A MARKOVIAN LOCAL RESAMPLING SCHEME FOR NONPARAMETRIC ESTIMATORS IN TIME SERIES ANALYSIS

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1 Econometric heory, 17, 2001, Printed in the United States of America+ A MARKOVIAN LOCAL RESAMPLING SCHEME FOR NONPARAMERIC ESIMAORS IN IME SERIES ANALYSIS EFSAHIOS PAPARODIIS University of Cyrus DIMIRIS N. POLIIS University of California, San Diego In this aer we study the roerties of a th-order Markovian local resamling rocedure in aroximating the distribution of nonarametric ~kernel estimators of the conditional exectation m~x;f E~f~X t 1 6Y t, x where $X t, t 1% is a strictly stationary rocess, Y t, ~X t, X t 1,+++, X t 1, and f~{ is a measurable real-valued function+ Under certain regularity conditions, asymtotic validity of the roosed resamling scheme is established for a class of stochastic rocesses that is broader than the class of stationary Markov rocesses+ Some simulations illustrate the finite samle erformance of the roosed resamling rocedure+ 1. INRODUCION Let $X t, t 1,2,+++% be a ~strictly stationary real-valued stochastic rocess and denote by F Yt, ~{ the marginal distribution of Y t, ~X t, X t 1,+++,X t 1 and by F Xt 1 6Y t, ~{6{ the conditional distribution of X t 1 given Y t, + In this aer we are interested in estimating the samling behavior of a nonarametric ~kernel estimator of the conditional exectation m~x;f E~f~X t 1 6Y t, x where f~{ is a real-valued, measurable function+ Several conditional functionals of the rocess can be obtained by aroriately secifying the function f+ For instance, for f 1 ~z z and f 2 ~z z 2 we have that m~x;f 1 is the conditional mean, whereas m~x;f 2 m 2 ~x;f 1 is the conditional variance of s X t 1 given that Y t, x+ Aart from the conditional moments E~X t 1 6Y t, x the function m~x;f can also be used to describe some other conditional functionals of the rocess that may be of interest+ As an examle, for f~z 1 ~ `, y# ~z and y a real number we have that m~x;f F Xt 1 6Y t, ~ y6x, that is, the conditional distribution function mentioned reviously+ We are grateful to the co-editor Joel Horowitz and to both referees for a number of useful comments+ his research has been suorted in art by National Science Foundation Grant DMS and by a University of Cyrus Research Grant+ Address corresondence to: Efstathios Paaroditis, Deartment of Mathematics and Statistics, University of Cyrus, P+O+ Box 20537, CY-1678 Nicosia, Cyrus; stathis@ucy+ac+cy Cambridge University Press $9+50

2 Given observations X 1, X 2,+++,X the nonarametric estimator of m~x;f, which we consider in this aer, is given by m[ h ~x;f MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 541 f~x t 1 K h ~x Y t, t K h ~x Y t, t + 1.1) In this notation K h ~{ h K~{0h where K : R r R is a nonnegative kernel satisfying K~udu 1, uk~u du 0 and h h~ 0 is the bandwidth used to smooth the observations+ Note that ~1+1 is the common Nadaraya Watson estimator of m~x;f+ Nonarametric estimators such as the one considered in this aer are frequently used as an end roduct in time series analysis for modeling or redictive uroses or, erhas more imortant, as a guide in identifying a arametric model to be used in a subsequent stage+ For instance, a general univariate model that contains many of the linear and nonlinear classes discussed in the literature, including the oular nonlinear conditional heteroskedastic ~ARCH model, is given by X t 1 g~x t, X t 1,+++,X t 1,«t where g : R 1 r R is a measurable function+ Here $«t % is an indeendent and identically distributed ~i+i+d+ sequence of random variables with mean zero and finite variance and «t is assumed to be indeendent of X s for s t ~cf+ ong, 1990, Ch+ 3+ One way to gain insight into the nonlinearity features of such models given a set of observations is to estimate nonarametrically functionals such as the conditional mean or the conditional variance and to comare the estimates obtained with those exected under a articular hyothesis about the underlying model+ Such an aroach of detecting and modeling nonlinearity in time series analysis has been roosed by several authors; see among others ong ~1990, Auestad and jøstheim ~1990, and jøstheim and Auestad ~1994+ Under certain regularity conditions, strong consistency and asymtotic normality of the estimator m[ h ~x;f have been established by several authors; we mention here among others the aers by Robinson ~1983, Roussas ~1990, and Masry and jøstheim ~1995+ he asymtotic normality of the estimator ~1+1 is, for instance, useful in constructing ointwise confidence intervals for m~x;f+ However, there is an inherent difficulty in using the limiting Gaussian distribution for such uroses that is due to the fact that the mean and the variance of this distribution deend on unknown ~and difficult to estimate characteristics of the rocess+ he aim of this aer is to show that a Markovian local bootstra rocedure roosed by Paaroditis and Politis ~1997 and which generates bootstra relicates X 1, X 2,+++,X that reroduce correctly the conditional distribution of X t 1 given the ast Y t, leads to an asymtotically valid aroximation of the distribution of L ~x h ~ m[ h ~x;f m~x;f+ 1.2)

