Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 RESEARCH Open Access Strong consistency of estimators in partially linear models for longitudinal data with mixingdependent structure Xing-cai Zhou1,2 and Jin-guan Lin1* * Correspondence: jglin@seu.edu.cn Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of China Full list of author information is available at the end of the article Abstract For exhibiting dependence among the observations within the same subject, the paper considers the estimation problems of partially linear models for longitudinal data with the -mixing and r-mixing error structures, respectively The strong consistency for least squares estimator of parametric component is studied In addition, the strong consistency and uniform consistency for the estimator of nonparametric function are investigated under some mild conditions Keywords: partially linear model, longitudinal data, mixing dependent, strong consistency Introduction Longitudinal data (Diggle et al [1]) are characterized by repeated observations over time on the same set of individuals They are common in medical and epidemiological studies Examples of such data can be easily found in clinical trials and follow-up studies for monitoring disease progression Interest of the study is often focused on evaluating the effects of time and covariates on the outcome variables Let tij be the time of the jth measurement of the ith subject, xij Ỵ Rp and yij be the ith subject’s observed covariate and outcome at time tij respectively We assume that the full dataset {(xij, yij, tij), i = 1, , n, j = 1, , mi}, where n is the number of subjects and mi is the number of repeated measurements of the ith subject, is observed and can be modeled as the following partially linear models yij = xT β + g(tij ) + eij , ij (1:1) where b is a p × vector of unknown parameter, g(⋅) is an unknown smooth function, eij are random errors with E(eij) = We assume without loss of generality that tij are all scaled into the interval I = [0, 1] Although the observations, and therefore the eij, from the different subjects are independent, they can be dependent within each subject Partially linear models keep the flexibility of nonparametric models, while maintaining the explanatory power of parametric models (Fan and Li [2]) Many authors have studied the models in the form of (1.1) under some additional assumptions or restrictions If the nonparametric component g(⋅) is known or not present in the models, © 2011 Zhou and Lin; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 they become the general linear models with repeated measurements, which were studied under Gaussian errors in a amount of literature Some works have been integrated into PROC MIXED of the SAS Systems for estimation and inference for such models If g(⋅) is unknown but there are no repeated measurements, that is m1 = ⋅ ⋅ ⋅ = mn = 1, the models (1.1) are reduced to non-longitudinal partially linear regression models, which were firstly introduced by Engle et al [3] to study the effect of weather on electricity demand, and further studied by Heckman [4], Speckman [5] and Robinson [6], among others A recent survey of the estimation and application of the models can be found in the monograph of Häardle et al [7] When the random errors of the models (1.1) are independent replicates of a zero mean stationary Gaussian process, Zeger and Diggle [8] obtained estimators of the unknown quantities and analyzed time-trend CD4 cell numbers among HIV sero-converters; Moyeed and Diggle [9] gave the rate of convergence for such estimators; Zhang et al [10] proposed the maximum penalized Gaussian likelihood estimator Introducing the counting process technique to the estimation scheme, Fan and Li [2] established asymptotic normality and rate of convergence of the resulting estimators Under the models (1.1) for panel data with a one-way error structure, You and Zhou [11] and You et al [12] developed the weighted semiparametric least square estimator and derived asymptotic properties of the estimators In practice, a great deal of the data in econometrics, engineering and natural sciences occur in the form of time series in which observations are not independent and often exhibit evident dependence Recently, the non-longitudinal partially linear regression models with complex error structure have attracted increasing attention by statisticians For example, see Schick [13] with AR(1) errors, Gao and Anh [14] with long-memory errors, Sun et al [15] with MA(∞) errors, Baek and Liang [16] and Zhou et al [17] with negatively associated (NA) errors, and