Large-sample inference for nonparametric regression with dependent errors.

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(1)LSE Research Online Article (refereed). Large-sample inference for nonparametric regression with dependent errors Peter M. Robinson. LSE has developed LSE Research Online so that users may access research output of the School. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy of any article(s) in LSE Research Online to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. You may freely distribute the URL (http://eprints.lse.ac.uk) of the LSE Research Online website.. You may cite this version as: Robinson, P.M. (1997). Large-sample inference for nonparametric regression with dependent errors [online]. London: LSE Research Online. Available at: http://eprints.lse.ac.uk/archive/00000302. This is an electronic version of an Article published in Annals of statistics, 25 (5). pp. 2054-2083 © 1997 Institute of Mathematical Statistics. http://www.imstat.org/aos/. http://eprints.lse.ac.uk Contact LSE Research Online at: [email protected].

(2) The Annals of Statistics 1997, Vol. 25, No. 5, 2054 ] 2083. LARGE-SAMPLE INFERENCE FOR NONPARAMETRIC REGRESSION WITH DEPENDENT ERRORS1 BY P. M. ROBINSON London School of Economics A central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum. We apply the result to kernel nonparametric fixed-design regression, giving a single central limit theorem which indicates how error spectral behavior at only zero frequency influences the asymptotic distribution and covers long-range, short-range and negative dependence. We show how the regression estimates can be Studentized in the absence of previous knowledge of which form of dependence pertains, and show also that a simpler Studentization is possible when long-range dependence can be taken for granted.. 1. Introduction. This paper justifies approximate normal inference on fixed design nonparametric regression in the presence of dependent observations. The dependence structures covered are unusually diverse, because the stationary errors can exhibit dependence of short-range, long-range, or negative type. Also, unusually for the time series regression literature, we give a single central limit theorem which simultaneously covers all three cases. The limiting covariance structure of the estimates depends on the nature of the dependence through only a self-similarity parameter, as well as a scale factor, and we indicate how to validly Studentize the regression estimates by estimating these parameters, without prejudging whether there is shortrange, long-range, or negative dependence. The Studentization is based on residuals from the regression model, but we also show that when long-range dependence can be taken for granted, the raw data can be used for this purpose. The paper clarifies the feature of serial correlation which is really relevant, namely the behavior of the spectral density at only zero frequency. In fact, the same point applies in other problems, to Gasser]Muller regres¨ sion estimates as well as the kernel ones we study, to certain wavelet regression problems, and to versions of parametric regression; Lemmas 1 and 2 below can be checked to provide analogous central limit theorems in these problems.. Received March 1995; revised March 1997. 1 Research supported by ESRC Grant R000235892. AMS 1991 subject classifications. Primary 62G07, 60G18; secondary 62G20. Key words and phrases. Central limit theorem, nonparametric regression, autocorrelation, long range dependence.. 2054.

(3) NONPARAMETRIC REGRESSION. We consider the model. Ž 1.1.. yt s r. ž / t. n. 2055. t s 1, 2, . . . ,. q ut ,. where yt is observed for t s 1, . . . , n, and u t is an unobservable error. As always in this model, yt must be viewed as a triangular array Žas can be u t , if so desired. as n increases, but we suppress reference to n here. To estimate r Ž x ., x g Ž0, 1., we consider. Ž 1.2.. ˆr Ž x . s. 1. Ýk nb t. ž. nx y t nb. /. yt ,. where b is a positive bandwidth number, k is a kernel function such that. Hy`k Ž v . dv s 1 `. Ž 1.3.. and Ý t will always denote a sum over t from 1 through n. Taking for granted covariance stationarity of u t , implying the mean and autocovariances are time invariant, then Eu1 s 0 with no loss of generality, and we denote by g j s Eu1 u1qj the lag-j autocovariance of u t . We introduce the following assumption. ASSUMPTION 1. The process u t is covariance stationary with absolutely continuous spectral distribution function, its spectral density, f Ž l., defined p Ž . Ž . by g j s Hy p f l cos j l d l, being of form. Ž 1.4.. f Ž l. s g Ž l. h Ž l. ,. yp - l F p ,. where Ži. g is an even nonnegative function that is continuous and positive at l s 0, and we denote G s g Ž0.; Žii. h is an integrable function such that. dj s. Ž 1.5.. Hyph Ž l. cos Ž jl. d l p. ; u Ž H . j 2 Hy2 s 2p D j0 ,. as j ª `,. H g Ž 0, 1 . _ 12 4 ,. H s 12 ,. where D a b is the Kronecker delta, u Ž H . s 2 G Ž2 y 2 H .cosp Ž1 y H .4 and also h Ž 0 . s 0, 0 - H - 12 . Ž 1.6. The cases H s 21 and 0 - H - 21 referred to here are termed, respectively, ‘‘short-range dependence’’ and ‘‘negative dependence,’’ the complementary case, 12 - H - 1, on Ž0, 1. being ‘‘long-range dependence.’’ Then u Ž H . is negative for 0 - H - 1r2 and positive for 1r2 - H - 1, so that d j is, respectively, eventually negative and positive. For H s 1r2 we have prescribed d j to thus interpolate between these other two cases, but there is no loss of.

(4) 2056. P. M. ROBINSON. generality relative to assuming only that hŽ l. is continuous and positive at l s 0. We formally identify the case H s 1r2 with a constant hŽ l. instead of taking d j ; u Ž1r2. jy1 so as to avoid the discontinuity in convergence rates resulting from the latter specification w see Hall and Hart Ž1990.x , thereby to simplify Studentization Žsee Section 4. and also to explicitly include shortrange dependence. In practice, the scale factor G is unknown, and the reason for incorporating the factor u Ž H . in Ž1.5. is so that Assumption 1 corresponds approximately to a simple local parameterization in the frequency domain,. Ž 1.7.. f Ž l . ; Gl1y 2 H. as l ª 0q,. for all H g Ž0, 1.. Assumption 1Ži. and. Ž 1.8.. h Ž l . ; l1y 2 H. as l ª 0q,. together imply Ž1.7.. For H g Ž0, 12 ., the Fourier series of hŽ l. converges absolutely and Ž1.6. implies. Ž 1.9.. `. Ý. d j s 0,. jsy`. and thence Ž1.8. from Theorem III-31 of Yong Ž1974.. For H s 12 , Ž1.8. obviously holds. For H g Ž 21 , 1., Ž1.8. is equivalent to Ž1.5. if the d j are quasimonotonically convergent to zero, that is, there exists C - ` such that d jq1 F d j Ž1 q Crj . for all sufficiently large j w Yong Ž1974., Theorem III-14x . Whatever the value of H g Ž0, 1., Assumption 1 effectively imposes no restrictions on f away from zero frequency, apart from integrability implied by covariance stationarity: it can be infinite or zero at any other frequencies, for example. This contrasts with assumptions made in previous work on central limit theory for nonparametric regression. The case H s 1r2 was in effect addressed by Roussas, Tran and Ioannides Ž1992., Csorgo ¨ ˝ and Mielniczuk Ž1995a., and Tran, Roussas, Yakowitz and Truong Van Ž1996.. They modelled u t as a strongly mixing process with at least summable mixing numbers, a nonlinear function, with Hermite number m, of a Gaussian process with autocovariances whose mth absolute powers are summable, and a linear process with absolutely summable weights Žcf. Assumption 2., respectively. All these assumptions imply that f is bounded Žand indeed satisfies a Lipschitz continuity condition of degree greater than 1r2. on Žyp , p x . Csorgo ¨ ˝ and Mielniczuk Ž1995b, c. addressed the case H g Ž1r2, 1.. Their assumption on g j is more general in that they allow for a slowly varying factor in the right-hand side of Ž1.5.: we could incorporate this, but as Csorgo ¨ ˝ and Mielniczuk’s Ž1995b, c. work indicates, it will then arise in the norming for asymptotic normality unless it satisfies very restrictive conditions. We stress Studentization later in the paper, in which practical circumstances it seems unlikely that the applied worker would incorporate any such factor. Ignoring this aspect, Assumption 1 is more general than Csorgo ¨ ˝ and Mielniczuk’s Ž1995b, c., because they effectively take g Ž l. ' G; the consequent require-.