3 542 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS he roosed rocedure rovides an alternative ~as comared to the normal aroximation of the distribution of L ~x and at the same time leads to consistent estimation of quantities such as the mean and the variance of the distribution of interest+ Although the local bootstra rocedure generates a seudoseries X 1, X 2,+++,X with a Markov deendence structure, we show in the resent aer that the validity of this resamling scheme for the class of nonarametric estimators considered is not limited to the Markov case only+ o understand this fact intuitively note that the information regarding estimation of F Xt 1 6Y t, ~{6{ and its moments tyically lies in the scatterlot of X t 1 vs+ Y t, for t 1,2,+++, ; the crucial feature of this scatterlot is the joint distribution of X t 1 and Y t, that is determined by the conditional distribution F Xt 1 6Y t, ~{6{ and the marginal distribution F Yt, ~{+ herefore, caturing F Xt 1 6Y t, ~{6{ and F Yt, ~{ by a bootstra rocess may give valid distributional conclusions for general ~not necessarily Markov stationary rocesses+ Bootstraing nonarametric estimators in time series analysis has received considerable interest in recent years+ Franke, Kreiss, and Mammen ~1996 roose some alternative aroaches for bootstraing the ~ointwise distribution of nonarametric estimators under the assumtion that the rocess considered is generated by the nonlinear autoregressive rocess X t 1 f ~X t s~x t «t 1 with $«t % an i+i+d+ sequence+ hese aroaches include a wild bootstra and a residual-based bootstra+ Under the assumtion that $X t % is a Markov rocess, Neumann and Kreiss ~1998 show validity of the wild bootstra for local olynomial estimators of the conditional mean E~X t 1 6X t x and for suremum tye statistics+ Comared to these aroaches, our Markovian local resamling scheme avoids any kind of reliminary nonarametric estimation of unknown functions to generate the bootstra seudoseries+ Furthermore, its realm of validity includes but is not restricted to the Markov class+ he aer is organized as follows+ In Section 2 the Markovian local resamling scheme is briefly described and the bootstra aroximation to the statistic L ~x is given+ Section 3 is devoted to the ability of the Markovian local resamling rocedure to aroximate the distribution of L ~x+ After stating the set of technical assumtions imosed on the class of stochastic rocess considered, the main result is given+ he finite samle erformance of the method roosed is illustrated by some simulated examles in Section 4, and all roofs are collected in Section HE MARKOVIAN LOCAL RESAMPLING SCHEME he Markovian local resamling scheme introduced by Paaroditis and Politis ~1997 generates bootstra relicates by reshuffling the original data oints according to a articular robability mechanism+ his local resamling scheme can be described by the following three stes+

4 1+ Select a resamling width b b~ 0 and a set of starting values Y, ~X, X 1,+++,X 1 + In ractice, a simle choice is Y, Y, + Select further a resamling kernel W that is a robability density on R satisfying W 0 and uw~udu For any time oint t 1 $ 1, 2,+++,% suose that Y t, has been already generated+ Let J be a discrete random variable taking its values in the set N, $, 1,+++, 1% with robability mass function given by P~J s W b ~Y t, Y s, W b ~Y t, Y m, 2.3) m N, for s N, + he bootstra relicate X t 1 is then defined by X t 1 X J If t 1 then go to ste 2+ At comletion of the algorithm, a new bootstra seudoseries X 1, X 2,+++,X is created with distributional roerties that mimic those of F Xt 1 6Y t, of the original series+ Note that the Markovian local resamling scheme described earlier works by assigning at each ste of the resamling rocess the resamling robabilities ~2+3 to each of the observed original values X s 1 $X 1, X 2,+++,X %+ By this algorithm the robability that the bootstra random variable X t 1 takes the value X s 1 deends on how close Y s, is to the already generated segment of bootstra values Y t, + he closer Y s, is to Y t, the larger the resamling robability ~2+3 will be rovided the mass of the resamling kernel W is concentrated around its mean value zero+ It is easily seen that by the ositivity of the resamling kernel P~X t 1 X s 1 6Y t, 0+ Furthermore, s P~X t 1 X s 1 6Y t, 1+ Our rocedure rovides a natural extension of Efron s ~1979 classical i+i+d+ bootstra to the Markovian case+ In effect, Efron s ~1979 i+i+d+ bootstra can be considered a secial case of our local bootstra if we allow the Markovian order to be zero, in which case all observations have the same resamling robability, that is, P~J s 10~ + Given a bootstra series X 1, X 2,+++,X generated by the Markovian local bootstra, we roose to aroximate the distribution of the statistic L ~x by the distribution of the bootstra statistic L ~x h ~ m[ h ~x;f m b ~x;f, 2.4) where MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 543 m[ h ~x;f f~x j 1 K h ~x Y j, K h ~x Y j, 2.5)

5 544 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS and m b ~x;f f~x t 1 W b ~x Y t, t + 2.6) t W b ~x Y t, Note that if x X, where X is the -times Cartesian roduct of the set of observed values X $X 1, X 2,+++,X %, then m b ~x;f E ~f~x t 1 6Y t, x, that is, the conditional exectation of the bootstra rocesses $X t, t 1% is a kernel estimator of m~x;f with a smoothing kernel W and a smoothing bandwidth b+ In the following discussion we assume throughout that the set of oints x at which the unknown function m~x;f is estimated is a subset of X + Note further that in obtaining m[ h ~x;f the same formula, that is, the same kernel and the same bandwidth as in ~1+1, is used with the only difference that the observations X t, t 1,2,+++, are relaced by the bootstra relicates X t, t 1,2,+++,+ 3. VALIDIY OF HE MARKOVIAN LOCAL BOOSRAP FOR KERNEL ESIMAORS o state the main result of this aer the following set of technical assumtions is needed+ ~A1+ ~i F Yt, ~{ and F Xt 1 6Y t, ~{6x are absolutely continuous with resect to Lebesgue measure on R and R and have bounded densities f Yt, ~{ and f Xt 1 6Y t, ~{6x, resectively+ ~ii For all x 1, x 2 R, and y R $`%, y ` y f Xt 1, Y t, ~z, x 1 dz f Xt 1,Y t, ~z,x 2 dz ` L~y7x 1 x 2 7, where f Xt 1 Y t, ~z, x f Xt 1 6Y t, ~z6x f Yt, ~x, inf y L~ y 0 and su y L~ y L `+ ~iii For all y 1, y 2 R, 6f Xt 1 6Y t, ~y 1 6x f Xt 1 6Y t, ~y 2 6x6 C~x6y 1 y 2 6where C~x 0 and su x R C~x C `+ ~iv A comact subset S of R exists such that X t S a+s+ Furthermore, f Xt 1 6Y t, ~{6x 0 for every x S, where S denotes the -fold Cartesian roduct of S+ Assumtions ~A1~ii and ~iii are smoothness assumtions on the joint and conditional densities and are common in nonarametric estimation roblems such as those discussed here ~cf+ Robinson, 1983; Masry and jøstheim, he assumtion of comactness of the suort of f Xt ~{ in ~A1~iv is of a rather technical nature+ It is imosed to simlify the technical arguments and

