For integral functionals of the Gasser–Muller regression function and its derivatives, we consider the plug-in estimator. The consistency and asymptotic normality of the estimator are shown.Для інтегральних функцiоналiв Функції регресії Гассера-Мюллера та їх похідних розглядається оцінка, що підключається. Встановлено обґрунтованість та асимптотичну нормальність цієї оцінки

Arabidze, D.

Babilua, P.

Nadaraya, E.

Sokhadze, G.

Tkeshelashvili, A.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 519.21D. Arabidze, P. Babilua, E. Nadaraya (I. Javakhishvili Tbilisi State Univ., Georgia),G. Sokhadze, A. Tkeshelashvili (I. Vekua Inst. Appl. Math. I. Javakhishvili Tbilisi State Univ., Georgia)INTEGRAL FUNCTIONALS OF THE GASSER – MULLERREGRESSION FUNCTION*IНТЕГРАЛЬНI ФУНКЦIОНАЛИ ФУНКЦIЇ РЕГРЕСIЇ ГАССЕРА – МЮЛЛЕРАFor integral functionals of the Gasser – Muller regression function and its derivatives, the plug-in estimator is considered.The consistency and asymptotic normality of the estimator are shown.Для iнтегральних функцiоналiв функцiї регресiї Гассера – Mюллера та їх похiдних розглядається оцiнка, що пiд-ключається. Встановлено обґрунтованiсть та асимптотичну нормальнiсть цiєї оцiнки.1. Introduction. The study of functionals of a probability distribution density function or a regressionfunction and its derivatives is an interesting task and attracts an active interest on the part of researchers(see, e.g., [4 – 9]). There are detailed studies functionals of a probability distribution density functionand its derivatives (see [6 – 8] and the references therein). Investigations of functionals of a regressionfunction and its derivatives are more modest [4, 5].In the present paper we investigate the integral functional of a regression function and its deriva-tives. In our investigation we use the Gasser – Muller regression function introduced and studied in[1 – 3].As it follows from these works consideration of these types of problems are important, particularly,while choosing asymptotical optimal bandwidth (see formula (5) in [5, p. 2584]). Our approach inthis paper is based on the derivation of a representation theorem which we further use to obtainthe results connected with asymptotic properties, in particular with consistency and the central limittheorem.Suppose we have n measurements taken at the pointst1, t2, . . . , tn (0 ≤ t1 ≤ t2 ≤ . . . ≤ tn ≤ 1),where the tk, k = 1, . . . , n, depend on n. The model considered is the following:Y (tk) = a(tk) + εk, k = 1, . . . , n,εk i.i.d. with E(εk) = 0, D(εk) = σ2 <∞.The estimator of the unknown regression function a(t) was introduced by Gasser and Muller [1]and defined by the expressionân(t) =1hnn∑i=1si∫si−1W(t− uhn)du · Y (ti),* The work is supported by the Shota Rustaveli National Scientific Foundation, Project No. FR/308/5-104/12.c© D. ARABIDZE, P. BABILUA, E. NADARAYA, G. SOKHADZE, A. TKESHELASHVILI, 2015ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4 435436 D. ARABIDZE, P. BABILUA, E. NADARAYA, G. SOKHADZE, A. TKESHELASHVILIwhere 0 = s1 ≤ s2 ≤ . . . ≤ sn = 1, ti ≤ si ≤ ti+1, i = 1, 2, . . . , n − 1, and maxi |si − si−1| == O(1n); {hn, n = 1, 2, . . . } is a sequence of positive numbers monotonically tending to zero.W (u) is the function with probability density properties.In the same paper [1], Gasser