Estimating the Density of the Residualsin Autoregressive ModelsEckhard LiebscherTechnical University Ilmenau,Institute of Mathematics,D-98684 Ilmenau/Thür.GermanySeptember 23, 1999AbstractWe consider a (nonlinear) autoregressive model with unknown parameters(vector �). The aim is to estimate the density of the residuals by a kernel esti-mator. Since the residuals are not observed, the usual procedure for estimatingthe density of the residuals is the following: �rst, compute an estimator � for�; second, calculate the residuals by use of the estimated model; and third, cal-culate the kernel density estimator by use of these residuals. We show that theresulting density estimator is strong consistent at the best possible convergencerate. Moreover, we prove asymptotic normality of the estimator.Mathematics Subject Classi�cation (1991): 62G07Key words: autoregressive models, density estimators, residuals

1. IntroductionLet fXk; k = 1; 2 : : :g be a strictly stationary sequence of real random variablesfollowing the modelXk = g (Xk�1; : : : ;Xk�p j �) + Zk for k = p+ 1; p + 2 : : : (1.1)where fZkg is a sequence of i.i.d. random variables (r.v.) and the autoregressionfunction g : Rp � ��! R is a measurable (linear or nonlinear) function. Here � =(�1; : : : ; �q)> 2 � � Rq is the vector of parameters of the autoregressive model.Assume that the density f of the residuals Zi exists and is bounded. We wantto estimate the density f by means of the data X1; : : : ;Xn. The main problem isthat we do not observe the residuals Zi. So the usual procedure for nonparametricestimation of the density f is the following: �rst, compute an estimator � for �; second,estimate the residuals by use of the �tted model; and third, calculate the kerneldensity estimator by use of these residuals. The goal of this paper is to show that thisprocedure leads to consistent kernel estimators for f and the estimator has the bestpossible convergence rate (w.r.t. strong convergence). Moreover, we provide a resulton asymptotic normality of the density estimator (see Section 2). The proofs of theresults of this paper are found in Section 3. It should be pointed out that we obtainthe same results as we would get for kernel estimators by using the sample Z1; : : : ; Zn.A similar approach is described in the paper by Kreiÿ (1991) where the estimationof the distribution function of the residuals in an ARMA-model is examined. Weshould remark that extending the results of this paper to the multivariate case andto �-mixing sequences fZkg (cf. Liebscher, 1996) is straightforward.There is a wide range of literature concerning kernel density estimators for sam-ples of i.i.d. random variables. Among the many papers about strong convergenceof these estimators, we should mention the paper by Stute (1982) where a law oflogarithm is derived. Asymptotic normality is already proved in Parzen (1962).The monograph by Tong (1990) represents an excellent account of nonlineartime series models of type (1.1). Parametric estimation in such models was studiedby Klimko and Nelson (1978), Hall and Heyde (1980) and Tjøstheim (1986). Theseauthors derive LIL-type results about estimators for � in the model (1.1). We utilizesuch results in the proofs in an essential way. Nonparametric estimation of function1

g is studied in papers by Masry and Tjøstheim (1995) and by the author (1998a).Under mild additional assumptions, the stationary sequence fXtg of model (1.1)is absolutely regular (and also �-mixing) with exponentially decaying mixing coe�-cients (see also Section 2). If one is interested in estimating the density of Xt, thenwe refer to the excellent monograph by Bosq (1996), and to papers by Ango Nzeand Rios (1995) and by the author (1996, 1998a,b), where strong convergence andconvergence in distribution of kernel estimators are studied.2. Main resultsSuppose that (�n1; : : : ; �nq)> = (�1; : : : ; �q)> = � is a strong consistent estimator for� which ful�lls a law of iterated logarithm. This law implieslimsupn!1 r nln ln(n) j�nj � �jj � C1 a:s: (j = 1; : : : ; q) (2.1)with an appropriate constant C1 > 0. Such a result holds true for least squareestimators under certain regularity conditions (cf. Klimko and Nelson, 1978). Asimilar result holds true for other types of estimators, for example the maximum-likelihood one (cf. Hall and Heyde, 1980).On the basis of the estimator �, we calculate the estimated residualsZk = Xk � g �Xk�1; : : : ;Xk�p j �� (k � p+ 1): (2.2)Zp+1; Zp+2; : : : ; Zn is now the sample for estimating the density f by the kernel esti-mator fn de�ned byfn(x) = n�1b(n)�1 nXi=p+1K �(x� Zi)=b(n)� (x 2 R):In this paper we study properties of fn(x) with bandwidth b(n) and kernel functionK : R! R. Suppose that the bandwidth b(n) satis�eslimn!1 b(n) = 0; b(n) � const �n�1=5: (2.3)We assume that, for some even r � 2, the kernel function ful�lls the following condi-tion: 2