(5) 2057. NONPARAMETRIC REGRESSION. ment that. Ž 1.10.. g j ; Gu Ž H . j 2 Hy2 as j ª `,. rules out such models covered by Assumption 1 as. Ž 1.11.. f Ž l . A <1 y e i l < 1y 2 H <1 y 2 cos v e i l q e 2 i l < 1r2yJ. for 0 - v F p and 0 - H F J if J ) 1r2, so there is a singularity around frequency l s v , of magnitude that at least matches that Žif H ) 1r2. at l s 0. ŽThe second factor in Ž1.11. has been discussed in detail by Gray, Zhang and Woodward, 1989.. At about the same time and independently of our work, Deo Ž1997. has also considered the case H g Ž1r2, 1., but requires that f Ž l. is positive and continuous at all l g Žyp , p . _ 04 . The fact that spectral behavior matters only at frequency 0 is familiar from theory for partial sums of weakly autocorrelated series, corresponding to the case H s 1r2. In particular, V Ž ny1 r2 Ý t u t . ª 2p f Ž0. if f is continuous at l s 0, where V denotes the variance operator. For H g Ž1r2, 1., however, Ž1.10. has been stressed Žagain with a slowly varying factor. in theory for partial sums and other statistics, following Taqqu Ž1975.. By arguing as in the proof of Lemma 3 below, we can partially extend Lemma 3.1 of Taqqu Ž1975. and Theorem 2.1 of Robinson Ž1993. w who took, respectively, 12 - H - 1 and 0 - H - 21 in Ž1.10. and allowed for a slowly varying factorx to obtain under Assumption 1,. Ž 1.12.. ž. V nyH. Ý ut t. / ª Gu Ž H . rH Ž 2 H y 1. ,. H g Ž 0, 1 . ,. taking sin 0r0 s 1. Likewise, under Assumption 1 we can justify the formula in Yajima Ž1988. for the asymptotic covariance matrix of the least squares estimate in polynomial time series regression with errors u t , which he obtained under the assumption that g is everywhere continuous. The following section gives some central limit results for weighted partial sums of a covariance stationary process that is linear in martingale differences with weights that are only square summable and is relevant to Assumption 1 for all H g Ž0, 1.. The results can apply to various problems, but in Section 3 we check them in the case of estimate Ž1.2. of Ž1.1.. Section 4 justifies the same normal approximation for suitably Studentized estimates. Section 5 contains an empirical application to the series of annual minimum levels of the Nile River. 2. Central limit theorem for weighted sums of linearly dependent variates. This section considers central limit theory for weighted partial sums of a sequence u t satisfying the following. ASSUMPTION 2.. Ž 2.1.. `. ut s. Ý. jsy`. `. a j « tyj ,. Ý. jsy`. a j2 - `,.

(6) 2058. P. M. ROBINSON. where the « t2 are uniformly integrable and E Ž « t2 < Fty1 . s 1 a.s. t s 0, "1, . . . ,. E Ž « t < Fty1 . s 0,. where Ft is the s-field of events generated by « s , s F t 4 . For a triangular array wt n , t s 1, . . . , n; n s 1, 2, . . . 4 , write S s Ý t wt n u t . The following result is related to ones of Eicker Ž1967., Ibragimov and Linnik Ž1971. and Hannan Ž1979.. LEMMA 1.. Let Assumption 2 hold and `. Ý. Ž 2.2.. 2 vjn s1. for all n,. jsy`. Ž 2.3.. max < vjn < s 0,. lim. nª` y`-j-`. where vjn s Ý t wt n a tyj . Then. Ž 2.4.. S ªd N Ž 0, 1 .. as n ª `.. yN y1 N q SyN q S`Nq1 , where S pq s Ý qjsp vjn « j . PROOF. For any N G 1, S s Sy` Thus. Ž 2.5. V Ž SyNy1 . q V Ž S`Nq1 . s y`. Ý. 2 vjn F. < j <)N. ž. Ý < wt n < t. / žÝa 2. j)N. 2 j. q. Ý. /. a j2 .. j-nyN. The squared factor depends on n only and in view of Ž2.1. we can choose N s Nn as a function of n such that Ž2.5. ª 0 as n ª `. For such N write N Nq1 SyN s Ý2ts1 x t n where x t n s vtyNy1, n « tyNy1. Thus EŽ x t n < FtyNy2 . s 0, 2 2 Ž . EŽ x t n < FtyNy2 . s vtyNy1, n a.s., by Assumption 2. It follows from 2.2 and N from Corollary Ž3.8. of McLeish Ž1974. that SyN ªd N Ž0, 1. as n ª ` if, for all h ) 0, 2 Nq1. Ý. lim. nª`. ts1. E Ž x t2n I Ž x t2n ) h . . s 0.. By Ž2.2. the sum on the left-hand side is bounded by max t EŽ « t2 I Ž « t2 ) 2 .. hrmax j vjn ª 0 as n ª ` by uniform integrability and Ž2.3.. I Condition Ž2.2. is merely a normalization. Sufficient conditions for Ž2.3. are given in the following lemma. Write. sn2 s V. ž. Ý ut t. /. `. s. Ý Ž a ty1 q ??? qa tyn . 2 ,. tsy`. and introduce the difference operator ^, such that ^ wt n s wt n y wtq1, n ..

(7) 2059. NONPARAMETRIC REGRESSION. LEMMA 2. Let Assumption 2 hold. If either Ži. there exists a positive-valued sequence a s a n such that, as n ª `,. ž. Ž 2.6.. Ý wt2n Ý t. a j2. < j <)a. /. 1r2. q max < wt n < 1FtFn. Ý. < j <Fa. < a j < ª 0,. I or if Žii. for I - `, wt n s Ý is1 wi t n and for all i s 1, . . . , I there exist sequences pi s pi n and qi s qi n such that 1 F pi - qi F n for all n, and, as n ª `,. piy1. Ý. Ž 2.7.. qi. n. wi2t n q. ts1. Ý. tsq iq1. wi2t n q. Ý. tsp i. 1r2 <^ wi t n < Ž styp q1 q 1 . i. q 1 . q sny1ª0, q < wi q i n < Ž sq1r2 i yp i q1 then Ž2.3. holds. PROOF. Ži. It is easily seen that max j < vjn < is bounded by the left-hand side Ž of 2.6.. Žii. We check Ž2.7. for an arbitrary i, dropping the i subscript in wi t n , pi , qi . We can easily account for the contribution from the summands for t - p and t ) q in vjn , and then by summation by parts,. Ž 2.8.. q. max j. qy1. Ý wt n a tyj. Ý. F. tsp. <^ wt n < max < Atyj < < < < qyj < pyj q wq n max A pyj ,. tsp. j. j. where Ats s Ýtjss a j . As in Ibragimov and Linnik Ž1971., we have the identity. žA / tyi py i. 2. ž. y Atq1yi pq1yi. /. 2. s 2 Atq1yi pq1yi Ž a pyi y a tq1yi . q Ž a pyi y a tq1yi . . 2. Summing over i s h q 1, . . . , j, we obtain. Ž 2.9.. ž. Atyj py j. /. 2. y. ž. Atyh pyh. /. 2. F 4stypq1. ž. `. Ý. isy`. a i2. /. 1r2. `. q4. Ý. a i2 ,. isy`. 2 .2 s Ý`y` Ž Atyi because stypq1 pyi . The last relation indicates that we can choose < Ž 1r2 h sufficiently negative, as a function of t y p, such that < Atyh py h - K stypq1 q 1., say, for all t y p G 0, where K is a generic positive constant. It follows < Ž 1r2 . Ž . from Ž2.9. that max j < Atyj pyj F K stypq1 q 1 , and then ii is established by reference to Ž2.8.. I. To illustrate the usefulness of both conditions Ži. and Žii. in case Assumption 1 is also imposed, take the simple case of least squares polynomial time series regression, mentioned in Section 1. We have wt n s f s Ž t .rn sq H , where f s is an sth degree polynomial. For H G 1r2 we easily check Ži.. For 0 - H - 1r2, we can check Ži. if we also assume a j s oŽ j Hy 1 ., as is readily.

(8) 2060. P. M. ROBINSON. seen. Although this does not entail absolute summability of the a j , it does, for example, imply g j s oŽ j 2 Hy1 ., and thus typically rule out the possibility that f Ž l. ; < l y v < 1y 2 H 9 as l ª v , for some v / 0 Žmod 2p ., unless H9 - H q 1r2; a spectrum can be zero at l s 0 but elsewhere unbounded. However, the polynomial structure of f s and Ž1.12. enables us to check Žii. when 0 - H 1r2 without any assumption on the a j besides the square summability in Ž2.1., which is merely equivalent to finite variance of u t . 3. Central limit theorem for nonparametric regression estimates. Now consider the estimate Ž1.2. for r Ž x . based on the model Ž1.1.. We impose first a condition on the kernel k. ASSUMPTION 3. Let k Ž v . be even, eventually monotone nonincreasing in < v <, differentiable with derivative k9Ž v ., satisfying Ž1.3. and. ž. k Ž v. s O Ž1 q v2 .. Ž 3.1.. y1. /,. ž. k9 Ž v . s O Ž 1 q < v < 1q h .. y1. /. for some h ) 0. It would be possible to establish Theorem 1 under somewhat milder conditions on k, whose strength decreases as H increases. However, we prefer the simpler condition above, which we motivate by a worker willing to contemplate an unknown H that is anywhere in Ž0, 1., and thus wishing to choose k accordingly. Kernels used in practice Žincluding typical higher-order kernels. are eventually monotonically decreasing. We have avoided compact support assumptions on k, imposing instead tail conditions. The differentiability condition does strictly exclude kernels such as the uniform, but such kernels, which are smooth almost everywhere, could be covered by a slight modification of our proofs. The following lemma estimates the covariance structure of ˆ r. LEMMA 3.. Let Ž1.1. and Assumptions 1 and 3 hold, and let. Ž 3.2.. Ž nb .. y1. q n1y2 H b 3y2 H ª 0. as n ª `.. Then for all x, y g Ž0, 1.. Ž 3.3.. Ž nb .. 2y 2 H. cov rˆŽ x . , rˆŽ y . 4 ª Gr Ž H . D x y. as n ª `,. where. ¡u Ž H . HH. ~. ` y`. k Ž v . k Ž w . y k Ž v . 4 < v y w < 2 Hy2 dv dw,. 2 ` r Ž H . s 2p Hy` k Ž v . dv,. ¢u Ž H . HH. ` y`. k Ž v . k Ž w . < v y w < 2 Hy2 dv dw,. 0 - H - 12 , H s 12 , 1 2. - H - 1..