Li and Liu [18], Chen and Cui [19] and Liang and Jing [20] with martingale difference sequence, among others For longitudinal data, an inherent characteristic is the dependence among the observations within the same subject Some authors have not considered the with-subject dependence to study the asymptotic behaviors of estimation in the semipara-metric models with assumption that the mi are all bounded, see, for example, He et al [21], Xue and Zhu [22] and the references therein Li et al [23] and Bai et al [24] showed that ignoring the data dependence within each subject causes a loss of efficiency of statistical inference on the parameters of interest Hu et al [25] and Wang et al [26] took into consideration within-subject correlations for analyzing longitudinal data and obtained some asymptotic results based on the assumption that max 1≤i≤n m i is bounded for all n Chi and Reinsel [27] considered linear models for longitudinal data that contain both individual random effects components and with-individual errors that follow an (autoregressive) AR(1) time series process and gave some estimation procedures, but they did not investigate asymptotic properties of estimations In fact, the observed responses within the same subject are correlated and may be represented by a sequence of responses {yij, j ≥ 1} for the i-individual with an intrinsic dependence structure, such as mixing conditions For example, in hydrology, many measures may be represented by a sequence of responses {yij, j ≥ 1} for the ith year at tij, where tij represents the time elapsed from the beginning of the ith year, and {eij, j ≥ 1} are the measurements of the deviation from the mean {xT β + g(tij ), j ≥ 1} It is not reasonable ij that E(eij1 eij2 ) = for j1 ≠ j2 In practice, {eij, j ≥ 1} may be “weak error’s structure”, Page of 18 Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page of 18 such as mixing-dependent structure In this paper, we consider the estimation problems for the models (1.1) with the -mixing and r-mixing error structures for exhibiting dependence among the observations within the same subject respectively and are mainly devoted to strong consistency of estimators Let {Xm , m ≥ 1} be a sequence of random variables defined on probability space l ( , F , P), Fk = σ (Xi , k ≤ i ≤ l) be s-algebra generated by X k , , X l , and denote l l L2 (Fk ) be the set of all Fk measurable random variables with second moments A sequence of random variables {Xm, m ≥ 1} is called to be -mixing if ϕ(m) = sup k ∞ k≥1,A∈F1 ,P(A)=0,B∈Fk+m |P(B|A) − P(B)| → 0, as m → ∞ A sequence of random variables {Xm, m ≥ 1} is called to be r-mixing if maximal correlation coefficient ρ(m) = |cov (X, Y)| sup k ∞ k≥1,X∈L2 (F1 ),Y∈L2 (Fk+m ) Var(X) · Var(Y) → 0, as m → ∞ The concept of mixing sequence is central in many areas of economics, finance and other sciences A mixing time series can be viewed as a sequence of random variables for which the past and distant future are asymptotically independent A number of limit theorems for -mixing and r-mixing random variables have been studied by many authors For example, see Shao [28], Peligrad [29], Utev [30], Kiesel [31], Chen et al [32] and Zhou [33] for -mixing; Peligrad [34], Peligrad and Shao [35,36], Shao [37] and Bradley [38] for r-mixing Some limit theories can be found in the monograph of Lin and Lu [39] Recently, the mixing-dependent error structure has also been used to study the nonparametric and semiparametric regression models, for instance, Roussas [40], Truong [41], Fraiman and Iribarren [42], Roussas and Tran [43], Masry and Fan [44], Aneiros and Quintela [45], and Fan and Yao [46] The rest of this paper is organized as follows In Section 2, we give least square estiˆ mator (LSE) βn of b based on the nonparametric estimator of g(·) under the mixingdependent error structure and state some main results Section is devoted to sketches of several technical lemmas and corollaries The proofs of main results are given in Section We close with concluding remarks in the last section Estimators and main results For models (1.1), if b is known to be the true parameter, then by Eeij = 0, we have g(tij ) = E(yij − xT β), ij ≤ i ≤ n, ≤ j ≤ mi Hence, a natural nonparametric estimator of g(·) given b is n ∗ gn (t, β) = mi Wnij (t)(yij − xT β), ij (2:1) i=1 j=1 where Wnij (t) = Wnij (t, t11 , t12 , , tnmn ) is the weight function defined on I Now, in order to estimate b, we minimize n mi SS(β) = i=1 j=1 ∗ yij − xT β − gn (tij , β) ij Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page of 18 The minimizer to the above equation is