Condition K(r). K is symmetric and vanishes outside the interval [�1; 1].Suppose that the second derivative K 00 exists on [�1; 1]. Moreover, assume thatC2 := sup�1�t�1 jK 00(t)j < +1; 1Z�1K(t) dt = 1;1Z�1 tkK(t) dt = 0 for all even k; 0 < k < r.Here K(j)(1) = K(j)(�1) := limh!0�0[K(j�1)(1 + h)�K(j�1)(1)]=h (j = 1; 2). 2The Epanechnikov kernel satis�es condition K(2). Now we introduce some as-sumptions on the model (1.1).Condition M. Let U � � be a neighbourhood of �. We assume that, for ally 2 Rp, �� 2 U , j; k = 1; : : : ; q,���� @@�j g �yj������� �M(y) and ���� @@�j@�k g �yj������� � �M (y)where EM (Xk�1; : : : ;Xk�p) < +1 with some > 2:5, E �M (Xk�1; : : : ;Xk�p) < +1.� Condition G. The function g is nonperiodic and bounded on compact sets. Zthas a density function that is positive on R. Moreoverg(y) = a>y + o(kyk) as kyk !1 (a; y 2 Rq), A = (a; e1; e2; : : : ; eq�1) 2 Rq;q,ei is the ith unit vector of Rq and the spectral radius � of A satis�es �(A) < 1. �Condition G is Assumption 3.2 of Masry and Tjøstheim (1995) and ensures thatthe sequence fXtg is geometrically ergodic in its Markovian representation. Then thestationary sequence fXtg is absolutely regular with mixing coe�cients �k = O(�k)(0 < � < 1). The monograph by Doukhan (1994) includes some further discussionson mixing sequences. Now we formulate the main statements.Theorem 2.1. Suppose that, for some even integer r � 2, the derivative f (r)exits and is continuous on R. Then, under the assumptions K(r), M, G, we havesupx2D ���fn(x)� f(x)��� = O �pln(n) n�1=2b(n)�1=2 + br(n)� a:s: (2.4)3

for any compact set D � R. The optimized rate reads as followssupx2D ���fn(x)� f(x)��� = O �(ln(n)=n)r=(2r+1)� a:s:provided that b(n) = const �(ln(n)=n)1=(2r+1): (2.5)Assumption (2.3) is not a strong restriction because it is ful�lled for optimalbandwidths (2.5) (r � 2) in Theorem 2.1. By means of Theorem 3.1, we can applyTheorem 1.3 of Stute (1982) and obtain the following result:Theorem 2.2. Let f be uniformly continuous on R and Lipschitz continuouson every compact set D � R. Then, under the assumptions K(2), M, G, we havelimn!1s nb(n)2 ln(b(n)�1) supx2D ���fn(x)� E fn(x)���pf(x) =sZ 1�1K2(t) dt a:s:for any compact set D � R with 0 < m � f(x) �M <1 8x 2 D.The following two results concerning asymptotic normality may be applied toconstruct con�dence intervals for f .Theorem 2.3. Assume that f is Lipschitz continuous in a neighbourhood ofx 2 R and f(x) > 0. Then, under the assumptions K(r), M, G, we havepnb(n) �fn(x)� E fn(x)� D�! N (0; �2)where �2 := f(x) R 1�1K2(t) dt.Corollary 2.4. Assume that b(n) = C3 � n�1=(2r+1) (C3 is a positive constant)and f (r) exists and is continuous at x 2 R for some even r � 2. Suppose thatf(x) > 0. Then, under the assumptions K(r), M, G, we havepnb(n)�fn(x)� f(x)� D�! N (�; �2)with � := Cr+1=23 (r!)�1 R 1�1 trK(t) dt � f (r)(x), �2 as above.4

