Two Samples Tests for Functional Data

Data in many experiments arises as curves and therefore it is natural to use a curve as a basic unit in the analysis, which is in terms of functional data analysis (FDA). Functional curves are encountered when units are observed over time. Although the whole function curve itself is not observed, a sufficiently large number of evaluations, as is common with modern recording equipment, is assumed to be available. In this article, we consider the statistical inference for the mean functions in the two samples problem drawn from functional data sets, in which we assume that functional curves are observed, that is, we consider the test if these two groups of curves have the same mean functional curve when the two groups of curves without noise are observed. The L 2-norm based and bootstrap-based test statistics are proposed. It is shown that the proposed methodology is flexible. Simulation study and real-data examples are used to illustrate our techniques.

[1]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[2]  A. M. Aguilera,et al.  Principal component estimation of functional logistic regression: discussion of two different approaches , 2004 .

[3]  Henry W. Altland,et al.  Applied Functional Data Analysis , 2003, Technometrics.

[4]  Bernard W. Silverman,et al.  Incorporating parametric effects into functional principal components analysis , 1995 .

[5]  Jane-ling Wang,et al.  Functional linear regression analysis for longitudinal data , 2005, math/0603132.

[6]  P. Diggle.  Analysis of Longitudinal Data , 1995 .

[7]  J. Ramsay.  When the data are functions , 1982 .

[8]  J. Ramsay,et al.  Principal components analysis of sampled functions , 1986 .

[9]  J. Hart,et al.  Kernel Regression Estimation Using Repeated Measurements Data , 1986 .

[10]  Gareth M. James.  Generalized linear models with functional predictors , 2002 .

[11]  Ricardo Fraiman,et al.  An anova test for functional data , 2004, Comput. Stat. Data Anal..

[12]  J. Faraway.  Regression analysis for a functional response , 1997 .

[13]  H. Muller,et al.  Generalized functional linear models , 2005, math/0505638.

[14]  P. Diggle,et al.  Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. , 1994, Biometrics.

[15]  B. Silverman,et al.  Canonical correlation analysis when the data are curves. , 1993 .

[16]  P. Sarda,et al.  Functional linear model , 1999 .

[17]  Alois Kneip,et al.  Nonparametric-estimation of Common Regressors for Similar Curve Data , 1994 .

[18]  Jianqing Fan,et al.  Test of Significance When Data Are Curves , 1998 .

[19]  B. Silverman,et al.  Estimating the mean and covariance structure nonparametrically when the data are curves , 1991 .

[20]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[21]  Sd Pezzulli,et al.  Some properties of smoothed principal components analysis for functional data , 1993 .

[22]  André Mas,et al.  Testing hypotheses in the functional linear model , 2003 .

[23]  J. Overstreet,et al.  Relationship of serum estradiol and progesterone concentrations to the excretion profiles of their major urinary metabolites as measured by enzyme immunoassay and radioimmunoassay. , 1991, Clinical chemistry.

[24]  Jin-Ting Zhang.  Approximate and Asymptotic Distributions of Chi-Squared–Type Mixtures With Applications , 2005 .

[25]  J. Rice,et al.  Smoothing spline models for the analysis of nested and crossed samples of curves , 1998 .

[26]  Jin-Ting Zhang,et al.  Statistical inferences for functional data , 2007, 0708.2207.

[27]  James O. Ramsay,et al.  A Functional Data Analysis of the Pinch Force of Human Fingers , 1995 .

[28]  J. Ramsay,et al.  Some Tools for Functional Data Analysis , 1991 .