found to be ⎛ ˆ βn = ⎝ n ⎞−1 mi ˜ ˜ xij yij , i=1 j=1 (2:2) i=1 j=1 mi l=1 n k=1 ˜ where xij = xij − mi n ˜ ˜ ij xij xT ⎠ n k=1 ˜ Wnkl (tij )xkl and yij = yij − mi l=1 Wnkl (tij )ykl So, a plug-in estimator of the nonparametric component g(·) is given by n mi ˆ gn (t) = ˆ Wnij (t)(yij − xT βn ) ij (2:3) i=1 j=1 In this paper, let {eij,1 ≤ j ≤ mi} be -mixing or r-mixing with Eeij = for each i(1 ≤ i ≤ n), and {e i, ≤ i ≤ n} be mutually independent, where ei = (ei1 , , eimi )T For each i, denote i(·) and ri(·) be the ith mixing coefficients of the sequence of -mixing and rmixing, respectively Define S2 = n n i=1 mi ˜ ˜T ˜ j=1 xij xij , g (t) = g(t) − n k=1 mi l=1 Wnkl (t)g(tkl ), denote I(·) be the indicator function, || · || be the Euclidean norm, and set ⌊z⌋ ≤ z < ⌊z⌋ + for the integer part of z In the sequence, C and C1 denote positive constants whose values may vary at each occurrence For obtaining our main results, we list some assumptions: A1 (i) {eij, ≤ j ≤ mi} are -mixing with Eeij = for each i; (ii) {eij, ≤ j ≤ mi} are r-mixing with Eeij = for each i A2 (i) max1≤i≤n mi = o(nδ) for some < δ < r−2 and r > 2; 2r S = , where Σ is a positive definite matrix and N(n) = N(n) n (iii) g(·) satisfies the first-order Lipschitz condition on [0, 1] (ii) limn→∞ n i=1 mi A3 For n large enough, the probability weight functions Wnij(·) satisfy n i=1 mi j=1 Wnij (t) = for each t Ỵ [0, 1]; ⎛ 1⎞ − (ii) sup0≤t≤1 max1≤i≤n,1≤j≤mi Wnij (t) = O ⎝n ⎠ ; (i) (iii) sup0≤t≤1 n i=1 mi j=1 n i=1 (iv) max1≤k≤n,1≤l≤mi || (v) sup0≤t≤1 n i=1 Wnij (t)I(|tij − t| > ε) = o(1) for any  > 0; mi j=1 mi j=1 Wnij (tkl )xij || = O(1), Wnij (t)xij = O(1), (vi) max1≤i≤n,1≤j≤mi Wnij (s) − Wnij (t) ≤ C|s − t| uniformly for s, t Ỵ [0, 1] Remark 2.1 For obtaining the asymptotic properties of estimators of the models (1.1), many authors often assumed that {mi, ≤ i ≤ n} are bounded Under the weak condition A2(i), we obtain the strong consistency of estimators of the models (1.1) with Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page of 18 mixing-dependent structure The condition of {m i , ≤ i ≤ n} being a bounded sequence is a special case of A2(i) Remark 2.2 Assumption A2(ii) implies that ⎛ 1⎞ n mi ||˜ ij || = O(1) and x max ||˜ ij || = o ⎝N(n) ⎠ x 1≤i≤n,1≤j≤mi N(n) i=1 j=1 Remark 2.3 As a matter of fact, there exist some weights satisfying assumption A3 For example, under some regularity conditions, the following Nadaraya-Watson kernel weight satisfies assumption A3: t − tij hn Wnij (t) = K mi n K k=1 l=1 t − tkl hn −1 , where K(·) is a kernel function and hn is a bandwidth parameter Assumption A3 has also been used by Hardle et al [7], Baek and Liang [16], Liang and Jing [20] and Chen and You [47] Theorem 2.1 Suppose that A1(i) or A1(ii), and A2 and A3(i)-(iii) hold If max 1≤i≤n,1≤j≤mi E(|eij |p ) ≤ C, a.s (2:4) for p > 3, then ˆ βn → β, (2:5) a.s Theorem 2.2 Suppose that A1(i) or A1(ii), and A2, A3(i-iv) and (2.4) hold For any t Ỵ [0, 1], we have ˆ gn (t) → g(t), a.s (2:6) Theorem 2.3 Suppose that A1(i) or A1(ii), and A2, A3(i-iii), A3(v-vi) and (2.4) hold We have sup |ˆ n (t) − g(t)| = o(1), g a.s (2:7) 0≤t≤1 Several technical lemmas and corollaries In order to prove the main results, we first introduce some lemmas and corollaries Let Sj = j l=1 Xl for j ≥ 1, and Sk (i) = k+i j=k+1 Xj for i ≥ and k ≥ Lemma 3.1 (Shao [28]) Let {Xm, m ≥ 1} be a -mixing sequence (1) If EXi = 0, then ⎧ ⎫ ⎨ log i ⎬ ES2 (i) ≤ 8000i exp ϕ 1/2 (2j ) max EX2 k ⎩ ⎭ k+1≤j≤k+i j j=1 (2) Suppose that there exists an array {c km } of positive numbers such that max1≤i≤m ES2 (i) ≤ ckmfor every k ≥ 0, m ≥ Then, for any q ≥ 2, there exists a positive k constant C = C(q, (·)) such that Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page of 18 q/2 E max |Sk (i)|q ≤ C ckm + E max |Xi |q 1≤i≤m k 1,0 If max max E(|eij |α ) ≤ C, a.s., 1≤i≤n 1≤j≤mi (3:3) we have ∞ mi |e ij | < ∞, a.s i=1 j=1 ⎞ mi |eij | = |eij |I ⎝|eij | > εi r ⎠ ⎛ Proof Note that ⎛ ξi = mi j=1 |eij |, ξ i = mi j=1 |eij | · I ⎝ mi j=1 ⎞ |eij | ≤ εi r mi ⎠ , ξ ⎛ i = ξi − ξ i = mi j=1 |eij |I ⎝ mi j=1 ⎞ |eij | > εi r mi ⎠ , Let and |ξi |d = |ξi |I(|ξi | ≤ d) for fixed d > First, we prove ∞ |ξ i | < ∞, a.s (3:4) i=1 Note that {|ξi | > d} = ⎧ ⎨ ⎩ mi j=1 ⎛ |eij |I ⎝ mi j=1 ⎫ ⎧ ⎞ ⎬ ⎨ |eij | > εi r mi ⎠ > d = ⎭ ⎩ mi j=1 ⎫ ⎬ |eij | > εi r mi ⎭ (3:5) Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page of 18 for i large enough By Markov’s inequality, Cr-inequality, and (3.3), we have ⎛ ⎞ mi ∞ ∞ eij > εi r mi ⎠ P( ξ i d) ≤ C P⎝ i=1 i=1 j=1 α − i r m−α E i ∞ ≤C i=1 α mi eij j=1 ≤ C lim n→∞ − ≤C α − i r max max E eij n α i r m−1 i ∞ i=1 mi E eij α