(9) 2061. NONPARAMETRIC REGRESSION. PROOF. For future use we observe that Assumption 3 implies. Ý < k x t < a s O Ž nb . ,. Ž 3.4.. a G 1,. t. where k x t s k ŽŽ nx y t .rnb ., so that k effectively behaves like a compactly supported kernel. The left-hand side of Ž3.3. is. Ž 3.5.. Ž nb .. y2 H. Hyp f Ž l. kˆ p. x. Ž l . kˆ y Ž yl . d l ,. ˆ x Ž l. s Ý t k x t e i t l. The difference between Ž3.5. and G times where k Ž 3.6.. Ž nb .. y2 H. Hyph Ž l. kˆ p. x. Ž l . kˆ y Ž yl . d l. is bounded in absolute value, due to the triangle and Schwarz inequalities, by. Ž nb .. y2 H. max g Ž l . y G < l <- «. =. ½H. q2 Ž nb .. Ž 3.7.. q2 Ž nb .. Ž 3.8.. p. yp. ˆ x Ž l. h Ž l. k. y2 H. y2 H. G. ½. H« h Ž l. p. ½H. p. «. 2. Hyp p. dl. 2. ˆ x Ž l. k. ˆ x Ž l. f Ž l. k. 2. ˆ y Ž l. h Ž l. k. H« h Ž l.. dl. H«. p. dl. p. 2. dl. 5. ˆ y Ž l. k. ˆ y Ž l. f Ž l. k. 2. 1r2. 2. dl. dl. 5. 5. 1r2. 1r2. for « g Ž0, p ., where h is nonnegative because. Ž 3.9.. 2. `. Ý. f Ž l. s. a j e i jl. 2p. jsy`. is, and because of Assumption 1. Consider Ž3.8.. By summation by parts. Ž 3.10.. H«. p. ˆ x Ž l. f Ž l. k. 2. dl s. H«. p. f Ž l.. ny1. Ý. 2. ^ k x t Dt Ž l . q k x n Dn Ž l .. dl,. ts1. where Dt Ž l. s Ýtss1 e i s l. Because w see Zygmund Ž1977., page 51x. Ž 3.11.. Dt Ž l . F. 2. l. ,. 0 - l F p for all t G 1,. 1 < <. 2 q k 2x n 4 r« 2 . Choose it follows that Ž3.10. is bounded by 4g 0 ŽÝ ny ts1 ^ k x t M ) 0 such that k Ž v . is monotone nonincreasing in < v < for < v < G M. For n sufficiently large nx y nbM G 2 and nŽ1 y x . y nbM G 2, and then. w nxynbM x y1. Ý. ts1. ny1. <^ k x t < q. Ý. ts w nxqnbM x q1. <^ k x t < F 4 k Ž M . ..

(10) 2062. P. M. ROBINSON. On the other hand, bounded differentiability of k implies w nxqnbM x. Ý. ts w nxynbM x. <^ k x t < F KM.. It follows that Ž3.10. s O Ž «y2 . so the contribution of Ž3.8. is negligible because nb ª ` and H ) 0. The term Ž3.7. can be handled in the same way. Because « is arbitrary and g is continuous at l s 0, it clearly remains to be shown that. Ž 3.12.. Ž 3.6. ª r Ž H . D x y as n ª `,. because sup H r Ž H . - `. Now Ž3.6. is. Ž 3.13.. Ž nb .. y2 H. Ý Ý dtys k x t k y s . s. t. First assume x ) y, and put z s Ž x y y .r4. Write AŽ x . s t: 1 F t F n, < t y nx < F nz 4 and B Ž x . s t: 1 F t F n, < t y nx < ) nz 4 , and note that < t y nx < F nz and < s y ny < F nz implies < s y t < G 2 nz. Thus because of d j s O Ž j 2 Hy2 . and Ž3.4., 1. Ž nb .. 2H. Ý. Ý. tgA Ž x . sgA Ž y .. Kn2 Hy2. d sy t k x t k y s F. Ž nb .. 2H. Ý < k x t < Ý < k y s < s O Ž b 2y 2 H . ª 0. s. t. On the other hand, for 0 - H - 1r2, we use Ž1.9. to write 1. Ž nb . Ž 3.14.. 2H. s. Ý. Ý. tgA Ž x . sgB Ž y .. 1. Ž nb . y. Ý. 2H. sgB Ž y .. 1. Ž nb .. d sy t k x t k y s k ys. Ý. 2H. sgB Ž y .. Ý. tgA Ž x .. k ys k xs. d syt Ž k x t y k x s .. Ý. tgB Ž x .. d syt .. The first term on the right-hand side is bounded in absolute value by K. Ž nb .. 2 Hq1. Ý. sgB Ž y .. ž. nb ny y s. /. 2. Ý. < s y t < 2 Hy1 s O. < syt <Fn. ž. Ž nb . Ž nb .. 2. n2 H. 2 Hq1. n. s O Ž b 1y 2 H . . The second term is bounded by K. Ž nb .. 2H. Ý. sgB Ž y .. ž. nb ny y s. /. 2. `. < kxs<. Ý. tsy`. < d syt < s O Ž n1y 2 H b 3y2 H . ,. /.

(11) 2063. NONPARAMETRIC REGRESSION. using Ž3.4.. Thus in view of Ž3.2. it follows that Ž3.14. tends to 0. For 1r2 F H - 1, 1. Ž nb .. 2H. Ý. Ý. tgA Ž x . sgB Ž y .. d sy t k x t k y s F. K. Ž nb .. 2H. Ý. sgB Ž y .. ž. /Ý 2. nb ny y s. < d syt <. t. s O Ž b 2y 2 H . , which also tends to 0. Replacing the double summation on the left of Ž3.14. by Ý t g BŽ x . Ý s g AŽ y . gives the same result. Finally 1. Ž nb .. 2H. Ý. Ý. d sy t k x t k y s. Ý. ž. tgB Ž x . sgB Ž y .. K. F. Ž nb .. 2H. tgB Ž x .. nb ny y t. /. 2. Ý. sgB Ž y .. ž. nb ny y s. /. 2. < d sy t < .. For 0 - H - 12 , this is O Ž n1y 2 H b 4y2 H . ª 0, and for 12 F H - 1 it is O Ž b 4y2 H . ª 0. Thus the proof of Ž3.12. for x ) y, and thus x - y, is completed. Now assume x s y. First let 0 - H - 12 . In view of Ž1.9., Ž3.13. can then be written 1. Ž nb .. Ž 3.15.. 2H. Ý Ý dsy t k x t Ž k x s y k x t . s. t. y. 1. Ž nb .. 2H. ž. Ý k 2x t Ý dsyt q Ý dsyt t. sF0. s)n. /.. The second term is. ž. Ž 3.16.. O Ž nb .. y2 H. Ý t 2 Hy1 q Ž n y t q 1. 2 Hy1 4 k 2x t t. /. .. For M g Ž0, xr2 b . 1. Ž nb .. 2H. F. Ý t 2 Hy1 k 2x t t. w nbM x. 1. Ž nb .. 2H. max. 1FtFnbM. k 2x t. Ý. ts1. t 2 Hy1 q. Ž nbM . Ž nb .. 2 Hy1 2H. n. Ý. ts w nbM x q1. k 2x t .. By Ž3.4. the second term on the right is O Ž M 2 Hy1 . as n ª `, and can be made arbitrarily small on making M large. For any M the first term approaches 0 as n ª ` because Ž nx y nbM .rnb ª ` and k Ž v . ª 0 as v ª `. The other part of Ž3.16. can be treated in the same way, so that Ž3.16. is oŽ1.. For « ) 0 put d jX s u Ž H .< j < 2 Hy2 if 0 - < j < - nbr« , and d jX s d j otherwise..

(12) 2064. P. M. ROBINSON. Then the first part of Ž3.15. differs from 1. Ž 3.17.. Ž nb .. by O. ž. 2H. 1. Ž nb .. 2 Hq1. sO. ž. X k xtŽ k xs y k xt . Ý Ý dtys. s. t. Ý < k xt < t. Ý. 2H. z syt < s y t < 2 Hy1. 0- < tys <Fnbr «. w nbr « x. 1. Ž nb .. Ý. js1. /. /. z j j 2 Hy1 ª 0,. where z j ª 0 as j ª ` and we use the Toeplitz lemma. Then Ž3.17. differs ` f nŽ v, w . dv dw, where from r Ž H . by H Hy` f n Ž v, w . s. X d tys. Ž nb .. 2 Hy2. k x t Ž k x s y k x t . y u Ž H . < v y w < 2 Hy2 k Ž v . k Ž w . y k Ž v . 4. for nx y t y 1 nb. -vF. nx y t. and. nb. nx y s y 1 nb. -wF. and. nx y s. , nb s, t s 1, . . . , n,. f n Ž v, w . s yu Ž H . < v y w < 2 Hy2 k Ž v . k Ž w . y k Ž v . 4 for vF. xy1. wF. y. n xy1 b. 1. or. nb 1 y nb. v). or. x n. w). y x b. 1. or. nb y. 1 nb. .. For almost all v, lim. max. nª` t : 0F ŽŽ nxyt .rnb .yv- Ž1rnb .. k xt y k Ž v. s 0. from Assumption 3 and Ž3.2., while from Assumption 1, for all v / w, lim. max. nª` s, t : <ŽŽ syt .rnb .qwyv <F1r Ž nb .. Ž nb .. 2y 2 H. X d tys y u Ž H . < v y w < 2 Hy2 s 0,. so that, for all fixed M, f nŽ v, w . ª 0 as n ª ` for almost all v, w such that < v y w < F M. For all sufficiently large n, < f nŽ v, w .< F K < v y w < 2 Hy1 from Assumption 3, and so. HH< vyw <GM f. n. Ž v, w . dv dw F. K. Ž nb .. 2H. Ý < k xt < t. Ý. X < d syt <. < syt <)nbMr2. HH< vyw <)M k Ž v . < v y w <. qK. 2 Hy2. dv dw,.