6 can be weakened by simultaneously strengthening other requirements ~for details, cf+ Paaroditis and Politis, ~A2+ ~i f is a real-valued, measurable function such that E6f~X t 6 m ` for some m 4+ ~ii For all z 1 ~ y 1, x 1, z 2 ~ y 2, x 2 R 1, 6f~y 1 f Xt 1 Y t, ~y 1,x 1 f~y 2 f Xt 1 Y t, ~y 2,x 2 6 C7z 1 z 2 7, where 0 C `+ he next assumtion deals with the deendence roerties of $X t %+ Recall the definition of the strong mixing ~a-mixing coefficient, that is, a~n su j ` A B 1, B Bj n 6P~A B P~AP~B6, n N, where B j 1 and B ` j n denote the s-algebras generated by the sets of random variables $X 1, X 2,+++,X j % and $X j n, X j n 1,+++%, resectively+ Based on this definition, $X t, t 1% is a-mixing if a~n r 0asnr`+ In the subsequent discussion we make the following assumtions+ ~A3+ $X t, t 1% is a-mixing, and the mixing coefficient satisfies a~n Cr n for some ositive constants C and r ~0,1+ Remark 3+1+ Note that ~A3 is fulfilled if the rocess $X t, t 1% is a thorder strictly stationary and geometrically ergodic Markov rocess and that this Markov class of stochastic rocesses contains several of the commonly used arametric and nonarametric models in time series analysis ~for several examles, see ong, 1990, Ch+ 3+ It is known that if $X t, t 1% belongs to this Markov class then it is geometrically absolutely regular ~i+e+, b-mixing, which imlies geometrically a-mixing ~see Doukhan, However, the class of stochastic rocesses satisfying ~A3 is broader than the Markov class described reviously+ he following conditions are imosed on the smoothing and resamling arameters+ ~A4+ MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 545 ~i K~{ is a roduct kernel; that is, for every x ~x 1, x 2,+++,x R we have K~x ) i 1 k~x i where k~{ is a bounded, Lischitz continuous, and symmetric robability density the suort of which is the 1,1# + ~ii he smoothing bandwidth h satisfies h r 0 and h r ` as r `+

7 546 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS ~A5+ ~i he resamling kernel W is a bounded, Lischitz continuous, and symmetric robability density on R satisfying W 0, uw~udu 0 and 7u7W~udu `+ ~ii he resamling bandwidth b satisfies b O~ d for some 0 d 10~2~ 2+ he next assumtion is needed to get ~asymtotic exressions for the bias term in estimating m~x;f using the kernel estimator given in ~1+1+ ~A6+ ~i W is two times continuously differentiable+ Furthermore, for i, j $1,2,+++,% we have u s i W~udu 0 for s odd and u i u j W~udu d i, j W 2 where d i, j is Kronecker s d and 0 W 2 ` for all i $1,2,+++,%+ ~ii f Xt 1 Y t, ~ y, x is two times continuously differentiable with resect to x, and the ~x functions c~yf l ~x Xt 1 Y t, ~y,xdy and c~yf l1,x l2 Xt 1 Y t, ~y,xdy are Lischitz continuous for c 1 and c f and every l 1, l 2 $1,2,+++,%+ Here, f l ~x Xt 1 Y t, ~ y, x and ~x f l1, x l2 Xt 1 Y t, ~ y, x denote first and second order artial derivatives of f Xt 1 Y t, ~ y,{ evaluated at x+ Given the receding set of assumtions, large samle validity of the roosed Markovian local resamling scheme in aroximating the distribution of L ~x can be established+ By large samle validity we mean that the law of L ~x ~conditionally on the data is close to the law of L ~x with high robability, if the samle size is large enough+ Our main theorem makes the receding statement recise; here Kolmogorov s distance d 0 ~P, Q su x R 6P~X x Q~X x6 between robability measures P and Q is emloyed+ HEOREM 3+1+ Suose A1) A6) hold and let 102 h ~ 402 r C h 0. We then have that, as r `, d 0 $L~L ~x6x 1, X 2,+++,X,L~L ~x% r 0 in robability. Note that if 102 h ~ 402 r 0 then the bias term h ~E m[ h ~x;f m~x;f is asymtotically negligible, whereas for 102 h ~ 402 r C h 0 this term converges weakly to B~x;f C h u 2 k~udu l 1 m ~x l ~x ~x;f f l Yt, ~x 1 2 m~x l x l ~x;f f Yt, ~x, 3.7) where m ~x l ~x ~x;f, f l Yt, ~x and m ~x l x l ~x;f denote first and second order artial derivatives of m~x;f and f Yt, ~x with resect to x l ~cf+ Auestad and jøstheim, he case 102 h ~ 402 r C h actually corresonds to otimal smoothing ~minimizing the mean square error of m[ h ~x, f, where the case

8 MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS h ~ 402 r 0 corresonds to undersmoothing that is subotimal+ In most cases where resamling is used to estimate the distribution of a kernel smoothed estimator, subotimal smoothing is used so that the bias term becomes negligible+ Quite remarkably, the Markovian local bootstra rocedure also aroximates correctly the asymtotic mean ~bias term of the statistic L ~x+ 4. NUMERICAL EXAMPLES wo simle numerical examles are resented in this section to demonstrate the finite samle erformance of the bootstra method roosed+ We consider realizations of length 500 from the random coefficient autoregressive model ~cf+ ong, 1990, X b t #X t 1 «t 4.8) and from the moving average model X t «t 0+7«t 1, 4.9) where $«t % and $f t % are i+i+d+ sequences indeendent from each other and satisfying «t ; N~0,1 and f t ; N~0,0+81+ For model ~4+8 we are interested in estimating the distribution of the kernel estimator of the conditional variance v~x Var ~ X t 1 6 X t x x 2 + Using a smoothing bandwidth h 0+7 and Eanechnikov s kernel, such an estimator of v~x is given by v[ h ~x m[ h ~x;f 2 m[ 2 h ~x;f 1 + For model ~4+9 we are interested in estimating the distribution of the kernel estimator m[ h ~x of the conditional mean m~x E~X t 1 6X t x+ Note that under the Gaussianity assumtion the conditional mean of X t «t u«t 1 is linear and given by m~x $u0~1 u 2 %x+ he 5%, 50%, and 95% ercentage oints of the exact distributions of the statistics considered have been estimated using the corresonding ercentage oints of the bootstra statistics h~ v[ h ~x v b ~x and h~ m[ h ~x m b ~x for several values of x in the interval ~ 3,3+ Note that v[ h ~x m[ h ~x, f m[ h ~x, f 1 # 2, v b ~x m b ~x;f b ~x,f 1 # 2, m[ h ~ x m[ h ~ x, f 1, and m b ~x m b ~x, f 1 + he results of this simulation examle are resented in Figure 1 for model ~4+8 and in Figure 2 for model ~4+9+ he bootstra estimates, shown in these figures by crosses, are mean estimates over 100 indeendent reetitions of the corresonding models+ For each reetition the distribution of the bootstra statistic has been evaluated using 1,000 bootstra relications, the Gaussian resamling kernel, and resamling width b 1+0+ he ercentage oints of the exact distribution, shown in these figures by circles, have been estimated using 10,000 relications of each of the models considered+ As these figures show, the bootstra rocedure roosed gives a very satisfactory estimation of the exact ercentage oints of the distribution of interest over the whole range of x-values considered+ We mention here that the accu-