and Muller introduced the estimator of the k th derivative of theregression function a(k)(t)â (k)n (t) =1hk+1nn∑i=1si∫si−1W (k)(t− uhn)du · Y (ti) (1)for all k = 0, 1, . . . ,m. It was assumed that â (0)n (t) + ân(t).In the works, which we have referred to above, the theorems of consistency and asymptoticnormality of estimators were obtained by imposing certain conditions.Let ϕ : Rm+2 → R be a continuous bounded smooth function. Consider an integral functional ofthe formI(a) =∞∫−∞ϕ(t, a(t), a′(t), . . . , a(m)(t))dt.We have the (ti, Yi), i = 1, 2, . . . , n. This means thatYi = Y (ti) = a(ti) + εi.To estimate I(a), we use the plug-in estimator, i.e., consider the functionalI(ân) =∞∫−∞ϕ(t, ân(t), â′n(t), . . . , â(m)n (t))dt.Here â (k)n (t)is defined from (1).2. Representation theorem. Our consideration is based on the representation theorem whichwill lead to obtaining the results we are interested in. Let us list the conditions which the consideredvariables are supposed to satisfy:Conditions on a :(a1) The function a = a(t) is defined and continuous on [0, 1] and takes its values in the interval[−k;k].(a2) The function a(t) has continuous derivatives up to order m inclusive.Conditions on εk :(ε1) Random values εk, k = 1, 2, . . . , are independent and equally distributed.(ε2) Eεk = 0, Dε2k = σ2 <∞.(ε3) The growth condition is P{|εk| > n} < e−n.In the sequel, for brevity, we will use the notation∂ϕ∂xi= ϕ(i) for i = 0, 1, . . . ,mandISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4INTEGRAL FUNCTIONALS OF THE GASSER – MULLER REGRESSION FUNCTION 437∂2ϕ∂xi∂xj= ϕ(ij) for i, j = 0, 1, . . . ,m.Conditions on ϕ :(ϕ1) The function ϕ : Rm+2 → R is continuous, bounded, integrable and has bounded continuousderivatives up to second order, inclusive, in some open convex domain A which contains the domainR× [−k;k]m+1.(ϕ2) All first and second derivatives of the function ϕ are uniformly bounded in the domain Aby a constant Cϕ > 0.Therefore, by this condition, for the function ϕ we have for all i, j = 0, 1, . . . ,m:sup{|ϕ(ij)|(s, s0, s1, . . . , sm) : (s, s0, s1, . . . , sm) ∈ A}≤ Cϕ. (2)Conditions onW :(w1)∫ ∞−∞W (t) dt = 1.(w2) The function W (t) has continuous derivatives up to order m, m ≥ 1.(w3) Function W (t) has the compact support [−τ, τ ].W (−τ) =W (τ) = 0;(w4) For any i = 0, 1, . . . ,m, W (i) ∈ L1([−τ, τ ]).Denote by an(t) the mathematical expectation ân(t):an(t) = Eân(t) = E1hnn∑i=1si∫si−1W(t− uhn)du · Y (ti) = 1hnn∑i=1si∫si−1W(t− uhn)du · a(ti).Then we obtaina(k)n (t) = Eâ(k)n (t) =1hi+1nn∑i=1si∫si−1W (k)(t− uhn)du · a(ti).Let us ascertain that there also exist expressions I(a), I(an) and I(ân) and they are finite. Usingthe Taylor formula, for any point (s, s0, s1, . . . , sm) ∈ A and some s˜m ∈ A we can write|ϕ|(s, s0, s1, . . . , sm) =∣∣∣∣∣∣m∑i=0ϕ(i)(s, 0, 0, . . . , 0)si +12m∑i,j=0ϕ(ij)(s, s˜0, s˜1, . . . , s˜m)sisj∣∣∣∣∣∣ .Accordingly, there exists a constant C such that|ϕ|(s, s0, s1, . . . , sm) ≤ C(m∑i=0|si|+m∑i=0|si|2).Hence it follows that for any bounded measurable functions f0(t), f1(t), . . . , fm(t) from L1(R) wehaveISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4438 D. ARABIDZE, P. BABILUA, E. NADARAYA, G. SOKHADZE, A. TKESHELASHVILI∞∫−∞|ϕ| (t, f0(t), . . . , fm(t)) dt <∞ (3)and therefore I(a) exists.The conditions which are imposed on