(13) 2065. NONPARAMETRIC REGRESSION. and this is O Ž M 2 Hy1 . ª 0 as M ª `. The proof is completed for 0 - H - 12 . When H s 21 , Ž3.13. with x s y is 2p times 1. Ý k 2x t s H nb. `. y`. t. k Ž v . dv q 2. Hxrbk Ž v .. s. Ž nxyt .rnb. nxyty1 .rnb. t. `. y. Ý HŽ. Hy`k Ž v . `. 2. 2. dv y. dv q O. k 2x t y k Ž v . 2 4 dv. Hy`xy1 rbk Ž v . Ž. ž. 1 nb. .. q b3. 2. dv. /. by straightforward use of Assumption 3. For 12 - H - 1, the proof of Ž3.12. is omitted because it is similar to, and simpler than, that already given for 0 - H - 12 , and the same type of result has been obtained previously, albeit under somewhat different conditions w see Hall and Hart Ž1990.; Csorgo ¨ ˝ and Mielniczuk Ž1995b, c.x . I In case H G 12 , Ž3.2. entails only nb ª ` and b ª 0. To estimate the bias of ˆ r we impose the following. ASSUMPTION 4. Either r Ž x . satisfies a Lipschitz condition of degree t , 0 - t F 1, or r Ž x . is differentiable with derivative satisfying a Lipschitz condition of degree t y 1, 1 - t F 2. The following lemma is standard and the proof is omitted. LEMMA 4. Under Ž1.1. with Eu t s 0, t s 1, 2, . . . , and Assumptions 3 and 4, for all x g Ž0, 1. E ˆ r Ž x . y r Ž x .4 s O Ž bt ., s O Ž bt q ny1 .,. 0-tF1 1 - t F 2.. In order that the bias be small enough to permit centering at r Ž x . in the central limit theorem, we impose the following. ASSUMPTION 5. For the same t as in Assumption 4,. Ž 3.18.. Ž nb .. y1. q n1yH b1yHq t ª 0 as n ª `.. For given t , the strength of Assumption 5 decreases in H, so that a given b requires less smoothness in r when H ) 12 compared to the usual case H s 21 , but more when H - 21 . THEOREM 1. Let Ž1.1. and Assumptions 1]5 hold. Then for any distinct x i , i s 1, . . . , I, in Ž0, 1., the Ž nb .1y H ˆ r Ž x i . y r Ž x i .4 , i s 1, . . . , I, converge to independent N Ž0, Gr Ž H .. variates as n ª `..

(14) 2066. P. M. ROBINSON. PROOF. From Lemmas 3 and 4 and Assumption 5 we have that Ž nb .1y H E rˆŽ x . y r Ž x .4 ª 0 and Ž3.3. holds for all x, y g Ž0, 1., noting that for all t g Ž0, 2x , Ž3.18. implies Ž3.2.. By the Cramer]Wold device, it then remains ´ to show that for all constants h1 , . . . , h I that are not all zero,. Ž nb .. ½. yH. I. Ý. Gr Ž H .. h2i. is1. 5. y1 r2. ž. I. Ý Ý hi k x t t. i. is1. /. u t ªd N Ž 0, 1 . .. By Lemma 3, the left-hand side differs by op Ž1. from Ý t wt n u t , where wt n s Ž nb .. I. Ý hi k x t ,. yH y1 nn. i. is1. I where nn2 s Ž nb .y2 H V ŽÝ t Ý is1 h i k x i t 4 u t .. Then Ž2.2. holds and we need to check Ž2.3.. From Assumption 3, the left-hand side of Ž2.6. is, for 12 - H - 1,. O. ž½. I. 1. Ž nb .. 2H. nn2. Ý k 2x i t. Ý. is1. t. 5. 1r2. 1. q. H Ž nb . nn. / ž. Ž nb .. sO. 1r2yH. nn. /. from the proof of Lemma 3 and choosing a ' 1, say, whereas for H s 12 we can choose a such that ay1 q arnb ª 0 as n ª ` so that the left-hand side of Ž2.6. is o. ž½. 1. 5 / ž / 1r2. I. ÝÝ. nbnn2 is1. t. k 2x i t. a. qO. s o Ž nny1 . .. nbnn. For 0 - H - 12 we consider Žii. of Lemma 2. Note first that sn2 ; n2 H Gu Ž H .rH Ž2 H y 1. ª ` w see Ž1.12.x . For each i, introduce an increasing integer-valued sequence z i s z i n - minŽw nx i x , n y w nx i x. , and put pi s w nx i y z i x , qi s w nx i q z i x . Then piy1. Ý. wi2t n. F. ts1. F. Ž nb .. ž. Ý. h2 2H 2 i nn ts1. K Ž nb .. sO. ž. p iy1. K. 4y 2 H. nx i y t. /. 4. `. Ý. nn2. Ž nb .. nb. jy4. js w nx iyp i x 4y 2 H. nn2 z i3. /. ,. with the same bound for Ý ntsq iq1 wi2t n . It follows from Assumption 3 that qi. Ý. <^ wi t n < s O. tsp i. ž. yH. Ž nb . nn. /. .. For t F qi , styp iq1 s O Ž sq iyp iq1 . s O Ž z iH . as n ª `, so q. Ý <^ wi t n < Ž. tsp. 1r2 styp i q1. q 1. s O. ž. Ž z i1r2rnb . nn. H. /. ..

(15) 2067. NONPARAMETRIC REGRESSION. q1. sO ŽŽ z i1r2rnb . H rnn . also. Taking z i ; Ž nb .Ž4yH .r3 , Clearly < wi q i n <Ž sq1r2 i yp i q1 Ž Ž . say which is o n under Assumption 5., it follows that the left-hand side of Ž2.7. is oŽ nny1 .. Now Lemma 3 implies that nn ª Gr Ž H . and so application of Lemma 2, for any H, requires r Ž H . ) 0, because G ) 0 is assumed. Clearly r Ž 12 . ) 0 from Ž1.3.. It is not immediately obvious that r Ž H . ) 0 for 12 - H - 1 because we have not assumed that k is nonnegative. However, noting that for v ) 0, v by1 s. 2. p. G Ž b . cos. ž / bp 2. H0 l `. yb. cos Ž l v . d l ,. 0 - b - 1,. and replacing v by < v y w < and b by 2 H y 1, we have after rearrangement and use of the reflection formula for the gamma function. rŽ H. s. Hy` `. < l < 1y 2 H. Hy` `. 2. k Ž v . e i v l dv. dl,. 1 2. - H - 1;. this is positive because Ž1.3. and Assumption 3 implies that the Fourier transform of k is not almost everywhere zero. To prove that r Ž H . ) 0 for 0 - H - 12 , note that. rŽ H. s y. uŽ H. 2. HHy` k Ž v . y k Ž w . 4 `. 2. < v y w < 2 Hy2 dv dw,. 0-H-. 1 2. ;. this is positive because u Ž H . - 0 for 0 - H - 12 , and Ž1.3. and Assumption 3 imply that k is not almost everywhere constant. I A similar end result was achieved by Roussas, Tran and Ioannides Ž1992., Csorgo ¨ ˝ and Mielniczuk Ž1995a., and Tran, Roussas, Yakowitz and Truong Van Ž1996., in case H s 12 , and by Csorgo ¨ ˝ and Mielniczuk Ž1995b, c., Deo Ž1997. in case 21 - H - 1. We previously indicated how their assumptions differ from ours in respect of their global implications for f. In other respects, their conditions are sometimes weaker, sometimes stronger than ours; there is substantial scope for trade-offs between the assumptions on u t , r, k and b. We can partially extend Theorem 2.2 of Hall and Hart Ž1990. by assuming that r is twice continuously differentiable with second derivative r 0 and impose Ž3.2. rather than Ž3.18.: an easy extension of Lemma 4 gives E ˆ r Ž x . y r Ž x . 4 ; Ž nb . 2. 2 Hy2. Gr Ž H . q b 4 k 2 r 0 Ž x . r4, 2. where k 2 s H v 2 k Ž v . dv, and thence the ‘‘optimal bandwidth’’ b̂ s Ž 2 y 2 H . Gr Ž H . rk 2 r 0 Ž x .. 2 1r Ž6y2 H .. 4. nŽ Hy1.rŽ3yH . .. The optimal rate has exponent which tends to y1r3 as H x0, and increases in H to 0 as H 1. Ray and Tsay Ž1996. discuss a practical procedure for the choice of b. 4. Studentization. In practice G and H will be unknown and estimates will have to be inserted in the approximate variance formula implied by.