3. ProofsAssume that fXkg follows the model (1.1) and f(Xk; Zk); k = p + 1; p + 2; :::g is anabsolutely regular sequence with �k = O(�k); 0 < � < 1. Moreover, suppose that (2.1)and (2.3) are ful�lled. Let D be any compact set, and let �D be an "-neighbourhoodof D.Theorem 3.1. Suppose that the conditions M and K(2) are satis�ed. Let f beLipschitz continuous on �D. Thensupx2D ���fn(x)� ~fn(x)��� = o �n�1=2b(n)�1=2� a:s:where ~fn(x) := n�1b(n)�1 nXi=p+1K ((x� Zi)=b(n)) (x 2 R):For the proof of this theorem, we need a series of auxiliary statements. By (2.2),we haveZk = Zk + g(Xk�1; : : : ;Xk�p j �)� g(Xk�1; : : : ;Xk�p j �) (k � p+ 1):By Taylor's expansion, we obtainZk � Zk = �(�1 � �1)Yk1 � : : :� (�q � �q)Ykq; Ykj = g�j(Xk�1; : : : ;Xk�p j ��)with �� = (��1; : : : ; ��q)> and an appropriate (random) 2 (0; 1); �� = � + (� � �).Here and in the sequel, g�j stands for @@�j g, and g�j�k stands for @2@�j@�k g. Let�Zk : = Zk � (�1 � �1)Uk1I (jUk1j � sn)� : : :� (�q � �q)UkqI (jUkqj � sn) ;Ukj : = g�j (Xk�1; : : : ;Xk�p j �); sn := pn(ln lnn)�1��b(n)3=2 (3.1)with some � > 0. Consequently,Zk � �Zk = � qXi=1 (�i � �i) (Yki � Uki + UkiI (jUkij � sn)) : (3.2)By (2.1), we havejZi � �Zij � q � maxj=1;:::;q j�j � �j j � sn � C4 �r ln ln(n)n sn =: �n a:s: for n � n0(!) (3.3)5

with C4 = qC1 + 1. Hence �n = C4 � (ln lnn)�1=2��b(n)3=2:Let f�ng be a sequence of positive real numbers such that �n � n�2. Since Dis compact, it can be covered with intervals �1; : : : ;�� with length �n and centresu1; : : : ; u�, resp., where � � const �n2. De�neGn(x) :=8<: K 0(x) for jxj < 1 � "n;0 otherwise, "n := �n=b(n):Now we obtainsupx2D ���fn(x)� ~fn(x)��� � maxk=1;:::;� ���fn(uk)� ~fn(uk)���++ maxk=1;:::;� supx2�k ����fn(x)� fn(uk)���+ ��� ~fn(x)� ~fn(uk)���� : (3.4)Under condition K(2), we havejK(v)�K(w)j � C5 � jv �wj for v;w 2 R: (3.5)with a positive constant C5. Thenmaxk=1;:::;� ���fn(uk)� ~fn(uk)���� maxk=1;:::;� �����n�1b(n)�2 nXi=p+1 �Zi � �Zi�Gn ((uk � Zi)=b(n))�����++C5 � n�1b(n)�2 � maxk=1;:::;� nXi=p+1 jZi � �Zij I (b(n)� �n < juk � Zij � b(n) + �n) ++12C2n�1b(n)�3 � maxk=1;:::;� nXi=p+1 �Zi � �Zi�2 I (juk � Zij � b(n)� �n) ++C5 � n�1b(n)�2 nXi=p+1 j �Zi � Zij= A1n +A2n +A3n +Bn: (3.6)Further, by (2.1),A1n � n�1b(n)�2 qXj=1 ����j � �j��� maxk=1;:::;� ����� nXi=p+1 �Vnijk + �Vnijk������� C1pln lnn n�3=2b(n)�2 qXj=1 maxk=1;:::;� ����� nXi=p+1 �Vnijk + �Vnijk������ (3.7)where 6