j=1 (3:6) α 1≤i≤n 1≤j≤mi i=1 α i r < ∞, ∞ − ≤C i=1 From (3.5), {|ξi | ≤ d} = ⎧ ⎨ mi j=1 ⎩ ⎫ ⎬ |eij | ≤ εi r mi for i large enough One gets ⎭ E(|ξ i |d ) = E(|ξ i |I(|ξ i | ≤ d)) ⎛ ⎛ = E⎝ mi |eij |I ⎝ j=1 mi ⎞ ⎛ |eij | > εi r mi ⎠I ⎝ j=1 mi ⎞⎞ |eij | ≤ εi r mi ⎠⎠ = j=1 and Var(|ξ i |d ) ≤ E(|ξ i |2 ) = E(|ξ i |I(|ξ i | ≤ d))2 d = E |ξ i |2 I(|ξ i | ≤ d) ≤ dE(|ξ i |I(|ξ i | ≤ d)) = for i large enough Therefore, ∞ ∞ E(|ξ i |d ) < ∞, Var(|ξ i |d ) < ∞ i=1 (3:7) i=1 Since {ξi , ≤ i ≤ n} is a sequence of independent random variables, (3.4) holds from (3.6) and (3.7) by Three Series Theorem Then, ⎛ ⎞ ∞ ∞ mi mi |eij | = i=1 j=1 |eij |I ⎝|eij | > εi r mi ⎠ i=1 j=1 ∞ mi ≤ ⎛ |eij |I ⎝ i=1 j=1 mi j=1 ⎞ |eij | > εi r mi ⎠ = ∞ |ξi | < ∞, a.s i=1 Thus, we complete the proof of Lemma 3.3 Lemma 3.4 Let {eij, ≤ j ≤ mi} be the -mixing with Eeij = for each i (1 ≤ i ≤ n) Assume that {anij(·), ≤ i ≤ n,1 ≤ j ≤ mi} is a function array defined on [0, 1], satisfying ⎛ 1⎞ ⎛ 1⎞ − − and max1≤i≤n,1≤j≤mi |anij (t)| = O ⎝n ⎠ any t Ỵ for max1≤i≤n,1≤j≤mi |anij (t)| = O ⎝n ⎠ [0, 1], and A2(i) and (2.4) hold Then, for any t Ỵ [0, 1] we have n mi anij (t)eij = o(1), i=1 j=1 a.s (3:8) Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page of 18 Proof Based on (3.1) and (3.2), we denote ζnij = eij − E(eij ), ηnij = eij − E(eij ) and take r satisfying ε ≤ ε−q E i=1 i=1 ⎛ ⎜ ≤ C⎜ ⎝ q ˜ ζni n n ˜ E|ζni |q + i=1 q⎞ 2⎟ ⎟ ⎠ ˜2 Eζni i=1 (3:11) =: A11n + A12n Note that i (m) ® as m ® ∞, hence log mi k=1 exp λ log mi k=1 1/2 ϕi (2k ) = o(log mi ) Further, 1/2 ϕi (2k ) = o(mτ ) for any l > and τ > i For A11n, by Lemma 3.1, A2(i) and (2.4), and taking q >p, we have q mi n A11n = C E i=1 n ≤C i=1 n ≤C anij (t)ζnij j=1 ⎡⎛ ⎧ ⎨ ⎢⎝ ⎣ mi exp ⎩ ⎡ ⎣ m1+τ n−1 i q/2 log mi 1/2 ϕi (2k ) k=1 mi + i=1 q − ≤ Cn ⎛ ≤ Cn ⎤ ⎞q/2 max E|anik (t)ζnik |2 ⎠ ⎭ 1≤k≤mi ⎤ q − p q−p ⎦ n E|ζnij | |ζnij | mi + ⎥ E|anij ζnij |q ⎦ j=1 j=1 n (τ + 1)q q − + Cn mi i=1 n mi q−p (i r mi ) i=1 j=1 ⎞ q (τ + 1)δq −⎝ − −1⎠ 2 q > max Take ⎫ ⎬ − + Cn q q p − + −(q−p+1)δ−1 r r 2r(2 + δ) , ,p r − 2rδ − − δ We have δq q − >2 2 and q q p − + − (q − p + 1)δ > 2 r r Next, take τ > small enough such that q (τ + 1)δq − > Thus, we have 2 ∞ A11n < ∞ n=1 (3:12) Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page of 18 For A12n, by Lemma 3.1 and (2.4), we have ⎧ ⎨ mi n A12n = C E ⎩ i=1 ⎧ ⎨ ⎧ ⎨ mi exp ⎩ n ≤C i=1 ⎧ ⎨ ⎩ anij (t)ζnij j=1 ≤C ⎩ q ⎬2 2⎫ n ⎭ log mi 1/2 ϕi (2k ) k=1 mi mτ +1 i E|anij (t)ζnij |2 i=1 ⎛ j=1 q ⎫ ⎬2 ⎫ ⎬ max E|anij (t)ζnij |2 ⎭ 1≤j≤mi ⎭ ⎫q ⎬2 ⎭ ⎞ q (τ + 1)δq ⎠ − −⎝ ≤ Cn r−2 q δq , we have − < Taking q > > Next, take τ > 2r − 2δ q (τ + 1)δq small enough such that − > Thus, we have 2 Note that δ < ∞ A12n < ∞ (3:13) n=1 Combining (3.11)-(3.13), we obtain (3.10) ⎛ 1⎞ By Lemma 3.3 and max1≤i≤n,1≤j≤mi |anij (t)| = O ⎝n ⎠ for any t Ỵ [0, 1], we have − n |A2n | ≤ Note that max i≤i≤n,1≤j≤mi mi |anij (t)| 1⎞ |eij | = O ⎝n ⎠ ⎛ − (3:14) i=1 j=1 p−1 > and δ > From (2.4), we have r mi n |A3n | = anij (t)E(eij ) i=1 j=1 ≤n mi n − i=1 j=1 =n − mi n ⎛ ⎞ E ⎝|eij |I(|eij | > εi r mi )⎠ ⎞⎞ E ⎝|eij |p |eij |1−p I ⎝|eij | > εi r mi ⎠⎠ ⎛ ⎛ (3:15) i=1 j=1 − ≤ Cn n mi ⎛ ⎞1−p − ⎝ i r mi ⎠ ≤ Cn i=1 j=1 − ≤ Cn n p−1 r m2−p i i − i=1 (p−2)δ+ = o(1) From (3.9), (3.10), (3.14) and (3.15), we have (3.8) Corollary 3.1 In Lemma 3.4, if {eij, ≤ j ≤ mi} are r-mixing with Eeij = for each i (1 ≤ i ≤ n), then (3.8) holds Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page 10 of 18 ∞ n=1 Proof From the proof of Lemma 3.4, it is enough to prove that ∞ n=1 A11n < ∞ and A12n < ∞ Note that r i (m) ® as m ® ∞, hence log mi exp λ log mi k=1 2/q ρi (2k ) = o(log mi ) Further, ρi (2k ) = o(mτ ) for any l > and τ > i 2/q k=1 For A11n, by Lemma 3.2 and (2.4), and taking q >p, we get q mi n A11n = C E i=1 anij (t)ζnij ⎛ j=1 ⎧ q ⎨ ⎜ ⎝mi exp C1 ⎩ n ≤C i=1 ⎧ ⎨ +mi exp C1 ⎩ ⎛ log mi ρ1 (2k ) k=1 log mi ⎫ ⎬ q max (E|anik ζnik |2 ) ⎭ 1≤k≤mi ⎫ ⎬ ⎞ max E|anik ζnik |q ⎠ ⎭ 1≤k≤mi k=1 ⎞ q q q ⎛ ⎞q−p τ+ − − ⎜ ⎟ τ +1 ⎝mi n + mi n ⎝i r mi ⎠ ⎠ n ≤C 2/q ρi (2k ) i=1 q − − ≤ Cn Take r+ q δ−1 − + Cn q q p − + −(q+p+r+1)δ−1 r r 2r(2 + δ) , ,p r − 2rδ − − δ q > max We qδ q − >2 2 have and q q p − + − (q − p + 1)δ > 2 r r Next, τ take > small enough such that q q δ>2 − τ+ 2 and q q p − + − (q + p + τ + 1)δ > Thus, ∞ A11n < ∞ n=1 r r For A12n, by Lemma 3.2 and (2.4), we have ⎧ ⎨ mi n A12n = C ⎩ ⎛ ≤ C⎝ E i=1 n i=1 ⎛ ≤ C⎝ n ≤ Cn anij (t)ζnij j=1 ⎧ ⎨ mi exp C1 ⎩ mτ +1 i i=1 ⎛ −⎝ q ⎬2 2⎫ mi j=1 ⎭ log mi ρ1 (2k ) k=1 ⎫ ⎬ ⎞ max E|anik ζnik |2 ⎠ ⎭ 1≤j≤mi q ⎞q E|anij ζnij |2 ⎠ ⎞ q (τ + 1)δq ⎠ − 4 δq q from A2(i) Taking q > , we have − > Next, take τ > − 2δ q (τ + 1)δq ∞ small enough such