Testing the Equality of Covariance Operators in Functional Samples

.  We propose a non-parametric test for the equality of the covariance structures in two functional samples. The test statistic has a chi-square asymptotic distribution with a known number of degrees of freedom, which depends on the level of dimension reduction needed to represent the data. Detailed analysis of the asymptotic properties is developed. Finite sample perfo-rmance is examined by a simulation study and an application to egg-laying curves of fruit flies.

[1]  D. Bosq.  Linear Processes in Function Spaces: Theory And Applications , 2000 .

[2]  Z. Q. John Lu,et al.  Nonparametric Functional Data Analysis: Theory And Practice , 2007, Technometrics.

[3]  Piotr Kokoszka,et al.  Inference for Functional Data with Applications , 2012 .

[4]  F. Ruymgaart,et al.  Application of a delta-method for random operators to testing equality of two covariance operators , 2011 .

[5]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[6]  Alois Kneip,et al.  Common Functional Principal Components , 2006, 0901.4252.

[7]  P. Vieu,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[8]  Matthew Reimherr,et al.  Two sample inference in functional linear models , 2009 .

[9]  F. Ferraty,et al.  The Oxford Handbook of Functional Data Analysis , 2011, Oxford Handbooks Online.

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

[11]  Mariela Sued,et al.  Testing the Equality of Covariance Operators , 2011 .

[12]  P. Reiss,et al.  Functional Principal Component Regression and Functional Partial Least Squares , 2007 .

[13]  Bernard W. Silverman,et al.  Functional Data Analysis , 1997 .

[14]  John H. Maddocks,et al.  Second-Order Comparison of Gaussian Random Functions and the Geometry of DNA Minicircles , 2010 .

[15]  Piotr Kokoszka,et al.  Tests for Error Correlation in the Functional Linear Model , 2010 .

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

[17]  Denis Bosq,et al.  Linear Processes in Function Spaces , 2000 .

[18]  Ron Reeder,et al.  Estimation of the mean of functional time series and a two‐sample problem , 2011, 1105.0019.

[19]  J. Dauxois,et al.  Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference , 1982 .

[20]  D. Gervini.  Robust functional estimation using the median and spherical principal components , 2008 .

[21]  Spencer Graves,et al.  Functional Data Analysis with R and MATLAB , 2009 .

[22]  H. Müller,et al.  Functional quadratic regression , 2010 .