Vnijk : = UijI (jUijj � sn) � (Gn ((uk � Zi)=b(n))� EGn ((uk � Zi)=b(n))) ;�Vnijk : = UijI (jUijj � sn) � EGn ((uk � Zi)=b(n)) (i = p + 1; : : : ; n; j = 1; : : : ; q):Next we need two general auxiliary lemmas. Subsequently, we establish convergencerates for A1n, A2n und A3n.Lemma 3.1. Let f�kg be a stationary �-mixing sequence of random variableswith mixing coe�cients f��k; k = 1; 2 : : :g. Moreover, assume that E� k = 0; D 2�k =�2 < +1 and j�kj � S a:s: Then, for every N � n=2; N 2 N, for every " > 4NS,P(����� nXk=1 �k����� > ") � 4 exp[�"2 �64�2nN + 3"NS��1] + 4 nN ��N :Proof. By D 2 NXi=1 �k! � N2�2;we can apply Theorem 2.1 of the author's (1996) paper. This leads to the lemma. 2Lemma 3.2. For n 2 N, let Kn be a bounded function with Kn(t) = 0 8t; jtj> 2,supjtj�1 jKn(t)j � �K <1 8n 2 N. Then, under condition M,maxk=1;:::;� ����� nXi=p+1 ~Unijk����� = O(pnb(n) ln(n)) a:s: andmaxk=1;:::;� ����� nXi=p+1 �Unijk����� = O(pnb(n) ln(n)) a:s: (j = 1; : : : ; q)with~Unijk : = UijI (jUijj � sn) (Kn ((uk � Zi) =b(n))� EKn ((uk � Zi) =b(n))) ;�Unijk : = jUijjI (jUij j � sn) (Kn ((uk � Zi) =b(n)) � EKn ((uk � Zi) =b(n))) :Proof. Note thatmaxi;j;k E ~U2nijk � EM 2(Xi�1; : : : ;Xi�p) maxk=1;:::;� EK2n ((uk � Zi) =b(n))� const � supx2RPfjZi � xj < b(n)g = O(b(n)):maxi;j;k denotes the maximum over i = p+ 1; : : : ; n; j = 1; : : : ; q; k = 1; : : : ; �. More-over, by de�nition, maxi;j;k ��� ~Unijk��� � 2sn � �K a:s:7

Let an :=pnb(n) ln(n)+ ln2(n)sn. Obviously, f(Zk;Xk�1; : : : ;Xk�p)g is a stationaryabsolutely regular sequence with mixing coe�cients f�Zk g which decay exponentially�Zk = O(�k). Hence, for �xed j; k; n, f ~Unijk; i = p + 1; p + 2 : : :g is a stationaryabsolutely regular sequence with mixing coe�cients f�Uk g and �Uk � �Zk . ApplyingLemma 3.1 with N := d5j ln(�)j�1 ln(n)e, we getP( maxk=1;:::;� ����� nXi=p+1 ~Unijk����� > "an)� �Xk=1 P(����� nXi=p+1 ~Unijk����� > "an)� C6� �exp[�C7"2a2n (n ln(n)b(n) + "an ln(n)sn)�1] + n�4� C8 exp[�C9"2 ln(n) (1 + ")�1]for " > 1 where C6 to C9 > 0 are constants not depending on n; ";N . An applicationof the Borel-Cantelli lemma leads tomaxk=1;:::;� ����� nXi=p+1 ~Unijk����� = O(an) a:s:Thus we get the lemma in view of (3.1). 2Lemma 3.3. Let f be Lipschitz continuous on �D. Suppose that condition K(2)is satis�ed. Thensupx2D jEGn ((x� Zi)=b(n))j = o �(ln lnn)�1=2b(n)3=2� a:s:Proof. Here we obtainsupx2D jEGn ((x� Zi)=b(n))j� b(n)�supx2DZ 1�1 jK 0(t)j jf(x� tb(n))� f(x)jdt+O("n)�= b(n) �O (b(n) + "n)since R 1�1K 0(t) dt = 0. 2Lemma 3.4. Let the condition M be ful�lled. Under the assumptions of Lemma3.3, we obtain A1n = o �n�1=2b(n)�1=2� a:s:8