that − > Thus, n=1 A12n < ∞ 2 Note that δ < Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page 11 of 18 So, we complete the proof of Lemma 3.4 Remark 3.1 If the real function array {anij(t),1 ≤ i ≤ n, ≤ j ε ≤ P ⎝ sup 0≤t≤1 mi n sup 0≤t≤1 u∈Dn (sn (t)) i=1 j=1 ⎞ anij (u)ζnij > ε⎠ ⎞⎞ ⎛ ⎞ ⎛ ⎛ q (r + 1)δq q (r + 1)δq q q p ⎠ −⎝ − −1⎠ −⎝ − ⎜ − − + −(q−p+1)δ−1 ⎟ 2 ⎟ ≤ Cn r ⎜n r r +n +n ⎝ ⎠ 2+ ⎛ ⎛ ⎞ ⎛ ⎞⎞ q (r + 1)δq q (r + 1)δq q q −⎝ − −4⎠ −⎝ − −3⎠ − −δd−δ−4 ⎟ 2 2 r ⎟ +n +n ⎠ ⎜ ⎜ ≤ C ⎝n − 2r(5 + δ) 16 q q δq q , , p We have − − δq − δ > 5, − >5 r − 2rδ − − 2δ r 2 δq q q (r + 1)δq and − > Next, take τ > small enough such that − > and 2 q (r + 1)δq ∞ − > Thus, we have n=1 P sup0≤t≤1 B3n (t) > ε < ∞ Thus, sup 0≤t≤1 B3n(t) = o(1),a.s Therefore, (3.16 ) holds Corollary 3.2 In Lemma 3.5, if {eij, ≤ j ≤ mi} are r-mixing with Eeij = for each i (1 ≤ i ≤ n), then (3.16) holds Proof By Corollary 3.1, with arguments similar to the proof of Lemma 3.5, we have (3.16) Take q > max Proof of Theorems Proof of Theorem 2.1 From (1.1) and (2.2), we have ⎛ ˆ βn − β = ⎝ mi n ⎞−1 ˜ ˜ ij xij xT ⎠ i=1 j=1 mi n ˜ y ˜ ij xij (˜ ij − xT β) i=1 j=1 mi n = S−2 n n i=1 j=1 Wnkl (tij )(ykl − xT β) kl k=1 l=1 mi n = S−2 n n i=1 j=1 ⎣ n n mi ˜ xij eij − S2 n N(n) −1 (4:1) ˜ Wnkl (tij )ekl + g(tij ) k=1 l=1 mi i=1 j=1 = mi ˜ xij eij − ⎡ Wnkl (tij )(g(tkl ) + ekl ) k=1 l=1 mi n = S−2 n = mi n ˜ xij (g(tij ) + eij ) − i=1 j=1 S−2 n mi ˜ xij (yij − xT β) − ij ⎡ ⎣ n ˜ xij i=1 j=1 n mi i=1 j=1 =: D1n + D2n + D3n mi ˜ xij eij − N(n) n mi Wnkl (tij )ekl + mi i=1 j=1 ˜ ˜ xij g(tij )⎦ i=1 j=1 k=1 l=1 n ⎤ ˜ xij N(n) n mi n mi Wnkl (tij )ekl + k=1 l=1 i=1 j=1 ⎤ ˜ xij ˜ g(tij )⎦ N(n) Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 n mi i=1 j=1 −1 S2 n N(n) From A2(ii), Page 13 of 18 = O(1) By Remark 2.2, we have ||˜ ij || x = O(1) N(n) and max 1≤i≤n,1≤j≤mi ⎛ 1⎞ ||˜ ij || x = o ⎝n ⎠ N(n) (4:2) According to (4.2) and Remark 3.1, we have mi n ||D1n || ≤ C i=1 j=1 ||˜ ij || x eij = o(1), N(n) (4:3) a.s By A3(i-ii), (4.2), Lemma 3.4 or Corollary 3.1, we have n ||D2n || ≤ C mi mi n Wnkl (tij )ekl · max 1≤i≤n,1≤j≤mi i=1 j=1 k=1 l=1 ||˜ ij || x = o(1) N(n) a.s (4:4) From A2(iii) and A3(iii), we obtain mi n max 1≤i≤n,1≤j≤mi ˜ g(tij ) = max 1≤i≤n,1≤j≤mi g(tij ) − n ≤ + mi Wnkl (tij )(g(tij ) − g(tkl ))I(|tij − tkl | > ε) max 1≤i≤n,1≤j≤mi (4:5) k=1 l=1 n mi Wnkl (tij )(g(tij ) − g(tkl ))I(|tij − tkl | ≤ε) = o(1) max 1≤i≤n,1≤j≤mi Wnkl (tij )g(tkl ) k=1 l=1 k=1 l=1 Together with (4.2), one gets n ||D3n || ≤ C max 1≤i≤n,1≤j≤mi mi ||˜ ij || x = o(1) N(n) |˜ (tij )| · g i=1 j=1 (4:6) By (4.1), (4.3), (4.4) and (4.6), (2.5) holds Proof of Theorem 2.2 From (1.1) and (2.3), we have mi n ˆ gn (t) − g(t) = ˆ Wnij (t)(yij − xT βn ) − g(t) ij i=1 j=1 mi n = ˆ Wnij (t) (yij − xT βn ) − (yij − xT β) + ij ij i=1 j=1 mi n = mi = mi Wnij (t)(yij − xT β) − g(t) ij i=1 j=1 ˆ Wnij (t)xT (β − βn ) + ij i=1 j=1 n n n mi Wnij (t)(g(tij ) + eij ) − g(t) (4:7) i=1 j=1 ˆ Wnij (t)xT (β − βn ) + ij i=1 j=1 n mi ˜ Wnij (t)eij − g(t) i=1 j=1 =: E1n + E2n + E3n By A3(iv) and (2.5), one gets n mi |E1n | ≤ ˆ Wnij (t)xij ||β − βn || = o(1), a.s (4:8) i=1 j=1 By Lemma 3.4 or Corollary 3.1, E2n = o(1), a.s.; With arguments similar to (4.5), we have E3n = o(1) Therefore, together with (4.7) and (4.8), (2.6) holds Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page 14 of 18 Proof of Theorem 2.3 Here, we still use (4.7), but Ein in (4.7) are replaced by Ein(t) for i = 1,2 and By A3(v) and (2.5), we get n mi sup |E1n (t)| ≤ sup 0≤t≤1 0≤t≤1 ˆ Wnij (t)xij ||β − βn || = o(1), a.s i=1 j=1 By Lemma 3.5 or Corollary 3.2, sup0≤t≤1 |E2n(t)| = o(1), a.s.; Similar to the arguments in (4.5), we have sup0≤t≤1 |E2n(t)| = o(1) Hence, (2.7) is proved Simulation study ˆ To evaluate the finite-sample performance of the least squares estimator βn and the ˆ nonparametric estimator gn (t), we respectively take two forms of functions for g(·): I g(t) = exp(3t); II 3π t , g(t) = cos consider the case where p = and mi = m = 12, and take the design points tij = ((i 1)m + j)/(nm), xij ~ N(1, 1) and the errors eij = 0.2ei, j-1 + ij, where ij are i.i.d N(0,1) random variables, and ei,0 ~ N(0,1) for each i The kernel function is taken as the Epanechnikov kernel K(t) = (1 − t2 )I(|t| ≤ 1), and the weight function is given by Nadaraya-Watson kernel weight t − tij t − tij mi The bandwidth h is selected by a “leaveWij (t) = K / n i=1 j=1 K hn hn one-subject-out” cross validation method In the simulations, we draw B = 1000 random samples of sizes 150,200,300 and 500 for b = 2, respectively We obtain the estiˆ ˆ (b) ˆ mators βn and gn (t) from (2.2) and (2.3), respectively Let βn be bth least squares ˆ estimator of b under the size n Some numerical results for βn are computed by ¯ βn = B B ˆ SD(βn ) = ˆ (b) βn , b=1 ˆ ¯ Bias(βn ) = β − β, B−1 ˆ MSE(βn ) = B 1/2 ˆ ¯ (βn − β) (b) , b=1 B−1 B ˆ (βn − β) , (b) b=1 which are listed in Table In addition, for assessing estimator of the nonparametric component g(·), we study the square root of mean-squared errors (RMSE) based on 1000 repetitions Denote (b) (b) ¯ ˆ ˆ ˆ gn (t) be the bth estimator of g(t) under the size n, and gn (t) = B gn (t)/B be the b=1 average estimator of g(t) We compute RMSEn = M M 1/2 ¯ ˆ (gn (ts ) − g(ts )) , s=1 and (b) RMSEn = M 1/2 M (b) (ˆ n (ts ) g s=1 − g(ts )) , b = 1, 2, , B, Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page 15 of 18 Table The estimators of b and some indices of their accuracy for the different sample size n and nonparametric function g(·) ¯ ˆ ˆ ˆ g(·) n βn MSE(βn ) SD(βn ) Bias(βn ) exp(3t) -0.00062 0.0257 0.00066 1.99960 -0.00040 0.0221 0.00049 300 500 1.99965 1.99963 -0.00035 -0.00037 0.0172 0.0133 0.00030 0.00018 150 1.99908 -0.00092 0.0238 0.00056 200 1.99945 -0.00055 0.0206 0.00043 300 500 3π t 1.99938 200 cos 150 1.99950 1.99969 -0.00050 -0.00031 0.0168 0.0131 0.00028 0.00017 where {ts, s = 1, , M} is a sequence of regular grid points on [0, 1] Figures and respectively provide the average estimators of the nonparametric function g(·) and RMSE n values for Cases I and II, respectively The boxplots for (b) RMSEn (b = 1, 2, , B) values for Cases I and II are presented in Figure ˆ ˆ ˆ From Table 1, we see that (i) |Bias(βn )|, SD(βn ) and MSE(βn ) decrease with ¯ increasing the sample size n; (ii) the larger the sample size n is, the closer the βn is to the true value From Figures 1, and 3, we observe that the biases of estimators of the nonparametric component g(·) decrease as the sample size n increases These show that, for semiparametric partially linear regression models for longitudinal data based on mixing error’s structure, the least squares estimator of parametric component b and the estimator of nonparametric component g(·) work well n=150 n=200 20 20 RMSE150=0.7530 10 RMSE200=0.6698 15 g(t) g(t) 15 10 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 n=300 0.7 0.8 0.9 0.6 0.7 0.8 0.9 20 RMSE =0.5676 300 RMSE500=0.4580 15 g(t) 15 g(t) 0.6 n=500 20 10 0.5 t 10 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 t Figure Estimators of the nonparametric component g(·) for the case I: gn (·) (dashed curve) and g ˆ (·) (solid curve) Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 Page 16 of 18 n=150 n=200 1 RMSE150=0.0860 −0.5 −1 RMSE =0.0760 200 0.5 g(t) g(t) 0.5 −0.5 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 −1 0.1 0.2 0.3 0.4 n=300 0.7 0.8 0.9 1 RMSE300=0.0618 RMSE500=0.0473 0.5 g(t) 0.5 g(t) 0.6 n=500 −0.5 −1 0.5 t −0.5 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 −1 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 Figure Estimators of the nonparametric component g(·) for the case II: gn (·) (dashed curve) and ˆ g(·) (solid curve) Concluding remarks An inherent characteristic of longitudinal data is the dependence among the observations within the same subject For exhibiting dependence among the observations within the same subject, we consider the estimation problems of partially linear models for longitudinal data with the -mixing and r-mixing error structures, respectively ˆ The strong consistency for least squares estimator βn of parametric component b is g(t)=cos(3πt/2) g(t)=exp(3t) 0.8 0.16 0.75 0.14 0.7 0.12 RMSE RMSE 0.65 0.6 0.1 0.08 0.55 0.06 0.5 0.04 0.45 n=150 n=200 n=300 n=500 (b) Figure The boxplots of RMSE(b) (b n n=150 n=200 n=300 = 1, 2, , B) values in the estimators of g(·) n=500 Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 studied In addition, the strong consistency and uniform consistency for the estimator ˆ gn (·) of nonparametric function g(·) are investigated under some mild conditions In the paper, we only consider (xT , tij ) are known and nonrandom design points, as ij Baek and Liang [16], and Liang and Jing [20] In the monograph of Hardle et al [7], they respectively considered the two cases: the fixed design and the random design, to study non-longitudinal partially linear regression models Our results can also be extended to the case of (xT , tij ) being random The interested readers can consider the ij work In addition, we consider partially linear models for longitudinal data with only -mixing and r-mixing In fact, our results with other mixing-dependent structures, such as a-mixing, *-mixing and r*-mixing, can