Oracle-efficient confidence envelopes for covariance functions in dense functional data

Abstract

Abstract:We consider nonparametric estimation of the covariance function for dense functional data using computationally efficient tensor product B-splines. We develop both local and global asymptotic distributions for the proposed estimator, and show that our estimator is as efficient as an "oracle" estimator where the true mean function is known. Simultaneous confidence envelopes are developed based on asymptotic theory to quantify the variability in the covariance estimator and to make global inferences on the true covariance. Monte Carlo simulation experiments provide strong evidence that corroborates the asymptotic theory. Two real data examples on the near infrared spectroscopy data and speech recognition data are also provided to illustrate the proposed method.

Multi‐step‐ahead prediction interval for locally stationary time series with application to air pollutant concentration data

“…B‐spline is widely used in nonparametrics for its computational simplicity and derivable asymptotic theory (Cai et al, 2019; Cao et al, 2012, 2016; Gu et al, 2014; Gu & Yang, 2015; Liu & Yang, 2010, 2016; Song & Yang, 2009; Wang & Yang, 2007, 2009; Wang et al, 2020; Xue & Yang, 2006) for its applications in different scenarios. To describe the spline functions, we denote by a sequence of equally spaced points, , called interior knots, which divides the interval [0, 1] into equal subintervals .…”

“…B-spline is widely used in nonparametrics for its computational simplicity and derivable asymptotic theory (Cai et al, 2019;Cao et al, 2012Cao et al, , 2016Gu et al, 2014;Gu & Yang, 2015;Liu & Yang, 2010Song & Yang, 2009;Wang & Yang, 2007, 2009Wang et al, 2020;Xue & Yang, 2006) for its applications in different scenarios. To describe the spline functions, we denote by t ℓ f g Js ℓ¼1 a sequence of equally spaced points,…”

Universal Local Linear Kernel Estimators in Nonparametric Regression

“…Moreover, in the problem of estimating the mean function, it is customary to subdivide design elements into certain types depending on the density of filling with the design points the regression function domain. The literature focuses on two types of data: or the design is in some sense "sparse" (for example, the number of design elements in each series is uniformly limited [37,38,40,56,57]), or the design is somewhat "dense" (the number of elements in each series grows with the number of series [37,40,44,57,58]). Theorem 2 considers the second of the specified types of design under condition (D 0 ) in each of the independent series.…”

“…, and (61):|E r 2 n,h ( f , t) − Er * 2 n,h ( f , t)| ≤ E| r n,h ( f , t) − r * n,h ( f , t)|| r n,h ( f , t) + r * n,h ( f , t)| ≤ C * 10 f 2 h −1 Eδ n ,which completes the proof.Proof of Proposition 3.From the definition of β n,i (t) in(32) it follows that, for any t ∈[0, 1], (t)(z n:i − t)K h (t − z n:i )∆z ni = 0, (t)(z n:i − t) 2 K h (t − z n:i )∆z ni = D −1 n (t)(w 2 n2 (t) − w n3 (t)w n1 (t)) =: B n (t),where D n (t) := w n0 (t)w n2 (t) − w 2 n1 (t). Expanding the function f (•) by the Taylor formula in a neighborhood of the point t (up to the second derivative), from the above identities we obtain, using(32),(58), and Lemma 1, that for any point t we haveBias f n,h (t) = EI(δ n ≤ c * h) {β n,i (t)( f (z n:i ) − f (t))K h (t − z n:i )∆z ni } + f (t)P(δ n > c * h) = f (t) 2 EI(δ n ≤ c * h)B n (t) + f (t)P(δ n > c * h) + o(h 2 ) = f (t) 2 B 0 (t) + O(Eδ n /h) + o(h 2 ); (62)moreover, the Oand o-symbols on the right-hand side of (62) are uniform in t. Note that B 0 (t) = O(h 2 ) holds for any t.Next, since for j = 1, 2 we have |w j (t)|w −1 0 (t) ≤ h j and |w nj (t)|w −1 n0 (t) ≤ h j for all natural n, the following asymptotic representation holds:(z n:i ) − f (t))K h (t − z n:i )∆z ni = − f (t)E w n1 (t) w n0 (t) I(δ n ≤ c * h) + f (t) 2 E w n2 (t) w n0 (t) I(δ n ≤ c * h) + O(hP(δ n > c * h)) + o(h 2 ) Eδ n ) + o(h 2 ). (63)Proof of Corollary 3.…”