the function W ensure its boundedness and membership inL1(R). Then condition (w4) and (2), (3) imply the finiteness of both variables I(an) and I(ân) forany n.By the Taylor formula we can writeI(ân)− I(an) = Sn(hn) +Rn, (4)where for any hn > 0, Sn(h) is the sum of independent random variablesSn(hn) =m∑i=01∫0ϕ(i)(t, an(t), a′n(t), . . . , a(m)n (t))(â (i)n (t)− a(i)n (t))dt. (5)A remainder Rn has the formRn =12m∑i,j=11∫0ϕ(ij)(˜bm(t))(â (i)n (t)− a(i)n (t))(â (j)n (t)− a(j)n (t))dt, (6)where b˜m(t) is the straight line connecting the points(t, an(t), a′n(t), . . . , a(m)n (t)) and (t, ân(t), â′n(t), . . . , â(m)n (t)).Let us estimate the remainder Rn. Applying the standard procedure, from (3) and (6) we obtain|Rn| ≤ Cϕ1∫0m∑i=0(â (i)n (t)− a(i)n (t))2dt.Let Cm[0, 1] denote the space of bounded real functions that are defined and continuous on [0, 1],having continuous derivatives of at least mth order. In this space we introduce the norm‖f‖m = m∑i=01∫0(difdti)2dt1/2 , f ∈ Cm[0, 1].The closure of Cm[0, 1] in this norm is defined by Wm2 and called the Sobolev space. This iscompleter separable Hilbert space with the scalar product〈f, g〉m =m∑i=01∫0difdtidigdtidt, f, g ∈ Cm[0, 1].Denotern(m) = ‖ân − an‖2m,ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4INTEGRAL FUNCTIONALS OF THE GASSER – MULLER REGRESSION FUNCTION 439then we can write|Rn| ≤ Cϕrn(m). (7)AssumeUk = Uk(t) =1hnsk∫sk−1W(t− uhn)du[Y (tk)− a(tk)], k = 1, 2, . . . , n,where a(tk) = EY (tk). Thenn∑k=1Uk =1hnn∑k=1sk∫sk−1W(t− uhn)du[Y (tk)− a(tk)]= ân(t)− an(t).Thereforern(m) =∥∥∥∥∥n∑k=1Uk∥∥∥∥∥2m. (8)Let us estimate the norm of one of the summands Uk in (8) for each k = 1, 2, . . . , n. We obtain‖Uk‖m = m∑i=01∫0∣∣∣∣ 1hi+1nsk∫sk−1W (i)(t− uhn)du[Y (tk)− a(tk)]∣∣∣∣2 dt1/2==m∑i=01h2i+2n1∫0∣∣∣∣∣∣∣∣hnt−sk−1hn∫t−skhnW (i)(t− uhn)d(t− uhn)∣∣∣∣∣∣∣∣2 ∣∣Y (tk)− a(tk)∣∣2 dt1/2≤≤ 2|εk|CW m∑i=01h2in1∫0∣∣∣∣ t− sk−1hn − t− skhn∣∣∣∣2 dt1/2 == 2|εk|CW |sk − sk−1|√1− h2m+2nhm+1n√1− h2n≤ L 1nhm+1n==Mm ∼ O(1nhm+1n)for sufficiently large L > 0. (9)To estimate rn(m), we use the McDiarmid’s inequality which we give here for convenience (fordetails see [10]).McDiarmid’s inequality. Let H(t1, . . . , tk) be a real function such that for each i = 1, . . . , k andsome ci, the supremum in t1, . . . , tk, t, of the differenceISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4440 D. ARABIDZE, P. BABILUA, E. NADARAYA, G. SOKHADZE, A. TKESHELASHVILI∣∣∣H(t1, . . . , ti−1, ti, ti+1, . . . , tk)−H(t1, . . . , ti−1, t, ti+1, . . . , tk)∣∣∣ ≤ ci.If X1, . . . , Xk are independent random variables taking values in the domain of the functionH(t1, . . . , tk), then for every ε > 0,P{∣∣H(X1, . . . , Xk)−EH(X1, . . . , Xk)∣∣ > ε} ≤ 2 exp− 2ε2∑ki=1c2i .Let us apply McDiarmid’s inequality for the functionsH(U1, . . . , Um) =∥∥∥∥∥n∑k=1Uk∥∥∥∥∥m.As ck we take ck ≡ 2Mm, k = 1, . . . , n. From (9), for any δ > 0 we obtainP{∣∣∣∣∣∥∥∥∥∥n∑k=1Uk∥∥∥∥∥m−E∥∥∥∥∥n∑k=1Uk∥∥∥∥∥m∣∣∣∣∣ ≥ δ}≤ 2 exp{−δ2nh2m+2n2M2m}.We substitute hereδ =√2 log n√nhm+1nand, by the Borel – Cantelli lemma, write∥∥∥∥∥n∑k=1Uk∥∥∥∥∥m= E∥∥∥∥∥n∑k=1Uk∥∥∥∥∥m+O( √log n√nhm+1n). (10)Using the Jensen’s inequalityE∥∥∥∥∥n∑k=1Uk∥∥∥∥∥2m≤ E∥∥∥∥∥n∑k=1Uk∥∥∥∥∥2m==n∑k=1m∑i=01∫0E∣∣∣∣∣∣∣1hi+1nsk∫sk−1W (i)(t− uhn)du[Y (tk)− a(tk)]∣∣∣∣∣∣∣2dt ≤≤ 2C2Wn∑k=1m∑i=01∫01h2i+2n∣∣∣∣∣∣∣sk∫sk−1du∣∣∣∣∣∣∣2E[Y (tk)− a(tk)]2dt == 2C2Wσ2 (1− h2m+2n )(1− h2n)h2m+2nn∑k=1(sk − sk−1)2 ≤ K · 1nh2m+2n, (11)from (7), (8), (10) and (11) we conclude thatRn = O(log nnh2m+2n).Therefore the following statement is true.ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4INTEGRAL FUNCTIONALS OF THE GASSER – MULLER REGRESSION FUNCTION 441Theorem 1. Assume that conditions (a1) – (a2), (ε1) – (ε3), (ϕ1) – (ϕ2) and (w1) – (w4) arefulfilled. Then representation (4), where the remainder with probability 1 has the orderRn = O(log nnh2m+2n)(12)is valid.3. Consistency. In this section of the paper we use Theorem 1 to prove that the estimator I(ân)is strictly consistent.Theorem 2. Let the conditions of Theorem 1 be fulfilled. If the positive sequence (hn)∞n=1,0 < hn < 1, is chosen so thatlog nnh2m+2n→ 0,then with probability 1 we haveI(ân)→ I(a) as n→∞.Proof. By Theorem 1 and formula (4)I(ân)− I(an) = Sn(hn) +Rn, Rn = o(1) a.e.,whereSn(hn) =m∑i=01∫0ϕ(i)(t, an(t), a′n(t), . . . , a(m)n (t))(â (i)n (t)− a(i)n (t))dt.By condition (a1),{(t, an(t), a′n(t), . . . , a(m)n): t ∈ [0, 1]}⊂ [0, 1]× [−k;k]m+1.This and condition (ϕ2) imply that there exists a constant Cϕ > 0, such thatsup{|ϕ(i)|(t, t0, . . . , tm) : (t, t0, . . . , tm) ∈ [0, 1]× [−k; k]m+1}≤ Cϕ.Keeping this in mind, we can write|Sn(hn)| ≤ Cϕm∑i=01∫01hi+1nn∑k=1sk∫sk−1∣∣∣∣W (i)( t− uhn)∣∣∣∣ du · [Y (tk)− a(tk)] dt ≤≤ 2CϕCWm∑i=01∫01hi+1nn∑k=1|εk| |sk − sk−1| dt ∼ (by (w2))∼M 1nhm+1n(for some M ). (13)ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4442 D. ARABIDZE, P. BABILUA, E. NADARAYA, G. SOKHADZE, A. TKESHELASHVILIHoeffding’s inequality. Suppose X1, X2, . . . , Xn are independent real-valued random variables,such that for each i, Xi takes values from the interval [ri, pi].Let Y +∑ni=1Xi. Then for all t > 0,P{|Y −E|Y || ≥ t} ≤ 2 exp− 2t2∑ni=1(pi − ri)2 . (14)As Xk we takeXk =m∑i=01∫01hi+1nsk∫sk−1∣∣∣∣W (i)( t− uhn)∣∣∣∣ du · [Y (tk)− a(tk)] dt.Analogously to (13), it can be shown that Xk takes its values in the interval[−M 1n2hm+1n,M1n2hm+1n].Thereforen∑i=1(pi − ri)2 = 2Mn3h2m+2n.Besides, we taket =2√M log nn3/2hm+1n.Then from (14) we obtainP{∣∣Sn(hn)∣∣ > 2√M log nn3/2hm+1n}≤ 2 exp{− 2 log n}by means of which, using the Borel – Cantelli lemma, we can conclude thatSn(hn) = O( √log nn3/2hm+1n).It is obvious that, for condition (12),√log nn3/2hm+1n, too, tends to zero. Thus we conclude that Sn(hn)→→ 0 as n→∞.By formula (6) from [2] we can writeEa(k)n (t) =τ∫−τW (u)a(k)(t− uhn) du+O(1nhkn).Hence we make the following conclusions:(i) for condition (12),1nhkn, too, tends to zero for any k = 0, 1, . . . ,m;(ii) Ea(k)n (t)→ a(k)(t) as n→∞.ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4INTEGRAL FUNCTIONALS OF THE GASSER – MULLER REGRESSION FUNCTION 443Summarizing the above discussion, we ascertain thatI(an) =1∫0ϕ(t, an(t), a′n(t), . . . , a(m)n (t))dt −→−→1∫0ϕ(t, a(t), a′(t), . . . , a(m)(t))dt = I(a) as n→∞.Since I(ân)− I(an) = o(1), we conclude thatI(ân)− I(a) −→ 0 a.e.The theorem is proved.4. Central limit theorem. Using our representation theorem we can obtain the limit distributionproperty for the integral functionalI(ân) =1∫0ϕ(t, ân(t), â′n(t), . . . , â(m)n (t))dt.Consider the difference (4), where for any hn > 0, Sn(h) is the sum of independent random variables(5). Rn is a remainder having the form (6). Clearly,ESn(hn) = 0 and ERn → 0 as n→∞. (15)Moreover,E(Sn(hn))2 = σ2m∑i=0 1∫0ϕ(i)(t, an(t), a′n(t), . . . , a(m)n (t)) dt2and DRn → 0 as n→∞.