(16) 2068. P. M. ROBINSON. Theorem 1. In order to ensure that such studentization does not affect the limiting distribution we first present a lemma. LEMMA 5. Under Assumption 3, r Ž H . is continuous on Ž0, 1.. PROOF. Write r Ž H ., for H / 1r2, as. rŽ H. s uŽ H.. Hy`k Ž v . < v < `. 2 Hy2. dv,. where. k Ž v. s s. Hy` k Ž v q w . y k Ž w . 4 k Ž w . dw, `. Hy`k Ž v q w . k Ž w . dw, `. 1 2. 0 - H - 12 ,. - H - 1.. Clearly u Ž H . is continuous on Ž0, 1.. Evenness of k implies evenness of k . Boundedness and integrability of k imply boundedness of k . For v ) 0, Ž4.1. Ž4.2.. k Ž v . v 2 Hy2 F Kv 2 Hy2 vI Ž0 F v F 1. q I Ž v ) 1.4 ,. 0 - H - 12 ,. k Ž v . v 2 Hy2 F Kv 2 Hy2 I Ž0 F v F 1.. H kŽ v q w. kŽ w.. dwI Ž v ) 1.,. q. 1 2. - H - 1,. where bounded differentiability of k is used in Ž4.1. and the right-hand sides of Ž4.1. and Ž4.2. are both integrable in view of the respective values of H concerned and the integrability of k. Because v 2 Hy1 is continuous in H for v ) 0, the lemma is proved for H / 1r2 by dominated convergence. To prove continuity at H s 1r2, note first that for H / 1r2,. uŽ H.. Ž 4.3.. H0 k Ž v . v `. 2 Hy2. dv. k Ž v . v 2 Hy1. suŽH.. 2H y 1. `. y 0. uŽ H.. H k 9Ž v . v 2H y 1 0 `. 2 Hy1. dv,. ` k9Ž v q w . k Ž w . dw satisfies where k 9Ž v . s Hy`. k 9Ž v . F. Hy`. yvr2. k9 Ž v q w . k Ž w . dw q. ž. s O Ž1 q v2 .. y1. q Ž 1 q < v < 1q h .. Hyvr2 k9 Ž v q w . k Ž w .. y1. `. dw. /.. Thus < k 9Ž v . v j < is bounded by the integrable function v jrŽ1 q v 1q h ., when < j < F hr2, say. Thus by dominated convergence, as H ª 12 ,. H0 k 9 Ž v . v `. 2 Hy1. dv ª. H0 k 9 Ž v . dv s yHy`k `. `. 2. Ž v . dv..

(17) NONPARAMETRIC REGRESSION. 2069. Because cos p Ž 1 y H . 4 r Ž 2 H y 1 . ª pr2 as H ª 12 ,. Ž 4.4.. it follows that the last term of Ž4.3. tends to r Ž1r2.r2. The proof is completed on noting that the first term on the right-hand side of Ž4.3. is zero, using Ž4.4. and also Ž4.1. in case of H 1r2 and. k Ž v. F. Hyvr2 k Ž v q w . k Ž w . `. dw q. Hy`. yvr2. k Ž v q w . k Ž w . dw s O Ž vy2 .. as v ª ` in case H x1r2. I. ˆ Hˆ satisfying the following assumption. Now suppose we have estimates G, ASSUMPTION 6.. Ž 4.5.. ˆ ªp G, G. Ž log nb . Ž Hˆ y H . ªp 0 as n ª `.. The following theorem is a simple application of Theorem 1, Lemma 5 and ˆ y H <log nbŽexp< Hˆ Slutsky’s lemma, and the inequality <Ž nb . Ĥy H y 1 < F < H y H <log nb4. for nb ) 1. THEOREM 2. Let Ž1.1. and Assumptions 1]6 hold. Then for any distinct x i , ˆr Ž Hˆ .4y1 r2 Ž nb .1yHˆ ˆr Ž x i . y r Ž x i .4, i s 1, . . . , I, coni s 1, . . . , I, in Ž0, 1. the G verge to independent N Ž0, 1. variables as n ª `. We now discuss estimation of H and G. The mildness of Assumption 6 indicates little incentive for basing estimates on a full parametric model for f Ž l. across Žyp , p x , such as a fractionally integrated autoregressive moving average ŽFARIMA. model, which would typically lead to estimates that are 'n -consistent if the autoregressive and moving average orders are correctly specified w Fox and Taqqu Ž1986.x , but inconsistent otherwise. As the discussion of Section 1 suggests, it is more appropriate to base estimates on Assumption 1 or Ž1.7.. Several such estimates Žbased on observable u t . have been justified as less than 'n-consistent but to have convergence rates which Ž1987. seems, can satisfy Assumption 6. The estimate suggested by Kunsch ¨ on the basis of results of Robinson Ž1995b., to have desirable large sample properties. To adapt this, we suggest as proxies for the u t :. Ž 4.6.. ž. u ˆ t s yt y Ž nc .. y1. Ý l tys ys s. /. it ,. where i t s I Ž nc - t F n y w nc x. , l tys s l ŽŽ t y s .rnc ., l Ž v . is a kernel function w not necessarily identical to k Ž v .x satisfying. Ž 4.7.. l Ž v . s 0 for < v < ) 1,. Hy1 l Ž v . dv s 1, 1.

(18) 2070. P. M. ROBINSON. and c is a positive bandwidth number Žnot necessarily equal to b .. Without taking the u ˆ t to be zero close to the boundaries, and the compact support assumption on l , an additional term would arise in the proof of Theorem 3 below which rules out the existence of a suitable c sequence when 0 - H 1r5, and entails a narrower band of acceptable c’s for other H. The fact that problems can arise with standard kernel estimates near boundaries is familiar since the work of Rice Ž1984a.. For sequences pt , qt , define. Ž 4.8.. wp Ž l . s. 1. Ž 2p n .. 1r2. Ý pt e i t l ,. Ip q Ž l . s wp Ž l . wq Ž yl . .. t. For l j s 2p jrn and integer m g w 1, nr2., let G̃ Ž c . s. Ž 4.9.. 1 m. m. Ý l2j cy1 Iuˆ uˆ Ž l j . ,. js1. ˜Ž c . s log G˜Ž c . y Ž 2 c y 1 . R. m. Ý log l j ,. js1. ˆ and Hˆ as and define G Ž 4.10.. ˆ s G˜Ž Hˆ . , G. ˆ s arg min R˜Ž c . , H cg w D 1 , D 2 x. for 0 - D 1 - D 2 - 1. In particular, one can take D 1 s 1 y D 2 s « for some small positive « if the admissible region includes negative-dependent and short-range dependent H values, as well as long-range dependent ones. We introduce the following further assumptions. ASSUMPTION 7. For finite constants m 3 and m4 , E Ž « t3 < Fty1 . s m 3. a.s.;. E Ž « t4 < Fty1 . s m4 ,. t s 0, "1, . . . .. ASSUMPTION 8. In a neighborhood Ž0, d . of the origin, a Ž l. s Ý`jsy` a j e i j l is differentiable and Ž drd l. a Ž l. s O Ž< a Ž l.<rl. as l ª 0q. ASSUMPTION 9. For some b g Ž0, 2x ,. Ž 4.11.. f Ž l . ; Gl1y 2 H Ž 1 q O Ž l b . .. as l ª 0q,. where G ) 0 and H g w D 1 , D 2 x . ASSUMPTION 10. Satisfying Ž4.7., l is even and boundedly differentiable..

(19) 2071. NONPARAMETRIC REGRESSION. ASSUMPTION 11. As n ª `,. Ž log n .. 4. ½ž. m n. /. 4b. q. Ž log m . m. 2. 1. q. cm. qcq. c 4 n2y2 H m1y 2 H. q. m2 Hy1 c 2 n2 H. 5. ª 0.. Assumptions 7]9 are taken from Robinson Ž1995b., where we note that EŽ « t4 . s m4 is insufficiently assumed, instead of EŽ « t4 < Fty1 . s m4 . Both Assumptions 8 and 9 hold Žwith b s 2. in case of FARIMA processes; these latter also satisfy Assumption 1, which the earlier discussion partially related to Ž4.11.. THEOREM 3. Let Ž1.1. and Assumptions 2, 4 with t s 2, and 7]11 hold, ˆ and Gˆ be given by Ž4.6., Ž4.9. and Ž4.10.. Then Ž4.5. follows. and let H The proof of this theorem is extremely technical, and is relegated to the Appendix. To interpret the joint impact of Assumptions 5 and 11 when Theorems 2 and 3 are combined, suppose that b ; nyh , c ; nyz , m ; n r. Then Assumption 5 for t s 2 and Assumption 11 hold when 1yH b ) 0, - h - 1, z - r - 1, 3yH 1 2. Ž 1 y H . q 14 Ž 2 H y 1 . r - z - H y 12 Ž 2 H y 1 . r .. Because H is unknown, it is useful to deduce conditions that hold for all H g Ž0, 1.:. b ) 0,. 1 3. F h - 1,. 2 3. F r - 1,. 1 2. Ž 1 y 12 r . F z F 12 r ,. implying that z s 1r3 when r s 2r3 and we cannot choose z outside Ž1r4, 1r2. for any r g w 2r3, 1.. The residual computation Ž4.6. is rather heavy, and also requires choice of c and l . In the context of independent u t , Rice Ž1984b. indicated that the error variance can be estimated without computation of nonparametric residuals, but by a differencing of the raw data, while Muller and Stadtmuller ¨ ¨ Ž1987, 1988. considered extensions to more general models. Perhaps more surprisingly, we now show that when long-range dependence, H ) 1r2, can ˆ and Hˆ are be taken for granted, it is possible to satisfy Ž4.5. even when G computed from the raw data yt ; that is, we take. Ž 4.12.. u ˆ t s yt ,. 1 F t F n.. ASSUMPTION 12. For H G D 1 ) 1r2 and some d ) 0, as n ª `. Ž log n .. 4. ž / m n. 4b. q. n2y2 H m1y 2 maxŽ d , HyD 1 .. ª 0.. THEOREM 4. For H g Ž 12 , 1., let Ž1.1. and Assumptions 2, 4 with t s 2, ˆ and Gˆ be given by Ž4.9., Ž4.10. with D 1 ) 1r2, 7]9, and 12 hold, and let H and Ž4.12.. Then Ž4.5. follows..