Proof. By the law of large numbers for strictly stationary ergodic sequences(i.e. the ergodic theorem, see Hall and Heyde (1980), p. 281, for example) andLemma 3.3, we obtainmaxk=1;:::;� ����� nXi=p+1 �Vnijk����� = o �n(ln lnn)�1=2b(n)3=2� n�1 nXi=p+1M(Xi�1; : : : ;Xi�p)= o �n(ln lnn)�1=2b(n)3=2� a:s:(j = 1; : : : ; q). Hence Lemma 3.4 is a consequence of (3.7) and Lemma 3.2. 2Lemma 3.5. Under the conditions M and K(2), we haveA2n = o �n�1=2b(n)�1=2� a:s:Proof. Observe thatA2n � n�1b�2(n) qXj=1 ����j � �j��� maxk=1;:::;� nXi=p+1 �V �nijk + �V �nijk�� C1pln lnn n�3=2b(n)�2 qXj=1 maxk=1;:::;� nXi=p+1 �V �nijk + �V �nijk� (3.8)whereV �nijk : = jUijj I (jUijj � sn) � (K�n ((uk � Zi)=b(n))� EK�n ((uk � Zi)=b(n))) ;�V �nijk : = jUijj I (jUijj � sn) � EK�n ((uk � Zi)=b(n)) ;K�n(t) = 1 for t 2 (1 � "n; 1 + "n] [ [�1 � "n;�1 + "n), K�n(t) = 0 otherwise. Anapplication of Lemma 3.2 leads tomaxk=1;:::;� ����� nXi=p+1V �nijk����� = O �pnb(n) ln(n)� a:s: (j = 1; : : : ; q): (3.9)Furthersupx2REK�n ((x� Zi)=b(n)) = supx2RPfb(n)(1� "n) < jx� Zij � b(n)(1 + "n)g= O("nb(n)) = O(�n):Applying the law of large numbers, we concludemaxk=1;:::;� ����� nXi=p+1 �V �nijk����� = O(n�n) n�1 nXi=p+1M(Xi�1; : : : ;Xi�p)= o �n(ln lnn)�1=2b(n)3=2� a:s: (3.10)9

(3.8)-(3.10) imply the lemma. 2Lemma 3.6. Assume that conditions M and K(2) are satis�ed. Then we haveA3n = o �n�1=2b(n)�1=2� a:s:Proof. By Cauchy's inequality, we deduceA3n � const �n�1b(n)�3sn qXj=1 ��j � �j�2maxk;j nXi=p+1 �Wnijk + �Wnijk�= o �n�3=2b(n)�3=2� maxk;j nXi=p+1 �Wnijk + �Wnijk� a:s: (3.11)(maxk;j denotes the maximum over k = 1; : : : ; �; j = 1; : : : ; q) whereWnijk : = jUijjI (jUij j � sn) (I (juk � Zij < b(n))� EI (juk � Zij < b(n))) ;�Wnijk : = jUijjI(jUijj < sn) EI (juk � Zij < b(n)) :From Lemma 3.2, it follows thatmaxk=1;:::;� ����� nXi=p+1Wnijk����� = O �pnb(n) ln(n)� a:s: (3.12)Moreover, by the law of large numbers,maxk=1;:::;� ����� nXi=p+1 �Wnijk����� = O(b(n)) � nXi=p+1M(Xi�1; : : : ;Xi�p) = O(nb(n)) a:s: (3.13)Lemma 3.6 is now a consequence of (3.11)-(3.13). 2Lemma 3.7. Suppose that condition M is satis�ed. Thenn�1 nXi=1 jUikj � I (jUikj > sn) = O �n�1=2 ln(n) + s� +1n � a:s: (k = 1; : : : ; q):Proof. Let k 2 f1; : : : ; qg be �xed. Since fUik; i = 1; 2 : : :g is a �-mixing se-quence with exponentially decaying mixing coe�cients, the law of iterated logarithmholds true (cf. Rio (1995), comment 3 on p. 1191). Therefore we haven�1 nXi=1 (jUikj � E jUik j) = O r ln ln(n)n ! a:s:10

Theorem 3.2 of Liebscher (1996) impliesn�1 nXi=1 (jUikj � I (jUikj � sn)� EjUik j � I (jUikj � sn))= O �n�1=2 ln(n) + n�1 ln2(n)sn� = O �n�1=2 ln(n)� a:s:Observe that E jUik j � I (jUikj > sn) � s� +1n E jM(Xi�1; : : : ;Xi�p)j which completes the proof. 2In the sequel, we derive convergence rates for Bn and prove Theorem 3.1.Lemma 3.8. Under conditions M and K(2), Bn = o �n�1=2b(n)�1=2� a:s.Proof. By (3.2), we obtainjZi � �Zij � qXj=1 j�j � �j j (jUijjI (jUij j > sn) + jUij � Yijj)� qXj=1 j�j � �j j jUijjI (jUijj > sn) + qXk=1 �M (Xi�1; : : : ;Xi�p)j�k � �kj! :Hence, by the law of large numbers and Lemma 3.7,Bn = O �pln lnn n�3=2b(n)�2�� maxj=1;:::;q nXi=p+1 jUijjI (jUijj > sn) + C1r ln ln(n)n nXi=p+1 �M(Xi�1; : : : ;Xi�p)!= n�1=2b(n)�1=2 �O �ln2(n)n�1=2b(n)�3=2 +pln lnn b(n)�3=2s� +1n � a:s:This proves the lemma since pln lnn s� +1n b(n)�3=2 = o(1) for > 2:5. 2Proof of Theorem 3.1. By the Lipschitz continuity (3.5), we deducemaxk=1;:::;� supx2�k ���fn(x)� fn(uk)���� n�1b(n)�1 nXi=p+1 ���K �(x� Zi)=b(n)��K �(uk � Zi)=b(n)����� C5b(n)�2 supx2�k jx� ukj � b(n)�2�n = O(n�1):An analogous inequality holds true with fn replaced by ~fn. Theorem 3.1 is now aconsequence of (3.4) and Lemmas 3.4-3.8. 211