9 548 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS Figure 1. Estimated exact and bootstra estimates of the 5%, 50%, and 95% ercentage oints of the distribution of h~ v[ h ~x v~x for model ~4+8+ racy of the bootstra estimator decreases as the value of 6x6 increases+ his can be seen in able 1, where the estimated standard deviation of the bootstra estimators is reorted for both models considered and for some different values of x+ able 2 gives ~for both models considered the emirical coverage robabilities of 90% bootstra confidence intervals and the corresonding mean lengths able 1. Standard errors of the bootstra estimates of the 5%, 50%, and 95% ercentage oints of the distributions considered in Figures 1 and 2 Model ~4+8 Model ~4+9 X 5% 50% 95% X 5% 50% 95%

10 [ [ MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 549 Figure 2. Estimated exact and bootstra estimates of the 5%, 50%, and 95% ercentage oints of the distribution of h~ m[ h ~x m~x for model ~4+9+ of these intervals based on 100 indeendent reetitions+ For model ~4+8 we calculate the bootstra confidence intervals for the conditional variance v~x and for model ~4+9 the bootstra confidence intervals for the conditional mean m~x using the values x 1+5,x 0,and x 1+5+ Also included in this table is a comarison of the Markovian local bootstra with the block bootstra method ~cf+ Künsch, 1989; Liu and Singh, Note that a ~1 a100% bootstra confidence interval for the conditional variance v~x is defined here v[ h ~x q[ 1 a02, v[ h ~x q[ a02 # where q[ denotes the th quantile of the bootstra estimate of the distribution of v[ h ~x v~x+ For the Markovian local resamling scheme this distribution is estimated using the distribution of v[ h ~x v b ~x and for the block bootstra method using the distribution of v h ~x v[ h ~x+ Here v[ h ~x is the kernel estimator of v~x based on the relicate X 1, X 2,+++,X obtained using the block bootstra method+ he bootstra confidence intervals for the conditional mean m~x have been obtained in an analogous way, that is, by relacing in the receding formulas v~x, v[ h ~x, v h ~x, v b ~x,and v[ h ~x by m~x, m[ h ~x, m[ h ~x, m b ~x, and m[ h ~x, resectively, where m[ h ~x is the kernel estimate of the conditional mean based on the block bootstra series+ Finally, and to see the effects of varying the resamling arameters, both methods have been alied using several values of the resamling width b and of the block size B+

11 550 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS able 2. Emirical coverage robabilities ~C+P+ and mean lengths of 90% bootstra confidence intervals for the conditional variance of model ~4+8 and for the conditional mean of model ~4+9 x 1+5 x 0+0 x 1+5 C+P+ Length C+P+ Length C+P+ Length Model 4.8) Block bootstra B B B B Markov local bootstra b b b b Model 4.9) Block bootstra B B B B Markov local bootstra b b b b As this table shows the local bootstra rocedure comares favorably with the block bootstra rocedure regarding the coverage robabilities of the confidence intervals for both classes of models considered+ In fact, both methods seem to behave very similarly taking into account the standard error ~1 [ 0100 [ of the estimated coverage robability + [ Aart from this overall behavior, it seems that in the case of the moving average model ~4+9 the block bootstra method is slightly more efficient than the local resamling method in terms of the average length of the bootstra confidence intervals obtained+ On the other hand, and in terms of the same quantity, the local bootstra method is quite a bit more efficient than the block bootstra method in the case of the random coefficient autoregressive rocess ~4+8; this is not surrising because model ~4+8 is a true first order Markov rocess+

12 5. DERIVAIONS MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 551 o rove heorem 3+1 we make use of some lemmas that are established in what follows+ We begin with two lemmas concerning some asymtotic roerties of the bootstra series $X t % and that are reroduced from and roved in Paaroditis and Politis ~1997+ o state these lemmas we fix some additional notation+ For x ~x 1, x 2,+++,x R let F ~x P~X t x 1, X t 1 x 2,+++, X t 1 x be the stationary distribution function of the bootstra Markov chain and F~x P~X t x 1, X t 1 x 2,+++,X t 1 x the stationary distribution function of the original rocess $X t, t 1%; that is, we set F~{ F Yt, ~{+ LEMMA 5+1+ Suose that Assumtions A1) A3) and A5) hold. hen conditionally on X 1, X 2,+++,X we have that, as r `, su6f ~x F~x6 r 0 x S a+s+ LEMMA 5+2+ Under the same assumtions as in Lemma 5.1 and conditionally on X 1, X 2,+++,X, 0 Nexists such that for all 0 the bootstra series $X t, t 1% is strongly mixing where the strong mixing coefficient a satisfies a ~n C n for some constants C 0 and ~0,1 that are indeendent of. LEMMA 5+3+ Suose that Assumtions A1) A5) hold and let f be a realvalued, continuous function, y a fixed real number, and s a ositive integer. We then have that, as r `, s 1 l su x S h f~x l 1 k h s ~y X l 1 W b ~x Y l, n W b ~x Y n, f~yf Xt 1 6Y t, ~y6x k s ~udu r0 in robability. Proof+ Let N~x;f f~yf Xt 1 Y t, ~y,xk s ~udu, N ~x;f h s 1 f~x l 1 k s h ~y X l 1 W b ~x Y l,, l

