Statistical inferences for functional data

By Jin-Ting Zhang, Jianwei Chen. 2007

267 citations recorded by the source corpus.

With modern technology development, functional data are being observed frequently in many scientific fields. A popular method for analyzing such functional data is ``smoothing first, then estimation.'' That is, statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another. However, little is known about this substitution effect on functional data analysis. In this paper this problem is investigated when the local polynomial kernel (LPK) smoothing technique is used for individual function reconstructions. We find that under some mild conditions, the substitution effect can be ignored asymptotically. Based on this, we construct LPK reconstruction-based estimators for the mean, covariance and noise variance functions of a functional data set and derive their asymptotics. We also propose a GCV rule for selecting good bandwidths for the LPK reconstructions. When the mean function also depends on some time-independent covariates, we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time. The LPK reconstruction-based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived. Moreover, we propose a $L^2$-norm-based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression. The effect of the bandwidths selected by the proposed GCV rule on the accuracy of the LPK reconstructions and the mean function estimator is investigated via a simulation study. The proposed methodologies are illustrated via an application to a real functional data set collected in climatology.

Subfields: Artificial Intelligence, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b22bda0357782d72b40ff9b86867fc8cac9ec828

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Citations in the corpus

Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis (2014, methodology): “The solid lines correspond to pLRT with a smooth covariance estimator, the dashed lines correspond to the LRT with known covariance matrix and the dotted lines correspond to the global L2-norm-based test of Zhang & Chen (2007), henceforth denoted ZC test.”
DECIDING THE DIMENSION OF EFFECTIVE DIMENSION REDUCTION SPACE FOR FUNCTIONAL AND HIGH-DIMENSIONAL DATA (2010, background): “In that situation, for most theoretical and practical purposes, one can fit continuous curves to the discrete-time data and then treat the fitted curves as the true functional data; see, for example, Hall, Müller and Wang (2006), Cai and Hall (2006) and Zhang and Chen (2007).”
STATISTICAL INFERENCES FOR LINEAR MODELS WITH FUNCTIONAL RESPONSES (2011, methodology): “Notice that the above ISH is the test statistic of the L2-norm based test of Zhang and Chen (2007), and”
Presmoothing in Functional Linear Regression (2012, methodology): “Another way to use presmoothing methods has been considered by Zhang and Chen (2007) in dealing with the regression model Yi(t) = Xtiθ(t) + Vi(t) + ²i(t), i = 1, . . . , n, where the covariate Xi is multidimensional and time-independent, the process Vi represents the ith individual variation, and…”
Two Samples Tests for Functional Data (2010, methodology): “With same common bandwidth and under some regular conditions, Zhang and Chen (2007) showed that the following properties are held for f̂i t i = 1 n.”
Adaptive Global Testing for Functional Linear Models (2014, methodology): “It can be shown (Hall, Müller, and Wang 2006; Zhang and Chen 2007) that when the observation is dense enough (Nhn−δ1 → ∞, N1−δ2h → ∞, with h = O(n−1/4) and positive constants δ1, δ2), the resulting estimators of individual curves and covariance operator are √ n-consistent under standard smoothness…”
One‐Way anova for Functional Data via Globalizing the Pointwise F‐test (2014, methodology): “…studies, aiming to check if the GPF test is comparable, in terms of size controlling and powers, with the L2-norm-based test (Faraway, 1997; Zhang & Chen, 2007) and the F -type test (Shen & Faraway, 2004; Zhang, 2011) adopted for the one-way ANOVA problem (2) as briefly described in…”