Statistical inferences for functional data

Abstract

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.Comment: Published at http://dx.doi.org/10.1214/009053606000001505 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A comparison of tests for the one-way ANOVA problem for functional data

“…The L 2 -norm-based test (Faraway 1997;Zhang and Chen 2007) adopted for (1) uses the test statistic S n = T SSR n (t)dt. Under the null hypothesis (1), it can be shown that S n ∼ βχ 2 d approximately, where s, u)γ (u, t)du.…”

“…5). When the k samples are not Gaussian and when the sample sizes are small, the L 2 -norm-based test and the F-type test are not preferred (see Zhang 2013). In this case, the bootstrap versions of these tests can be used to bootstrap the p-values of S n and F n (see Zhang 2013, p. 159 …”

“…However, the design time points may in some situations be different for different individual functions. To implement the tests numerically for such situations, one first has to reconstruct the k functional samples from the k observed discrete functional samples using some smoothing technique, then discretize each individual function of the k reconstructed functional samples on a common grid of time points, and finally apply the tests accordingly (see Zhang 2013, or Zhang andLiang 2014, for more details).…”

“…These data sets are commonly used to illustrate the use of statistical methods for real functional data (see Abramovich et al 2004;Ramsay and Silverman 2005;Zhang 2013;Zhang and Liang 2014). The Canadian temperature data are available in the R package fda, and the orthosis data can be downloaded from "http://www.stat.nus.edu.sg/~zhangjt/books/Chapman/FANOVA.…”

“…We would like to check statistically whether location has an effect on the mean temperature curves of the Eastern, Western and Northern weather stations. This problem was considered by Zhang (2013), and it is equivalent to the one-way ANOVA problem for functional data with k = 3. To solve this problem, we applied all of the tests under consideration, obtaining the results given in Table 5 (the estimates of the parameters of the L 2 N, L 2 B, FN, FB and GPF tests are given in Table 20 in the Supplementary Materials).…”