Now we proceed to prove the main results.Proof of Theorem 2.1. A standard argument (cf. Földes and Revesz (1974),for example) leads tosupx2D ��� ~fn(x)� f(x)��� = O �plnn n�1=2b(n)�1=2� a:s:for any compact set D. Moreover, a Taylor expansion yieldssupx2D ���E ~fn(x)� f(x)��� = O(br(n)):Since, under Condition G, the stationary sequence fXtg is absolutely regular with�k = O(�k) (0 < � < 1) and Zt is a measurable function of Xt�p;Xt�p+1 : : :Xt, weobtain that f(Xk; Zk); k = p + 1; p + 2; :::g is an absolutely regular sequence withexponentially decaying mixing coe�cients. Theorem 2.1 follows now from Theorem3.1. 2Proof of Theorem 2.3. Let D be a closed neighbourhood of x. Apply thestatement about asymptotic normality in the paper by Parzen (1962) and Theorem1 in Liebscher (1990) to get asymptotic normality of ~fn(x) � E ~fn(x). Theorem 3.1completes the proof. 24. AcknowledgementThe author wishes to thank the anonymous referee for his valuable hints.5. ReferencesP. Ango Nze and R. Rios (1995) Estimation L1 de la fonction de densité d'un proces-sus faiblement dépendant: les cas absolument régulier et fortement mélangeant, C.R.Acad. Sci. Paris, 320, Série I, 1259-1262D. Bosq (1996) Nonparametric Statistics for Stochastic Processes, Lecture Notesin Statistics 110, Springer-Verlag, New YorkP. Doukhan (1994) Mixing - Properties and Examples, Springer-Verlag New York12

A. Földes and P. Revesz (1974) A general method for density estimation, StudiaSc. Math. Hungarica 9, 81-92P. Hall and C.C. Heyde (1980) Martingale Limit Theory and Its Application,Academic PressL.A. Klimko; P.I. Nelson (1978) On conditional least squares estimation for sto-chastic processes, Ann. Statist. 6, 629-642J.-P. Kreiÿ (1991) Estimation of the distribution function of noise in stationaryprocesses, Metrika 38, 285-297E. Liebscher (1990) Kernel estimators for probability densities with discontinu-ities, Statistics 21, 185-196E. Liebscher (1996) Strong convergence of sums of �-mixing random variableswith applications to density estimation, Stoch. Proc. Appl. 65, 69-80E. Liebscher (1998a) Asymptotic normality of nonparametric estimators under�-mixing condition, Statistics & Probability Letters, to appearE. Liebscher (1998b) Estimation of the density and the regression function undermixing conditions, Statistics & Decisions, to appearE. Masry and D. Tjøstheim (1995), Nonparametric estimation and identi�cationof nonlinear ARCH time series, Econometric Theory 11, 258-289A. Mokkadem (1990), Study of risks of kernel estimators, Teoriya veroyatnosteii eyo primeneniya 35, 531-538E. Parzen (1962) On estimation of a probability density function and mode, Ann.Math. Statist. 33, 1065-1076E. Rio (1995) The functional law of the iterated logarithm for stationary stronglymixing sequences, Ann. Probab. 23, 1188-1203W. Stute (1982) A law of the logarithm for kernel density estimators, Ann.Probab. 10, 414-422H. Tong (1990) Non-linear Time Series: A Dynamical Approach, Oxford Uni-versity PressD. Tjøstheim (1986) Estimation in nonlinear time series models, Stoch. Proc.Appl. 21, 251-273 13