(20) 2072. P. M. ROBINSON. Again the proof is left to the Appendix. Taking b ; nyh , m ; n r as before, we now require Ž2 y 2 H .r 1 y 2 maxŽ d , H y D 1 .4 - r - 1. While such r exist whenever 1r2 - D 1 F H - 1, the admissible set is very narrow when H is close to 1r2. 5. Empirical application to Nile data. Some of the earliest theoretical development of long-range dependence was prompted by empirical studies of the Nile River data, which has since routinely illustrated new methods of estimating H. These data consist of readings of annual minimum levels at the Roda gorge near Cairo, commencing in the year 622; often only the first 663 observations are employed because missing observations occur after the year 1284 w see Toussoun Ž1925.x . It was one of the hydrological series examined by Hurst Ž1951. which led to his recognition of the ‘‘Hurst effect’’ and invention of the RrS statistic. The series provides evidence of long periods of unusually high or low precipitation, named the ‘‘Joseph effect’’ by Mandelbrot and Wallis Ž1968., who argued that stationary long-range dependent models like Ž1.7. are appropriate for such data, which are notably cyclic but not periodic. Subsequently, estimates of H for the Nile data were obtained by various methods by such authors as Graf Ž1983., Beran Ž1992. and Robinson Ž1995b.. On the other hand, it has been suggested that the phenomena noted by these authors could also be symptomatic of forms of nonstationarity; see, for example, Klemes Ž1974., Bhattacharya, Gupta and Waymire Ž1983. and Teverovsky and Taqqu Ž1997., who considered the possibility of a time-varying mean, and Beran and Terrin Ž1996., who considered piecewise stationarity with H varying over time. Application of our present methods to the Nile series between 622 and 1284 provides evidence of nonstationarity in the mean and also an illustration of the consequences of Studentizing with an estimated H rather than by the conventional method that assumes H s 1r2, and of the desirability of allowing for the possibility of negative dependent, as well as long-range dependent, H. We computed ˆ r Ž x . Ž1.2. for x s ir30, 1 F i F 29, with kŽ v. s. 1 2. 1 q cos Ž p v . 4 ,. s 0,. < v < F 1,. < v < ) 1,. which satisfies Assumption 3 w note that k Ž v . is differentiable at < v < s 1, k9Ž v . tending to 0 as < v <1 and being zero for < v < ) 1x . In the Studentization we took l Ž v . s k Ž v ., so Assumption 10 holds also. We took b s c, considering each of the values 0.05, 0.075 and 0.1 w these would all result in estimates ˆr Ž ir30. that would be exactly independent across i if yt were independent across t, and not merely asymptotically independent as Theorem 1 impliesx . ˆ given by Ž4.10., we took m s 82 w one of the values used by Robinson In H Ž1995b. for the same data and estimatesx . We tried both choices Ž4.6. and Ž4.12. of u ˆ t and computed for x s ir30, i s 1, . . . , 29, the 100Ž1 y 2 a .%.

(21) 2073. NONPARAMETRIC REGRESSION. pointwise confidence intervals. ˆr Ž x . " za Gˆr Ž Hˆ . 4. Ž 5.1.. 1r2. Ž nb . Ĥy 1. and. ˜r Ž x . " za G˜Ž 12 . r Ž 12 . 4. Ž 5.2.. 1r2. Ž nb .. y1r2. ,. for a s 0.05 and 0.025, where P Ž Z ) za . s a for a standard normal variate Z. In case u ˆ t s yt , we have Hˆ s 0.905 w as in Robinson Ž1995b.x and Gˆ s 0.076. For brevity we display only the results for b s c s 0.05 with 90% intervals; see Figure 1. The solid line is the interpolated ˆ r Ž x ., the broken lines indicate Ž . the interpolated 5.1 and the dotted lines the interpolated Ž5.2.; we stress that the latter are not simultaneous bands; these are far wider, replacing z 0.05 s 1.645 by z 0.002 s 2.88. There is clearly evidence of changes in level, though use of the conventional H s 1r2 interval Ž5.2. substantially overstates their significance relative to Ž5.1.. In case u ˆ t are the modified residuals ˆ and Gˆ vary with c: for c s 0.1, Hˆ s 0.6928, Gˆ s 0.094; for c s 0.075, Ž4.6., H ˆ s 0.614, Gˆ s 0.114; for c s 0.05, Hˆ s 0.407, Gˆ s 0.169. These estimates H ˆ decreasing with c thus vary greatly over the range of smoothing employed, H ˆ occurs when c s 0.05. Figto the extent even that a negative dependent H ure 2 displays the results in the latter case Žwith b s 0.05 and a s 0.05 again.. The intervals are far narrower Žnote the difference in scale from Figure 1. and now the ones for Ž5.1. are narrower than those for Ž5.2.. The conclusions suggested by the other choices of b, c and a are qualitatively similar, though of course the larger b produce smoother ˆ r Ž x .. This study highlights the need for developing methods for choosing b and c which respond automatically to the strength of the dependence in u t . APPENDIX Proofs of Theorems 3 and 4. PROOF. OF. THEOREM 3.. Ži. Plan of proof. We show first that it suffices to prove that, as n ª `,. Ž A.1.. ž / . Ý ž /. my1. Ý. is1. Ž A.2.. Ž log n. 2. i. 2 Ž DyH .q1. m. my1 is1. i. m. 1 i. 1y 2 d. 2. 1 i2. i. Ý dj. js1 i. Ý dj. js1. ªp 0, ªp 0,.

(22) 2074. P. M. ROBINSON. FIG. 1. Nonparametric regression and 90% interval estimates for Nile data based on b s c s 0.05 and Ž4.12.: ˆ r Ž x . }; Ž5.1. - - - - -; Ž5.2. ?????.. Ž log n .. Ž A.3. Ž A.4.. m. m. Ý d j ªp. 0,. Ý Ž a j y 1. d j ªp. 0,. m 1. 2. js1. m. js1. where D s D 1 when H - 12 q D 1 and D g Ž H y 12 , H x otherwise; d is an 2H arbitrarily small positive number; d j s Iuˆ uˆŽ l j . y Iu u Ž l j .4 rg j , g j s Gl1y ; j 2Ž D y H . 2Ž D 1 y H . Ž . Ž . a j s jrh , 1 F j F h; a j s jrh , h - j F m, h s.

(23) NONPARAMETRIC REGRESSION. 2075. FIG. 2. Nonparametric regression and 90% interval estimates for Nile data based on b s c s 0.05 and Ž4.6.: ˆ r Ž x . }; Ž5.1. - - - - -; Ž5.2. ?????.. . expŽ my1 Ý m ˆ t s u t q vt , where vt s xt q u t q z t q j t and xt js1 log j . Write u s u t Ž i t y 1., u t s yŽ nc .y1 Ý s l tys u s i t , z t s Ž nc .y1 Ý s Ž rt y r s . l tys i t , j t s rt 1 y Ž nc .y1 Ý s l tys 4 i t , where rt s r Ž trn.. By elementary inequalities 1r2 Ž A.5. < Iuˆ uˆ y Iu u < F 2 Iu u Iv v 4 q Iv v ,. Iv v F 4 Ž Ix x q Iuu q Izz q Ijj .. Žsuppressing reference to the argument l., where here and below subscripted I and w are defined as in Ž4.8.. We shall then estimate Iv v Ž l j .rg j , and thence verify ŽA.1. ] ŽA.4...