13 552 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS f b Z ~x 1 n W b ~x Y n,, and f ~x f Yt, ~x+ Note that h s 1 f~x l 1 k s h ~y X l 1 W b ~x Y l, l W b ~x Y n, n f~yf Xt 1 6Y t, ~y6x k s ~udu 1 ~x;f N~x;f6 f b Z ~x 6N 6N~x;f6 6 f b Z ~x f ~x ) f ~x In the following discussion we show that su x S 6N ~x;f N~x;f6 r 0in robability+ Because su x S 6 f b Z ~x f ~x6 r 0 follows by the same arguments, the lemma is established using ~5+10 and ~A1~iv+ Consider 6N ~x;f N~x;f6+ For this term we have 6N ~x;f N~x;f6 h s 1 f~z 1 k h s ~y z 1 W b ~x z 2 f Xt 1 Y t, ~z 1,z 2 dz 1 dz 2 f~yf Xt 1 Y t, ~y,x k s ~udu h s 1 1 l 1 ~x 2 ~x f~x l 1 k h s ~y X l 1 W b ~x Y l, f~z 1 k h s ~y z 1 W b ~x z 2 f Xt 1 Y t, ~z 1,z 2 dz 1 dz 2 with an obvious notation for 1 ~x and 2 ~x+ Consider first 1 ~x+ Using the substitutions z 1 y u 1 h and z 2 x u 2 b we have 1 ~x ~f~y u 1 hf Xt 1,Y t, ~y u 1 h,x u 2 b f~yf Xt 1,Y t, ~y,xk s ~u 1 W~u 2 du 1 du 2 f~y u 1 hf Xt 1,Y t, ~y u 1 h,x u 2 b f~yf Xt 1,Y t, ~y,x k s ~u 1 W~u 2 du 1 du 2 O~max$h, b% uniformly in x by ~A2~ii+ Consider next 2 ~x and observe that because f is continuous and K is bounded and comactly suorted, f~x l 1 k s ~~ y X l 1 0h is bounded+ Fur-

14 MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 553 thermore, instead of h s 1 1 l f~x l 1 k s h ~y X l 1 W b ~x Y l, we consider in the following discussion the asymtotically equivalent statistic R ~ y, x h s 1 ~ 1 l f~x l 1 k s h ~y X l 1 W b ~x Y l, + Observe that E~R ~ y, x h s 1 f~z 1 k h s ~y z 1 W b ~x z 2 f Xt 1 Y t, ~z 1,z 2 dz 1 dz 2 and that su x S 2 ~x su6r ~ y, x E~R ~ y, x6 o P ~1+ x S hus we have to show that su x S 6R ~ y, x E~R ~ y, x6 r 0 in robability+ o deal with this term divide S in a number of N cubes denoted by I i, with centers x i and length L + We then get su6r ~ y, x E~R ~ y, x6 x S max su 6R ~ y, x R ~ y, x i 6 1 i N x S I i, max 1 i N 6R ~ y, x i E~R ~ y, x i 6 max su 6E~R ~ y, x i E~R ~ y, x6 1 i N x S I i, M 1, M 2, M 3, + By the Lischitz continuity of K it is easily seen that M 1, O P ~L b ~ 1 and M 3, O P ~L b ~ 1 + Furthermore, P max 1 i N 6R ~ y, x i E~R ~ y, x i 6 L b ~ 1 where N 1 P i 1 ~ hb l v l, ~ y, x i f~x l 1 k s y X l 1 h E f~x l 1 k s y X l 1 h v l, ~ y, x i ~ L b 1, W x i Y l, b W x i Y l, b

15 J J J 554 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS and E~v 2 l, ~y,x s 2 C 2 hb with C 0 a generic constant indeendent of + o bound this robability let first vj l, ~ y, x i C 1 v l, ~x i and L ~log~ 3 b 2 e h 1 e for some 0 e 1+ Alying Proosition 1 of Doukhan ~1994, + 33, with s hb and log s 1 loghb 1 and using h0h 1 e 1 and b 2 e 0 b 2 1, we then have that constants a 0 and b 0 exist such that N 1 P v l, ~ y, x i 1 C L ~ b 1 i 1 ~ hb l O~N P v l, ~ y, x i l 1 ~log~ 3 loghb hb loghb 1 O~N P 1 v l, ~ y, x i l 1 C ~log~ 2 hb loghb O~N aex$ b log~ C% o~ ) hus for b O~ d and d 10 we get that L 0b ~ 1 r 0 and L r 0as r`+ he following result deals with a well known roerty of the kernel estimator m[ h ~x;f+ Its roof is omitted because it closely follows that of Lemma 5+3+ LEMMA 5+4+ Suose that Assumtions A1) A4) hold and let w be a realvalued continuous function. hen as r ` su6 m[ h ~x;w m~x;w6 r 0 x S in robability. Let f ~{6x denote the conditional density f Xt 1 6Y t, ~{6x and let f Z~ y6x be the kernel estimator of f ~ y6x given by Z k h ~ y X l 1 W b ~x Y l, f ~ y6x l, 5.12) n W b ~x Y n, where y S+ Note that in the notation of Lemma 5+3 and if we set f~{ [ 1 and s 1 we immediately get that su x S 6 f Z~ y6x f ~ y6x6 r 0 in robability as r `+

16 Z o simlify calculations and because the results obtained are asymtotic for r `, we assume in the following that Y, ; F, that is, we assume that the bootstra series starts with its stationary distribution+ Now, for k N let x jk ~x jk, x jk 1,+++,x jk 1 X and recall that $X t, t 1% is a discrete Markov chain with state sace X + Using iteratively the relation P~Y t, MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 555 x j1 j 1 1 P~X t x j1 6Y t 1, x j2 P~Y t 1, x j2 and the transition robability ~2+3 we get the following useful exression for the stationary robability mass function of Y t, + P~Y t, x j1 {{{ P~ X t x j1 6 X t 1 x j2, X t 2 x j3,+++,x t x j 1 j 1 j 2 j 2 P~X t 1 x j2 6X t 2 x j3,x t 3 x j4,+++,x t 1 x j 2 {{{ P~ X t 1 x j 6 X t x j 1, X t 1 x j 2,+++,X t 2 1 x j2 P~X t x j 1,X t 1 x j 2,+++,X t 2 1 x j2 ) i 1 ) i 1 P~X t i 1 x ji 6Y t i, x ji 1 df ~x j 1 W b ~x ji 1 Y ji 1, n W b ~x ji 1 Y n, df ~x j ) o roceed with the roof of heorem 3+1 we first note that L ~x can be exressed as L ~x 1 f h Z ~x h 1 f h Z ~x h ~f~x j 1 j 1 6Y j, #K h ~x Y j, j 1 6Y j, # m ~x;fk h ~x Y j, 1 f h Z ~x $L 1, ~x L 2, ~x%, where f h ~x 1 K h ~x Y j, and an obvious notation for L 1, ~x and L 2, ~x+ Using the receding exression, the assertion of the theorem is established if we show that f h Z ~x r f ~x weakly, that L 1, ~x converges to a Gaussian distribution with mean zero and the aroriate variance, and that L 2, ~x0f h Z ~x is a consistent estimator of the mean ~bias term of L ~x+ his is established in the following three lemmas+