(16)Using appropriate conditions, we have to prove that the variable√n (I(ân)− I(an))is asymptotically normal and calculate the limiting variance. For this, according to the theorem andformulas (4), (15) and (16), we have to show the asymptotic normality of the variable√nSn(hn).As follows from (6), in this case it suffices to study this property for the variablesdk = Y (tk)m∑i=01hi+1n1∫0sk∫sk−1W (i)(t− uhn)ϕ(i)(t, an(t), a′n(t), . . . , a(m)n (t))dt du.It can be easily verified thatISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4444 D. ARABIDZE, P. BABILUA, E. NADARAYA, G. SOKHADZE, A. TKESHELASHVILIEdk = a(tk)m∑i=01hi+1n1∫0sk∫sk−1W (i)(t− uhn)ϕ(i)(t, an(t), a′n(t), . . . , a(m)n (t))dt du.Thus we consider the sequence of independent random variablesfk(n) = α(n, k)(Y (tk)− a(tk)) = α(n, k)εk,whereα(n, k)=m∑i=01hi+1n1∫0sk∫sk−1W (i)(t− uhn)ϕ(i)(t, an(t), a′n(t), . . . , a(m)n (t))dt du.Let consider the sumSn(hn) =n∑k=1α(n, k)εk.Let Fk,n be the probability distribution function of a random variable α(n, k)εk, and Fε bethe distribution function of a random variable εk. The Lindeberg’s condition is written in the formlimn→∞ Ln(δ) = 0 ∀ δ > 0, whereLn(δ) =(σ2n∑k=1α2(n, k))−1 n∑j=1∫x2J|x| ≥ δσ( n∑k=1α2(n, k))1/2 dFk,n(x),where J(A) is the indicator function of the set A.It is easy to see thatLn(δ) ≤ 1σ2max0≤j≤n∫x2J (|x| ≥ δσv(n, j)) dFε,wherev(n, j) =|α(n, j)|(∑nj=1α2(n, j))1/2 .It remains to show thatmax1≤j≤nv(n, j)→ 0 as n→∞.But sincemax1≤j≤n|α(n, j)| = O(1nhm+1n),we havemax1≤j≤nv(n, j) = O(1√nhm+1n).Thus the Lindeberg’s condition is fulfilled and we can conclude that the theorem is valid.ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4INTEGRAL FUNCTIONALS OF THE GASSER – MULLER REGRESSION FUNCTION 445Theorem 3. Let the conditions of Theorem 1 be fulfilled. Then ifhn → 0 and nhm+1n →∞ as n→∞,we have √n (I(ân)− I(a)) d−→ N(0, r2),wherer2 = σ2m∑i=0 1∫0ϕ(i)(t, a(t), a′(t), . . . , a(m)(t))dt2 .5. Example. As an example we consider the problem of estimation of total curvature (see[11, p. 22]) of a regression function a:I =1∫0(a′′(t))2 dt.We obtainϕ(t, x0, . . . , xm) = x22.Then we haver2 = 4σ2(a′(1)− a′(0))2 .For hn → 0 and√nh6n →∞, we have the convergence√n(I2(a)− I2(ân)) d−→ N(0, r2).These considerations can be used for checking hypothesis about total curvature of regressionfunction.6. Iterated logarithm law. Applying the well-known iterated logarithm law from Kuelbs paper[12], we ascertain that the following statement is true.Theorem 4. If the sequence hn is chosen so thathn =(log n√n log log n) 12m,thenlim supn→∞±√n [I(ân)− I(an)]√2 log log n= r.Proof. Note that for this hn we haveRn = o(√log lognn).It can be easily verified thatlim supn→∞±√n [I(ân)− I(an)]√2 log log n= lim supn→∞±√n[α(n, k)Y (tk)− α(n, k)a(tk)]√2 log log n= r.1. Gasser T., Muller H.-G. Nonparametric estimation of regression functions and their derivatives. – Heidelberg, 1979.– (Preprint / Univ. 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Integral Functionals of the Gasser–Muller Regression Function

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Integral Functionals of the Gasser–Muller Regression Function

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