(24) 2076. P. M. ROBINSON. Žii. Proof of sufficiency of ŽA.1. ] ŽA.4.. This makes heavy use of the proofs of Theorems 1 and 2 of Robinson Ž1995b., which in turn use results of Robinson Ž1995a.. We have. ˆ y G < F G˜Ž Hˆ . y G˜Ž H . q G˜Ž H . y GˆŽ H . q GˆŽ H . y G , <G ˆŽ H . s my1 Ýnjs1 l2j Hy1 Iu u Ž l j .. The last term on the right is op Ž1. where G from Robinson Ž1995b., the middle term is op Ž1. by the remainder of the current proof, and the first term is bounded by max. 1FjFm. HyHˆ . ˜Ž H . s Op Ž log n < Hˆ y H < . l2Ž y1 G j. w cf. the proof of Theorem 5 of Robinson Ž1994a.x , so we shall actually show ˆ y H . ªp 0. that Žlog n.Ž H By a standard type of argument for proving consistency of implicitly defined extremum estimates,. ˆy H< G «. F P P Ž log n < H. Ž A.6.. ž inf S˜Ž c . F 0 / , QlM. where M s c : log n < c y H < ) « 4 for « g Ž0, 12 ˜Ž H .. As in Robinson Ž1995b., define Q1 s c : R c : D 1 F c F D4 when H G 12 q D 1 , and to d g Ž «rlog n, 12 ., define Nd s c : < c y H < - d 4 , ŽA.6. is bounded by. Ž A.7.. P. ž. / ž. inf S˜Ž c . F 0 q P. Q 1 lNd. inf. Q1lNd lM. ˜Ž c . y log n., and S˜Ž c . s R 4 D F c F D 2 and also Q 2 s be empty otherwise. For Nd s Žy`, `. _ Nd , so that. /. ž. /. S˜Ž c . F 0 q P inf S˜Ž c . F 0 . Q2. ˆŽ c . s my1 Ýmjs1 l2j cy1 Iu u Ž l j . replaced Now S˜ is S of Robinson Ž1995b. with G ˜Ž c .. By the arguments in Theorem 1 of Robinson Ž1995b., the first two by G probabilities in ŽA.7. tend to 0 if sup. ˜Ž c . y G Ž c . G GŽ c .. Q1. q Ž log n .. 2. ˜Ž c . y G Ž c . G. sup. GŽ c .. Q1lNd. ªp 0,. 2Ž c y1. . By the triangle inequality this is implied if where GŽ c . s Ž Grm. Ý m js1 l j. Ž A.8.. sup. ˆŽ c . y G Ž c . G GŽ c .. Q1. Ž A.9.. sup. ˜Ž c . y GˆŽ c . G GŽ c .. Q1. q Ž log n .. 2. q Ž log n .. 2. ˆŽ c . y G Ž c . G. sup. GŽ c .. Q1lNd. ˜Ž c . y GˆŽ c . G. sup. GŽ c .. Q1lNd. ªp 0, ªp 0.. Now ŽA.8. is proved with only minor modifications in Robinson Ž1995b., whereas ŽA.9. is implied if. Ž A.10.. sup Q1. 1 m. m. Ý. js1. ž / j. m. 2 Ž c yH .. d j q Ž log n .. 2. sup Q1lNd. 1 m. m. Ý. js1. j 2Ž cyH . d j ªp 0..

(25) 2077. NONPARAMETRIC REGRESSION. By summation by parts and the inequalities 2Ž c y H . q 1 G 2Ž D y H . y 1 ) 0 and, for r ) 0, <Ž1 q 1ri . 2Ž cyH . y 1 < F 2ri on Q1 , it follows that ŽA.10. is implied by ŽA.1. ] ŽA.3.. For H F 12 q D 1 , ŽA.1. ] ŽA.3. suffice. For H ) 12 q D 1 , the last probability in ŽA.7. can be nonzero, but following the argument in Robinson Ž1995b., it is bounded by P. ž. m. 1 m. Ý Ž a j y 1.. js1. FP. ž. qP. ž. Iuˆ uˆ Ž l j . gj. ½. m. 1 m. ½. Ý Ž a j y 1.. js1. 1 m. y1. 5 / G1. Iu u Ž l j .. y1. gj. m. Ý Ž a j y 1. d j. G. js1. 1 2. /. 5 / G. 1 2. .. The first of the last two probabilities tends to zero by the proof in Robinson Ž1995b., so that ŽA.1. ] ŽA.4. suffice for H ) 12 q D 1. Žiii. Estimation of Iv v Ž l j .rg j . Bearing in mind the second inequality in ŽA.5., we first consider Iuu . We have EIuu Ž l . s Ž 2p n3 c 2 .. Hyp f Ž m . p. y1. where JlŽ m . s Ý t i t Ý s l tys e i t lyi s m . Now Jl Ž m . s. Ý i t Ý l tys e i t lyi sm s s. t. ½. Ý l t e i tm. < t <Fnc. 2. Jl Ž m .. 5½. dm ,. Ý. nc-tFny w nc x. i t e i tŽ ly m .. 5. by Assumption 10 and the definition of i t , so that. Ž A.11.. Jl Ž m . F Knc Dny2w n cx Ž l y m . ,. where DnŽ l. is defined below Ž3.10.. Also. Hyp p. Ž A.12.. Jl Ž m .. 2. d m s 2p Ý s. Ý i t l tys e i t l. 2. ,. t. which is bounded by 2p Ý s. ny1. 2. t. Ý i t Ž l tys y l tq1ys . Ý i v e. i vl. vs1. ts1. q l nys. Ý it e. itl. t. FK. n. l2. for 0 - l - p from Ž3.11., and also by Kn3 c 2 because l has compact support. For l g Ž0, p ., EIuu Ž l. is bounded by. Ž A.13.. 1. Hlr2. 3 l r2. 3 2. 2p n c. f Ž m . y f Ž l.. Jl Ž m .. 2. dm.

(26) 2078. P. M. ROBINSON. Ž A.14.. q. Ž A.15.. q. 1. ½H. p. 3 2. 2p n c f Ž l. 3 2. pn c. q. 3 l r2. Hyp p. 5. Hyp. lr2. Jl Ž m .. 2. f Ž m . Jl Ž m .. 2. dm. dm.. It follows from both bounds for ŽA.12. that ŽA.15. is O Ž f Ž l.min 1, Ž nc l.y2 4. as l ª 0q. We split ŽA.14. into several components. For all H g Ž0, 1. and sufficiently small l, there exists « g Ž3 lr2, p . such that f Ž m .rm1y H s O Ž f Ž l.rl1y H ., for l - m - « . Thus, as l ª 0q, and using Ž3.11. and ŽA.11., 1. Ž A.16.. H3lr2 «. n2 c 2. 2. f Ž m . Jl Ž m .. dm s O sO. Ž A.17. Ž A.18.. 1. H«. p. 2 2. n c 1. f Ž m . Jl Ž m .. Hylr2 f Ž m . lr2. 2 2. n c. 2. dm s O. Jl Ž m .. 2. ž. f Ž l.. l1yH. H3 lr2 Ž m y l.. ž Ž ./ f l. žH. l. p. žH. l. yl. 2. dm. /. ,. f Ž m . d mr Ž « y l .. yp. dm s O. m1y H. p. /. 2. /. s O Ž «y2 . ,. f Ž m . d mrl2 s O Ž f Ž l . rl . ,. y« yl r2 using Ž3.11., ŽA.11. and Assumption 9, while Hy and Hy « p are treated like ŽA.16. and ŽA.17., so that ŽA.14. s O Ž f Ž l.rn l.. Finally, Assumption 8 and Ž3.9. imply that. sup lr2F m F3 l r2. f Ž l. y f Ž m .. r< l y m < 4 s O Ž f Ž l . rl .. as l ª 0q, so that from Ž3.11. and ŽA.11., ŽA.13. is bounded by Kf Ž l . nl. Hlr2. 3 l r2. Dny 2w n cx Ž l y m . d m F. Kf Ž l . nl. H0. l r2. Dny2w n cx Ž m . d m. and this is uniformly O Ž g j Ž1 q log j .rj . as n ª ` when evaluated at l s l j for j s 1, . . . , m, from Lemma 5 of Robinson Ž1994b.. It follows that uniformly in j s 1, . . . , m,. Ž A.19.. ž ½ ž. Iuu Ž l j . s Op g j min 1,. 1 2 2. c j. /. q. 1 q log j j. 5/. as n ª `.. Using the same decomposition of Žyp , p ., and by similar arguments to some of those just used, we deduce that EIx x Ž l . s F. 1. H f Ž m. 2p n yp 2. p. H f Ž m. p n yp p. D w n cx Ž l y m . q Dn Ž l y m . y Dnyw n cx Ž l y m . Dw n cx Ž l y m .. 2. dm.. 2. dm.

(27) 2079. NONPARAMETRIC REGRESSION. Expanding the latter integral as in ŽA.13. ] ŽA.18. and proceeding similarly we find that, uniformly,. ž ½. Ix x Ž l j . s Op g j. Ž A.20.. 1 q log j. qc. j. 5/. .. To deal with Izz , we have w see Ž4.8.x wz Ž l . F. Kny3r2. Ý i t < rtX <. c. Ý s. t. ž. tys n. /. l tys q. Kny3r2 c. ÝÝ t. s. ž. tys n. /. 2. < l tys <. by a two-term Taylor expansion. The last term is trivially O Ž n1r2 c 2 .. From Assumption 10 the first term on the right is bounded by Kny3 r2. Ý Ý. c. s. t. F. Kc. ž. tys n. /l. tys. y nc 2. Ý Ý HŽ. Ž tq1ys .rnc. 1r2. n. 1. ½ž / tys nc. tys .rnc. s. t. Hy1v l Ž v . dv. 5. l tys y v l Ž v . dv. for n large enough. By the mean value theorem this is bounded by Kc 1r2. n. Ý t. ½. 1. Ž nc .. 1. 2. Ý < l tys < q nc H. 1. y1. s. 5. < v < dv s O Ž ny1 r2 . .. Thus Izz Ž l j . s O Ž nc 4 q ny1 .. Ž A.21. uniformly. Finally,. Ž A.22.. Ijj Ž l . F. K n. ½. Ý it t. Ý HŽ. Ž tq1ys .rnc. tys .rnc. s. 5 ž / 2. l tys y l Ž v . 4 dv. sO. 1. nc 2. From ŽA.19. ] ŽA.22. it follows that for j s 1, . . . , m,. Ž A.23.. Iv v Ž l j . gj. ž ž. s Op min 1,. /. 1 2 2. c j. q. Ž 1 q log j . j. qcq. nc 4 gj. q. 1 nc 2 g j. .. /. uniformly, as n ª `. Živ. Verification of ŽA.1.. By changing the order of summation, the left-hand side of ŽA.1. is bounded by. Ž A.24.. Km2Ž HyD .y1. m. Ý. j 2ŽDyH . < d j < for D - H ,. js1. and by. Ž A.25.. K. log m m. m. Ý < dj <. js1. for D s H..