17 Z 556 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS LEMMA 5+5+ Under Assumtions A1) A5) we have conditionally on X 1, X 2,+++,X that for all x S, f h Z ~x r f ~x in robability. Proof+ Let x ~x 1, x 2,+++,x + Using ~5+13 we have E ~ f h Z ~x K h ~x x j1 P~Y t, x j1 j 1, j 2,+++,j o P ~1 ) j 1,j 2,+++,j m 1 ) j 2,+++,j m 2 ) j 2,+++,j m 2 ) j 3,+++,j m 3 ) i 2 ) i 2 ) i 1 k h ~x m x jm W b ~x ji 1 Y ji 1, df ~x j 1 n W b ~x ji 1 Y n, f~x 1 6x j2 k h~x m x jm W b ~x ji 1 Y ji 1, df ~x j 1 n W b ~x ji 1 Y n, f~x 6x kh~x x W ~x Y 1, m jm b ji 1 ji df 1 j2 ~x j 1 j 2 ) i 3 n W b ~x ji 1 Y n, f~x 1 6x j2 k h ~x 2 x j2 W b ~x j3 Y j2 1, n W b ~x j3 Y n, k h~x m x jm W b ~x ji 1 Y ji 1, df ~x j 1 o P ~1 n W b ~x ji 1 Y n, ) j 3,+++,j m 3 ) f~x 1 6x 2,x j3,+++,x j 1 f~x 2 6x j3 i 3 k h~x m x jm W b ~x ji 1 Y ji 1, df ~x j 1 o P ~1, n W b ~x ji 1 Y n, where the receding o P ~1 terms are due to the uniform ~over x convergence of f Z~{6x given in ~5+12 to f ~{6x and Lemma 5+3+ Continuing in this way we end u with the exression

18 E ~ f h Z ~x f ~x 1 6x 2,+++,x,x j 1 f~x 2 6x 3,+++,x,x j 1,x j 2 {{{ f ~ x 6 x j 1,+++,x j2 df ~x j 1 o P ~1 r f ~x 1 6x 2,+++,x,x j 1 f~x 2 6x 3,+++,x,x j 1,x j 2 {{{ f ~ x 6 x j 1,+++,x j2 df~x j 1 f ~x, 5.14) where the last convergence is due to Lemma 5+1 and Helly Bray s theorem+ o evaluate the variance of f h Z ~x, let Z h, j K h ~x Y j, and note that f h Z ~x# ~ 2 Var ~Z h, ~1 t0 Cov ~Z h,1, Z h,1 t ) t 1 Now, h h, j # h j1, j 2,+++,j K 2 h ~x x j1 P~Y t, x j1 and by the same arguments as those leading to ~5+14 we get h h, j # r f ~xk 2 ~udu, that is, 1 h, j # O P ~ 1 h + o handle the second term on the right hand side of ~5+15 note first that by Davydov s lemma ~Hall and Heyde, 1980, A+ we have 6Cov~Z h,1, Z h,1 t 6 8a ~t 1 20n ~E 6Z h,1 6 n 20n for some real n 2+ Because E 6Z h,1 6 n j1, j 2,+++,j K n h ~x x j1 P~Y t, x j1 O P ~h ~1 n by the same arguments as in establishing that ~5+14 is true, we get by Lemma 5+2 and arguments similar to those in Masry and jøstheim ~1995, that t 1 ~1 t0 Cov ~Z h,1, Z h,1 t 6 o P ~ 1 h + LEMMA 5+6+ Under Assumtions A1) A5) we have conditionally on X 1, X 2,+++,X that, as r `, h ~f~x j 1 m b ~Y j, ;fk h ~x Y j, n N~0, t 2 ~x;f in robability, where t 2 ~x;f K 2 ~udu f ~x Var@f~X t 1 6Y t, x#. Proof+ Define MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 557 D j 1 [ h ~f~x j 1 m b ~Y j, ;fk h ~x Y j and denote by F j s~x 1,X 2,+++,X j the s-algebra generated by $X 1, X 2,+++,X j %+ Because for x X, m b ~x;f t 1 6Y t, x# we have

19 558 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS by the Markov roerty that $D j 1, F j %, 1,+++ forms a martingale difference scheme+ hus to establish the desired result, we aly heorem 1 of Brown ~1971+ By this theorem we have to show that 2 j 1 6F j # r K 2 ~udu f ~xvar@f~x t 1 6Y t, x# in robability and that for every d 0 the Linderberg condition 5.16) j 1 I~D j 1 d6f j # r 0 in robability 5.17) is satisfied+ o simlify notation and to stress the essentials, we focus in what follows on the case 1+ he case of general can be handled along the same lines+ Consider first ~5+16+ Because 2 j 1 6F j # h ~f~x l m b ~X j ;f 2 K 2 h ~x X j P~X j 1 X l 6X j l 2 we have 2 j 1 6 F j # 1 j 1 h j 1 where v[ b ~z;f l 2 K 2 x X j v[ b ~X j ;f, h ~f~x l m b ~z;f 2 W b ~z X l 1 s 1 W b ~z X s Let v~z;f t 1 m~z; f 2 6X t z# and note that as in the roof of Lemma 5+5, ~h 1 j 1 K 2 ~~x X j 0h O P ~1+ hen j 1 2 j 1 6F j # 1 h j 1 1 h j 1 K 2 x X j h K 2 x X j + v~x j ;f ~ v[ b ~X j ;f v~x j ;f+ h