also be obtained by the same arguments in our paper At present, we have not given the asymptotic normality of estimators, since some details need further discussion We will devote to establish the ˆ ˆ asymptotic normality of βn and gn (·) in our future work Acknowledgements The authors are grateful to an Associate Editor and two anonymous referees for their constructive suggestions that have greatly improved this paper This work is partially supported by NSFC (no 11171065), Anhui Provincial Natural Science Foundation (no 11040606M04), NSFJS (no BK2011058) and Youth Foundation for Humanities and Social Sciences Project from Ministry of Education of China (no 11YJC790311) Author details Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of China 2Department of Mathematics and Computer Science, Tongling University, Tongling 244000, Anhui, People’s Republic of China Authors’ contributions The two authors contributed equally to this work All authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Received: 26 July 2011 Accepted: 17 November 2011 Published: 17 November 2011 References Diggle, LD, Heagerty, P, Liang, K, Zeger, S: Analysis of Longitudinal Data Oxford University Press, New York, (2002) Fan, J, Li, R: New estimation and model selection procedure for semiparametric modeling in longitudinal data analysis J Am Stat Assoc 99, 710–723 (2004) doi:10.1198/016214504000001060 Engle, R, Granger, C, Rice, J, Weiss, A: Nonparametric estimates of the relation between weather and electricity sales J Am Stat Assoc 81, 310–320 (1986) doi:10.2307/2289218 Heckman, N: Spline smoothing in a partly linear models J R Stat Soc B 48, 244–248 (1986) Speckman, P: Kernel smoothing in partial linear models J R Stat Soc B 50, 413–436 (1988) Robinson, PM: Root-n-consistent semiparametric regression Econometrica 56, 931–954 (1988) doi:10.2307/1912705 Härdle, W, Liang, H, Gao, JT: Partial Linear Models Physica-Verlag, Heidelberg (2000) Zeger, SL, Diggle, PL: Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters Biometrics 50, 689–699 (1994) doi:10.2307/2532783 Moyeed, RA, Diggle, PJ: Rate of convergence in semiparametric modeling of longitudinal data Aust J Stat 36, 75–93 (1994) doi:10.1111/j.1467-842X.1994.tb00640.x 10 Zhang, D, Lin, X, Raz, J, Sowerm, MF: Semiparametric stochastic mixed models for longitudinal data J Am Stat Assoc 93, 710–719 (1998) doi:10.2307/2670121 11 You, JH, Zhou, X: Partially linear models and polynomial spline approximations for the analysis of unbalanced panel data J Stat Plan Inference 139, 679–695 (2009) doi:10.1016/j.jspi.2007.04.037 12 You, JH, Zhou, X, Zhou, Y: Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors J Multivar Anal 101, 1079–1101 (2010) doi:10.1016/j.jmva.2010.01.003 13 Schick, A: An adaptive estimator of the autocorrelation coefficient in regression models with autocoregressive errors J Time Ser Anal 19, 575–589 (1998) doi:10.1111/1467-9892.00109 14 Gao, JT, Anh, VV: Semiparametric regression under long-range dependent errors J Stat Plan Inference 80, 37–57 (1999) doi:10.1016/S0378-3758(98)00241-9 15 Sun, XQ, You, JH, Chen, GM, Zhou, X: Convergence rates of estimators in partial linear regression models with MA(∞) error process Commun Stat Theory Methods 31, 2251–2273 (2002) doi:10.1081/STA-120017224 16 Baek, J, Liang, HY: Asymptotics of estimators in semi-parametric model under NA samples J Stat Plan Inference 136, 3362–3382 (2006) doi:10.1016/j.jspi.2005.01.008 17 Zhou, XC, Liu, XS, Hu, SH: Moment consistency of estimators in partially linear models under NA samples Metrika 72, 415–432 (2010) doi:10.1007/s00184-009-0260-5 Page 17 of 18 Zhou and Lin Journal of Inequalities and Applications 2011, 2011:112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112 18 Li, GL, Liu, LQ: Strong consistency of a class estimators in partial linear model under martingale difference sequence Acta Math Sci (Ser A) 27, 788–801 (2007) 19 Chen, X, Cui, HJ: Empirical likelihood inference for partial linear models under martingale difference sequence Stat Probab Lett 78, 2895–2910 (2008) doi:10.1016/j.spl.2008.04.012 20 Liang, HY, Jing, BY: Asymptotic normality in partial linear models based on dependent errors J Stat Plan Inference 139, 1357–1371 (2009) doi:10.1016/j.jspi.2008.08.005 21 He, X, Zhu, ZY, Fung, WK: Estimation in a semiparametric model for longitudinal data with unspecified dependence structure Biometrika 89, 579–590 (2002) doi:10.1093/biomet/89.3.579 22 Xue, LG, Zhu, LX: Empirical likelihood-based inference in a partially linear model for longitudinal data Sci China (Ser A) 51, 115–130 (2008) doi:10.1007/s11425-008-0020-4 23 Li, GR, Tian, P, Xue, LG: Generalized empirical likelihood inference in semipara-metric regression model for longitudinal data Acta Math Sin (Engl Ser) 24, 2029–2040 (2008) doi:10.1007/s10114-008-6434-7 24 Bai, Y, Fung, WK, Zhu, ZY: Weighted empirical likelihood for generalized linear models with longitudinal data J Multivar Anal 140, 3445–3456 (2010) 25 Hu, Z, Wang, N, Carroll, RJ: Profile-kernel versus backfitting in the partially linear models for longitudinal/clustered data Biometrika 91, 251–262 (2004) doi:10.1093/biomet/91.2.251 26 Wang, S, Qian, L, Carroll, RJ: Generalized empirical likelihood methods for analyzing longitudinal data Biometrika 97, 79–93 (2010) doi:10.1093/biomet/asp073 27 Chi, EM, Reinsel, GC: Models for longitudinal data with random effects and AR(1) errors J Am Stat Assoc 84, 452–459 (1989) doi:10.2307/2289929 28 Shao, QM: A moment inequality and its application Acta Math Sin 31, 736–747 (1988) 29 Peligrad, M: The r-quick version of the strong law for stationary φ-mixing sequences Proceedings of the International Conference on Almost Everywere Convergence in Probability and Statistics pp 335–348.Academic Press, New York (1989) 30 Utev, SA: Sums of random variables with φ-mixing Sib Adv Math 1, 124–155 (1991) 31 Kiesel, R: Summability and strong laws for φ-mixing random variables J Theor Probab 11, 209–224 (1998) doi:10.1023/ A:1021655227120 32 Chen, PY, Hu, TC, Volodin, A: Limiting behaviour of moving average processes under φ-mixing assumption Stat Probab Lett 79, 105–111 (2009) doi:10.1016/j.spl.2008.07.026 33 Zhou, XC: Complete moment convergence of moving average processes under φ-mixing assumptions Statist Probab Lett 80, 285–292 (2010) doi:10.1016/j.spl.2009.10.018 34 Peligrad, M: On the central limit theorem for p-mixing sequences of random variables Ann Probab 15, 1387–1394 (1987) doi:10.1214/aop/1176991983 35 Peligrad, M, Shao, QM: Estimation of variance for p-mixing sequences J Multivar Anal 52, 140–157 (1995) doi:10.1006/ jmva.1995.1008 36 Peligrad, M, Shao, QM: A note on estimation of the variance of partial sums for p-mixing random variables Stat Probab Lett 28, 141–145 (1996) 37 Shao, QM: Maximal inequalities for partial sums of p-mixing sequences Ann Probab 23, 948–965 (1995) doi:10.1214/ aop/1176988297 38 Bradley, R: A stationary rho-mixing Markov chain which is not “interlaced” rho-mixing J Theor Probab 14, 717–727 (2001) doi:10.1023/A:1017545123473 39 Lin, ZY, Lu, CR: Limit Theory for Mixing Dependent Random Variables Science Press/Kluwer Academic Publishers, Beijing/London (1996) 40 Roussas, GG: Nonparametric regression estimation under mixing conditions Stoch Process Appl 36, 107–116 (1990) doi:10.1016/0304-4149(90)90045-T 41 Truong, YK: Nonparametric curve estimation with time series errors J Stat Plan Inference 28, 167–183 (1991) doi:10.1016/0378-3758(91)90024-9 42 Fraiman, R, Iribarren, GP: Nonparametric regression estimation in models with weak error’s structure J Multivar Anal 37, 180–196 (1991) doi:10.1016/0047-259X(91)90079-H 43 Roussas, GG, Tran, LT: Asymptotic normality of the recursive kernel regression estimate under dependence conditions Ann Stat 20, 98–120 (1992) doi:10.1214/aos/1176348514 44 Masry, E, Fan, JQ: Local polynomial estimation of regression functions for mixing processes Scand J Stat 24, 165–179 (1997) doi:10.1111/1467-9469.00056 45 Aneiros, G, Quintela, A: Asymptotic properties in partial linear models under dependence Test 10, 333–355 (2001) doi:10.1007/BF02595701 46 Fan, JQ, Yao, QW: Nonlinear Time Series–Nonparametric and Parametric methods Springer, New York (2003) 47 Chen, GM, You, JH: An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models Stat Pap 46, 173–193 (2005) doi:10.1007/BF02762967 doi:10.1186/1029-242X-2011-112 Cite this article as: Zhou and Lin: Strong consistency of estimators in partially linear models for longitudinal data with mixing-dependent structure Journal of Inequalities and Applications 2011 2011:112 Page 18 of 18 ... consider partially linear models for longitudinal data with only -mixing and r-mixing In fact, our results with other mixing-dependent structures, such as a-mixing, *-mixing and r*-mixing, can... observations within the same subject For exhibiting dependence among the observations within the same subject, we consider the estimation problems of partially linear models for longitudinal data with. .. properties of the estimators In practice, a great deal of the data in econometrics, engineering and natural sciences occur in the form of time series in which observations are not independent and often