(28) 2080. P. M. ROBINSON. Applying ŽA.5., ŽA.24. is bounded by Km. ½. 2Ž HyD .y1. Ž A.26.. m. Ý. j. 2ŽDyH .. Iu u Ž l j . gj. js1. m. Ý. j. 2ŽDyH .. Iv v Ž l j . gj. js1 m. Ý. q. Iv v Ž l j .. j 2ŽDyH .. gj. js1. Because 2Ž D y H . q 1 ) 0, m. Ý. Ž A.27.. Iu u Ž l j .. j 2ŽDyH .. .. s Op Ž m2ŽDyH .q1 .. gj. js1. 5. 1r2. by Ž3.16. of Robinson Ž1995b.. From ŽA.23., for s g w 1, m y 1x , m. Ý. js1 Ž1 .. Iv v Ž l j . gj. ž. s Op s q. m. 1 c. 2. Ý. q nc 4 q. ž. s Op. 1. 1 nc 2. /. m. log j. Ý. js1. j. n1y2 H. Ý. jssq1. ž. Ž A.28.. jy2 q. q cm. m. ž. j 2 Hy1. js1. q Ž log m . q cm q nc 4 q 2. /. 1. /. /. n1y2 H m2 H , nc 2 on choosing s ; cy1 . Thus ŽA.25. is Op Žlog mŽ z n q z n1r2 .. s op Ž1. by Assumption 11, where 1. zn s. c. q. Ž log m .. 2. cm m Proceeding similarly, for D g Ž H y 12 , H . m. ŽA.29.. Ý. j 2ŽDyH .. js1. ž. q c q nc 4 q. 1 nc 2. /ž / n. 1y2 H. .. m. Iv v Ž l j . gj. ž. ž. s Op c 2Ž HyD .y1 q1qcm2ŽDyH .q1 q nc 4 q. 1 nc 2. /. n1y2 H m2 D. so that ŽA.24. is Op Ž z n q z n1r2 . s op Ž1.. Žv. Verification of ŽA.2.. The left-hand side of ŽA.2. is bounded by m. Ý. K Ž log n . m2 dy1 2. jy2 d < d j <. js1. Ž A.30.. F K Ž log n . m2 dy1 2. =. ½. m. Ý. js1. y4 d. j. Iu u Ž l j . gj. m. Ý. js1. Iv v Ž l j . gj. 5. 1r2. m. q. Ý. js1. jy2 d. Iv v Ž l j . gj. .. /.

(29) 2081. NONPARAMETRIC REGRESSION. Applying ŽA.27. ] ŽA.29. with d g Ž0, minŽ 14 , H .., this is. ž. Op Ž log n .. 2. ž. 1. Ž log m .. q. mc. ž. 2. q c q nc 4 q. m. /ž / /. 1. n. nc 2. 1y2 H. 1r2. m. /. s op Ž 1.. by Assumption 11. Žvi. Verification of ŽA.3.. In view of the bound for ŽA.25., the left-hand side of ŽA.3. is clearly Op ŽŽlog n. 2 Ž z n q z n1r2 .. ªp 0 under Assumption 11. Žvii. Verification of ŽA.4.. From Robinson Ž1995b. we have h ; mre as n ª ` and using a j s O Ž1. uniformly for j ) h and Žvi. the left-hand side of ŽA.4. is, as n ª `, bounded by 1 m. m. Ý a j < d j < q op Ž 1. s Op. js1. ž. m. 1 m. Ý. 2ŽDyH .q1. /. j 2Ž DyH . < d j < q op Ž 1 . s op Ž 1 .. js1. by Ži.. I PROOF OF THEOREM 4. It suffices to check ŽA.1. ] ŽA.3. only, with d j defined as before but with Iuˆ uˆ defined in terms of Ž4.12.. The first part of ŽA.5. still holds but now vt s rt , and so by Assumption 4, Ž3.11. and ŽA.11., wv Ž l . F. ny1. K. Ý. 1r2. n. < vt y vtq1 < Dt Ž l . q. ts1. It follows that Iv v Ž l j .rg j s O Ž n. ž. Op m2Ž HyD .y1. ½. 2y 2 H. rj. m2ŽDyH .q1 n2y2 H. 3y2 H .. K n. j 2 Dy3. js1. 5. 1r2. ž. log m m. n. Ý. j 2 Dy3. js1. s Op Ž mŽ HyD 1 .y1r2 n1y H q m2Ž HyD 1 .y1 n 2y 2 H . s op Ž 1 . ,. ½. Ž mn2y2 H .. 1r2. q n2y2 H. 5/. l. m. q n2y2 H. under Assumption 12, and because D s D 1. Likewise ŽA.25. is Op. K 1r2. ,. 0-l-p. uniformly. Next, ŽA.26. is. m. Ý. < vn < Dn Ž l . F. 1r2. ž ½. n1y H. s Op log m. m1r2. q. n 2y 2 H m. 5/. /. s op Ž 1. .. Thus ŽA.1. is checked. To check ŽA.2., we see that ŽA.30. is, with 0 - d - 14 ,. ž. Op Ž log n . m2 dy1 m1y4d n 2y2 H 4 2. ž. s Op Ž log n .. 2. ž. n1y H m1r2. q. n 2y 2 H m1y2 d. under Assumption 12. Finally, ŽA.3. is. ž. Op Ž log n .. 2. ž. n1y H m1r2. q. 1r2. n 2y 2 H m. //. q n2y2 H. //. /. s op Ž 1.. s op Ž 1. .. I.

(30) 2082. P. M. ROBINSON. Acknowledgments. I am very grateful for two comprehensive referee reports which uncovered numerous defects and led to significant improvements in presentation, and for the comments of an Associate Editor. I thank Luis Gil-Alana for carrying out the computations described in Section 5. REFERENCES BERAN, J. Ž1992.. A goodness-of-fit test for time series with long-range dependence. J. Roy. Statist. Soc. Ser. B 54 749]760. BERAN, J. and TERRIN, N. Ž1996.. Testing for a change in the long-memory parameter. Biometrika 83 627]638. BHATTACHARYA, R. N., GUPTA, V. K. and WAYMIRE, E. Ž1983.. The Hurst effect under trends. J. Appl. Probab. 20 649]662. CSORGO ¨ ˝, S. and MIELNICZUK, J. Ž1995a.. Close short-range dependent sums and regression estimation. Acta Sci. Math. Ž Szeged . 60 177]196. CSORGO ¨ ˝, S. and MIELNICZUK, J. Ž1995b.. Nonparametric regression under long-range dependent normal errors. Ann. Statist. 23 1000]1014. CSORGO ¨ ˝, S. and MIELNICZUK, J. Ž1995c.. Distant long-range dependent sums and regression estimation. Stochastic Process. Appl. 59 143]155. DEO, R. S. Ž1997.. Nonparametric regression with long memory errors. J. Time Ser. Anal. To appear. EICKER, F. Ž1967.. Limit theorems for regressions with unequal and dependent errors. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 1 59]82. Univ. California Press, Berkeley. FOX, R. and TAQQU, M. S. Ž1986.. Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517]532. GRAF, H.-P. Ž1983.. Long-range correlations and estimation of the self-similarity parameter. Ph.D. dissertation, ETH Zurich. GRAY, H. L., ZHANG, N.-F. and WOODWARD, W. A. Ž1989.. On generalized fractional processes. J. Time Ser. Anal. 10 233]257. HALL, P. G. and HART, J. D. Ž1990.. Nonparametric regression with long-range dependence. Stochastic Process. Appl. 36 339]351. HANNAN, E. J. Ž1979.. The central limit theorem for time series regression. Stochastic Process. Appl. 9 281]289. HURST, H. E. Ž1951.. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineeering 16 770]799. IBRAGIMOV, I. A. and LINNIK, Y. V. Ž1971.. Independent and Stationary Sequences of Random Variables. Walters-Noordhoff, Groningen. ¨ KLEMES, V. Ž1974.. The Hurst phenomenon: a puzzle? Water Resources Research 10 678]688. KUNSCH , H. R. Ž1987.. Statistical aspects of self-similar processes. In Proc. First World Congress ¨ of the Bernoulli Society 67]74. VNU Science Press. MANDELBROT, B. B. and WALLIS, J. R. Ž1968.. Noah, Joseph and operational hydrology. Water Resources Research 4 909]918. MCLEISH, D. L. Ž1974.. Dependent central limit theorems and invariance principles. Ann. Probab. 2 620]628. M¨ ULLER, H.-G. and STADMULLER , U. Ž1987.. Estimation of heteroscedasticity in regression analy¨ sis. Ann. Statist. 15 610]625. M¨ ULLER, H.-G. and STADMULLER , U. Ž1988.. Detecting dependencies in smooth regression models. ¨ Biometrika 75 639]650. RAY, B. and TSAY, R. Ž1966.. Bandwidth selection for nonparametric regression with long range dependent errors. In Athens Conference in Applied Probability and Time Series Analysis ŽP. M. Robinson and M. Rosenblatt, eds.. 2 339]351. Springer, New York. RICE, J. A. Ž1984a.. Boundary modification for kernel regression. Comm. Statist. Theory Methods 13 893]900..

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