Let X be a second order random process indexed by a compact interval [0,T]. Assume that n independent realizations of X are observed on a fixed grid of p time points. Under mild regularity assumptions on the sample paths of X, we show the asymptotic normality of suitable nonparametric estimators of the trend function mu = EX in the space C([0,T]) as n, p go to infinity and, using Gaussian process theory, we derive approximate simultaneous confidence bands for mu.
Deep Dive into Nonparametric estimation of a trend based upon sampled continuous processes.
Let X be a second order random process indexed by a compact interval [0,T]. Assume that n independent realizations of X are observed on a fixed grid of p time points. Under mild regularity assumptions on the sample paths of X, we show the asymptotic normality of suitable nonparametric estimators of the trend function mu = EX in the space C([0,T]) as n, p go to infinity and, using Gaussian process theory, we derive approximate simultaneous confidence bands for mu.
In various application fields such as internet traffic monitoring, medical imagery, or signal processing, modern technology has allowed to collect data routinely from population samples with a high temporal and/or spatial resolution. Indeed, such datasets should be viewed as (collections of) curves or functions rather than as high-dimensional vectors; they are thus commonly termed functional data. (See [7,13] for a comprehensive introduction to functional data analysis.) Typical functional data may be modeled as observations of independent realizations X 1 , . . . , X n of a second order random process X = {X(t), t ∈ D} at fixed design points t 1 , . . . , t p , where D denotes a continuous temporal and/or spatial domain. In this framework, the observed data are
where the ε ij are mean zero random variables (r.v.) representing potential measurement errors. The trend function µ = EX often appears as a population mean response function, which motivates its inference. The nonparametric regression literature contains several results on the asymptotic properties of estimators of µ as the sample sizes n and p go to infinity. For instance when D = [0, 1], mean-square convergence rates of kernel and spline estimators can be found in [2,3,6,9]. When D is a compact metric space, [5] gives a universal consistency result as well as the asymptotic normality of all usual regression estimators in the sense of finite dimensional distributions and of the space L 2 (D), with an application to simultaneous confidence intervals. The task of building (nonparametric and simultaneous) confidence bands for µ, which proves useful in various problems of prediction, model diagnostic, or calibration (e.g. [1,11]), has received considerable attention in the classical regression setting (e.g. [8,15]) but not, to our knowledge, for functional data.
In this Note, we study the model (1) in the case where the random process X is indexed by a compact interval D = [0, T ] and has mildly regular sample paths. In Section 2, we state the asymptotic normality of suitable nonparametric estimators of µ in the space C([0, T ]) of all continuous functions on [0, T ] as n, p → ∞. In Section 3, we use the former results to build approximate simultaneous confidence bands for µ. Finally in Section 4, some potential applications and extensions of our results are discussed.
We state here the assumptions made on the random process X and on the model (1) of Section 1.
- The random errors ε ij are mutually independent and independent of the X i ; they have mean zero and common variance σ 2 ≥ 0. (B.3) The t j are ordered (0 ≤ t 1 < . . . < t p ≤ T ) and they have a quasi-uniform repartition, i.e., writing t 0 = 0 and t p+1 = T , it holds that max 0≤j≤p (tj+1-tj) min 1≤j≤p-1 (tj+1-tj ) = O(1) as n, p → ∞. Note that (A.2’) implies (A.2). Also, (B.3) ensures that the t j are regularly spaced in [0, T ]; it is fulfilled e.g. when the t j are equally spaced or are generated by a regular probability density function (p.d.f.).
For each assumption (A.2) and (A.2’), we now introduce a suitable nonparametric estimator of µ and give its asymptotic distribution. Under (A.2), we use the interpolation-type estimator of [4], denoted by µ C , with a boundary correction. We recall here its definition. Let Y j = ( n i=1 Y ij )/n for 1 ≤ j ≤ p, and let Y (t) be the process obtained by linear interpolation of the (t j , Y j ) such that
For convenience we take K as a symmetric, compactly supported, Lipschitz-continuous p.d.f.. The real h > 0 is a fixed bandwidth. Following [12], we say that a sequence (Z n ) of random elements of C([0, T ]) converges weakly to a limit Z in C([0, T ]) if Eϕ(Z n ) → Eϕ(Z) as n → ∞ for all uniformly continuous functional ϕ on C([0, T ]) equipped with the sup-norm. We denote by R the covariance function of the process X and by G(0, C) any Gaussian process indexed by [0, T ] with mean zero and covariance C. We are now in position to state the weak convergence of µ C in C([0, T ]) as n, p → ∞ (recall that p = p(n)). Next, we address the case where X satifies (A.2’) (Hölder continuity). We consider the local linear estimator, denoted here by µ L and defined by
Remarks.
(1) The proofs of the former theorems are similar and rely on the following steps: (i) note that the estimator is linear in the data; (ii) use the functional central limit theorem 10.6 of [12] for the estimator applied to the data without noise X i (t j ); (iii) show that under condition ph 2 → ∞, the estimator applied to the errors ε ij becomes negligible in probability before n -1/2 as n, p → ∞, uniformly over [0, T ]; (iv) impose additional conditions on µ and on the joint rates of n, p and h to make the asymptotic bias of the estimator as o(n -1/2 ) uniformly over [0, T ] (in particular the rate n = o(p) used in Theorem 2.1 is only needed to control the boundary effects in the bias of µ C ). ( 2) Under (A.2), we can prove asymptotic normality in C(0, T ]) only for µ C . This i
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