Buckholtz. Concerning an approximation of Copson, Proc. Amer. Math. Soc. 14 (1961), Accessi~on For "SGRA&I DTIC TAB Lz "'n Ounee 0 d Justiticati 0 Dy JAvai~j and/or 'c

8 FILMED DTIC