20 Now, because v[ b ~z;f m b ~z;f 2 ~m ~z;f 2 is a Nadaraya Watson kernel estimator of v~z;f t 1 m~z; f 2 6X t z# we have for D h, x h# that su 6 v[ b ~z;f v~z;f6 su 6m b ~z;f 2 m~z;f 2 6 z D h z D h r 0 in robability by Lemma 5+4+ hus 1 h j 1 K 2 x X j su z D h 6m b ~z;f m~z;f6 su z D h 6m b ~z;f m~z;f6 6 v[ b ~X j ;f v~x j ;f6 h su 6 v[ b ~z;f v~z;f6o P ~1 r 0+ z D h o conclude the roof of ~5+16 we show that V ~x;f [ 1 h j 1 MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 559 K 2 x X j h v~x j ;f r t 2 ~x;f 5.18) in robability+ aking exectation of this term with resect to the bootstra distribution we get using the same arguments as in establishing ~5+14 that ~x;f# h 1 l 2 v~x l ;f~k 2 ~~x X l 0hW b ~X m X l 1 s 1 W b ~X m X s df ~X m r K 2 ~uduf ~xvar@f~x t 1 6X t x#, where the last convergence follows using Lemma 5+3+ Furthermore, along the same lines as in the roof of ~5+15 we get that ~x;f# O P ~ 1 h + o rove ~5+17 we use the inequality C# C 1 E6X6 2 where C 0 is a constant rovided the random variable X satisfies E6X6 2 `+ We then have for every d 0

21 560 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS 2 j 1 I~D 2 j 1 d6f j # j 1 1 d j 1 4 j 1 6F j # 1 E $f~x dh 2 2 j 1 j 1 m b ~X j ;f% 4 K 4 x X j h 6F j Let 1 K 4 ~~x X dh 2 2 j 0h j 1 l 2 ~f~x l m b ~X j ;f 4 W b ~X j X l 1 s 1 W b ~X j X s + R ~z;f l 2 ~f~x l m b ~z;f 4 W b ~z X l n 1 W b ~z X s which is a Nadaraya Watson estimator of R~z;f t 1 m~z; f 4 6X t x#, and note that su z Dh 6R ~z;f R~z;f6 r 0 in robability+ hus 2 j 1 I~D 2 j 1 d6f j # 1 K 4 ~~x X j 1 d 2 h 2 j 0hR ~X j ;f o P ~1 j 1 O P ~d 1 1 h 1, where the last equality follows by the same arguments as those used to rove ~5+18+ LEMMA 5+7+ Under Assumtions A1) A6) and if 102 h ~ 402 r C h 0, we have conditionally on X 1, X 2,+++,X that, as r `, h j 1 6Y j, # m b ~x;fk h ~x Y j, r C h u 2 k~udu in robability. l 1, m ~x l ~x ~x;f f l Yt, ~x 1 2 m~x l x l ~x;f f Yt, ~x 5.19) Proof+ Let m b ~Y t, ;f t 1 6Y t, # and x ~x 1, x 2,+++,x + By a aylor series exansion we get

22 I MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 561 h ~m b ~Y j, ; f m b ~x;fk h ~x Y j, m ~xl b ~x;f h l l 1 1 l 2 1 K h ~x Y j, ~X j l 1 x l K h ~x Y j, m ~xl1 xl2 b ~ x;f I h ~X j l1 1 x l1 ~X j l2 1 x l2 for some x between x and Y j, + Because under the assumtions made h ~E m[ h ~x;f m~x;f r B~x;f in robability, the assertions of the lemma will be established if we show that, in robability, h and h m b ~xl ~x;f r m ~x l ~x;f, 5.20) ~X j l 1 x l K h ~x Y j, r C h u 2 ~x k~uduf l Yt, ~x, 5.21) ~X j l1 1 x l1 ~X j l2 1 x l2 K h ~x Y j, r d l1,l 2 C h u 2 k~uduf Yt, ~x, 5.22) K su 6m ~xl1 xl2 b ~z;f m ~x l1 x l2 ~z;f6 r 0, 5.23) h, x h# where m K ~x i x j ~z;f 1 ~x i x j f Yt, ~z 1 2 ~1 d i, j f~yf Xt 1 Y t, ~x ~y,zdy m~z;f f i x j Yt, ~z m ~x i ~x ~z;f f j Yt, ~z m ~x j ~x ~z;f f i Yt, ~z ) he rest of the roof is devoted to showing that ~5+20 ~5+23 are true+ Consider ~5+20 and note that m ~xl b ~x;f 1 1 f b Z ~x f~x b 1 j 1 W ~xl ~~x Y j, 0b m b ~x;f 1 b 1 W ~x l ~~x Y j, 0b +

23 562 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS Recall that f b Z ~x r f Yt, ~x and m b ~x;f r m~x;f in robability+ Furthermore, by standard arguments we have 1 b 1 and 1 b 1 f~x j 1 W ~xl ~x ~~x Y j, 0b r f~yf l Xt 1 Y t, ~y,x dy 5.25) W ~x l ~x ~~x Y j, 0b r f l Yt, ~x+ 5.26) herefore, m b ~xl ~x;f r m ~x l ~x;f in robability+ he roof of ~5+21 and ~5+22 is very similar and makes exlicit use of ~5+13 and arguments like those alied in the roof of Lemma 5+3+ o avoid tedious maniulations of formulas we focus on the case 1+ Consider ~5+21 so we have E h j 1 ~X j xk h ~x X j ~ 1 h ~X l xk h ~x X l P~X t X l l 2 ~ 1 h l 2 ~X l xk h ~x X l W b ~X s X l 1 n 2 W b ~X s X n Now, to establish the desired result we have to show that df ~X s + su y S h ~X l xk h ~x X l W b ~ y X l 1 ~x l 2 f Xt X t 1 ~x, y C h u 2 f Xt ~ y W b ~ y X n 1 K~udu r 0+ n ) By ~A1~iv and the uniform convergence of f b Z ~ y to f ~ y, it suffices to show that su y S h ~x ~X l xk h ~x X l W b ~ y X l 1 C h f Xt X t 1 ~x, y u 2 K~udu l 2 r )

24 E For this note that E h l 2 ~X l xk h ~x X l W b ~ y X l 1 ~ h 3 uk~uw~r f X t X t 1 ~x uh, y rbdrdu ~ 1 h 5 u2 ~x K~uW~r f Xt X t 1 ~ x, I y rbdrdu, where 6xI x6 h+ hus by the Lischitz continuity of f ~x ~{,{ we get that ~ 1 h 5 u2 K~uW~r f ~x ~x ~ x, I y rbdrdu r C h f Xt X t 1 ~x, y u 2 K~udu uniformly in y; that is, we have shown that ~x su ~X l xk h ~x X l W b ~ y X l 1 C h f Xt X t 1 ~x, y u 2 K~udu y S E h l 2 r 0 in robability+ hus to establish ~5+28 it remains to show that su y S h $~X l xk h ~x X l W b ~ y X l 1 l 2 E~X l xk h ~x X l W b ~ y X l 1 % r 0 in robability+ o handle this term, we roceed as in the roof that M 1, r 0in robability in Lemma 5+3+ In articular, let Z l, ~ y ~X l xk x X l h E~X l xk x X l h W y X l 1 b W y X l 1 b and note that 6Z l, 6 C 1 h and that EZ 2 ~ y C 2 2 h 3 b for some generic ositive constants C 1 and C 2 + Using the slitting device alied in the roof of Lemma 5+3 we get by the Lischitz continuity of W that 1 1 Z l, ~ y hb l 2 max Z l, ~ y i hb l 2 O~h 102 L b 2 + Along the same lines as in dealing with ~5+11 we get for the first term on the right hand side of the receding inequality using the notation Z l, ~ y i h 1 C 1 2 Z l, ~ y i and for 0 e 1 and L ~log~ 3 b 302 e h 102 e that su y S MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS i N

25 E E 564 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS P max 1 1 i N Z l, ~ y i hb l 2 L h 102 b 2 N i 1 N i 1 P l 2 P l 2 Z l, ~ y i L 102 C 2 b Z l, ~ y i ~log~ 2 hb loghb 1 O~N aex$ b log~ C 2 % for some a 0 and b 0+ hus by the assumtions made on b and h we get that L r 0, L h 102 0b 2 r 0, and O~N aex$ b log~ 0C 2 % r 0asr`+ he roof of ~5+22 follows exactly the same lines as the roof of ~5+21 and is therefore omitted+ o establish ~5+23 verify first that m ~xl1, xl2 b ~z;f 1 1 f b Z ~z f~x b 2 j 1 W ~x l1, x l2 ~~z Y j, 0b m b ~xl2 ~z;f 1 b 2 C 2 W ~x l1, x l2 ~~z Y j, 0b m ~xl1 ~x b ~z;f f b Z l2 ~z m ~xl2 ~x b ~z;f f b Z l1 ~z ) We then have to show that every element on the right hand side of ~5+29 converges uniformly to the corresonding elements on the right hand side of ~5+24+ We show this for the first term on the right hand side of ~5+29, the other terms being handled in exactly the same manner+ For this note that su z S 1 ~x f~x j 1 W l1 x l2 b ~z Y j, 1 2 ~1 d ~x x l1 x l2 f~yf l1 x l2 Xt 1 Y t, ~y,z dy su z S 1 ~x ~f~x j 1 W l1 x l2 ~x b ~z Y j, Ef~X j 1 W l1 x l2 b ~z Y j, su z S ~x f~yw x l1 l2 b ~z rf Xt 1 Y t, ~y,r dydr 1 d x l1 x l2 ~x l1 x f~yf l2 Xt 1 Y 2 t, ~y,z dy B 1, B 2,, with an obvious notation for B 1, and B 2, + Now, by the same arguments used to handle the term 1 ~x in the roof of Lemma 5+3 and by ~A6, we get that B 2, O~b+ Furthermore, by the mean value theorem we have alying the

26 same slitting device as the one used in handling the term 2 ~x in the lemma mentioned earlier, that su z S 1 ~x ~f~x j 1 W l1 x l2 ~x b ~z Y j, Ef~X j 1 W l1 x l2 b ~z Y j, max 1 i N 1 O P ~L b ~ 3 + ~f~x j 1 W b ~x l1 x l2 ~z i Y j, Ef~X j 1 W b ~x l1 x l2 ~z i Y j, o conclude the roof verify that Var~f~X j 1 W ~x l1 x l2 ~~z i Y j, 0b Cb and let L ~log 3 b ~ 02 1 e for some 0 e 1+ Using the notation C j, ~z i f~x j 1 W ~x l1 x l2 ~~z i Y j, 0b E~f~X j 1 W ~x l1 x l2 ~~z i Y j, 0b and alying the exonential inequality for strongly mixing sequences used in the roof of Lemma 5+3 we then get that for some constants a 0 and b 0 N P i 1 1 b 2 N P i 1 C j, ~z i L b C 1 C j, ~z i ~ 3 O~N aex$ b log~ C% o~1+ REFERENCES MARKOVIAN LOCAL RESAMPLING OF KERNEL ESIMAORS 565 L 1b02 logb 1 02 Cb ~ 02 1 logb 02 Auestad, B+ &D+jøstheim ~1990 Identification of nonlinear time series: First order characterization and order determination+ Biometrika 77, Brown, B+M+ ~1971 Martingale central limit theorems+ Annals of Mathematical Statistics 42, Doukhan, P+ ~1994 Mixing: Proerties and Examles+ Lecture Notes in Statistics 85+ New York: Sringer-Verlag+ Efron, B+ ~1979 Bootstra methods: Another look at the jackknife+ Annals of Statistics 7, Franke, J+, J+-P+ Kreiss, &E+Mammen ~1996 Bootstra of Kernel Smoothing in Nonlinear ime Series+ Prerint+ Hall, P+ &C+C+Heyde ~1980 Martingale Limit heory and Its Alications+ San Diego: Academic Press+ Künsch, H+R+ ~1989 he jackknife and the bootstra for general stationary observations+ Annals of Statistics 17, Liu, R+ &K+Singh ~1992 Moving blocks jackknife and bootstra cature weak deendence+ In R+ LePage & L+ Billard ~eds+, Exloring the Limits of the Bootstra, New York: Wiley+ Masry, E+ &D+jøstheim ~1995 Nonarametric estimation and identification of nonlinear ARCH time series+ Econometric heory 11, Neumann, M+ &J+-P+ Kreiss ~1998 Regression-tye inference in nonarametric autoregression+ Annals of Statistics 26, Paaroditis, E+ &D+N+Politis ~1997 he local bootstra for Markov rocesses+ Journal of Statistical Planning and Inference, forthcoming+ Robinson, P+M+ ~1983 Nonarametric estimation for time series models+ Journal of ime Series Analysis 4,

27 566 EFSAHIOS PAPARODIIS AND DIMIRIS N. POLIIS Roussas, G+ ~1990 Nonarametric regression estimation under mixing conditions+ Stochastic Processes and heir Alications 36, jøstheim, D+ &B+Auestad ~1994 Nonarametric identification of nonlinear time series: Projections+ Journal of the American Statistical Association 89, ong, H+ ~1990 Nonlinear ime Series: A Dynamical Aroach+ New York: Oxford University Press+