On convergence of sample and population Hilbertian functional principal components

Document Type : Research Paper


1 Department of Statistics‎, ‎Shiraz University and Department of Statistics and Operations Research‎, ‎Kuwait‎ ‎University‎, ‎State of Kuwait.

2 Department of Statistics‎, ‎Shiraz University‎, ‎Shiraz‎, ‎Iran.


In this article we consider the sequences of sample and population covariance operators for a sequence of arrays of Hilbertian random elements. Then under the assumptions that sequences of the covariance operators norm are uniformly bounded and the sequences of the principal component scores are uniformly sumable, we prove that the convergence of the sequences of covariance operators would imply the convergence of the corresponding sequences of the sample and
population eigenvalues and eigenvectors, and vice versa. In particular we prove that the principal component scores converge in distribution in certain family of Hilbertian elliptically contoured distributions.


Main Subjects

‎R.A‎. ‎Al-Jarallah‎, ‎A.R‎. ‎Soltani and A.N‎. ‎Al-Kandari‎, ‎ On continuity of the Pearson statistic and sample quantiles‎, ‎ Ann‎. ‎Inst‎. ‎Statist‎. ‎Math.‎ 58 (2006)‎, ‎no‎. ‎3‎, ‎527--535‎ .
‎F‎. ‎Alqallaf‎, ‎A.R‎. ‎Soltani and N‎. ‎Al-Kandar‎, ‎ Continuity and analysis of sequences of principal components‎, ‎ Comm‎. ‎Statist‎. ‎Theory Methods39 (2010)‎, ‎no‎. ‎19‎, ‎3558--3567‎.
‎T.W‎. ‎Anderson‎, ‎ Asymptotic theory for principal component‎ ‎analysis‎, ‎ Ann‎. ‎Math‎. ‎Statist.‎ 34 (1963) 122--148‎.
‎D‎. ‎Bosq‎, ‎ Linear Process in Function Spaces‎, ‎Theory and Applications‎, ‎ Springer-Verlag‎, ‎Berlin‎, ‎2000‎.
‎L‎. ‎Debnath and P‎. ‎Mikusinski‎, ‎ Introduction to Hilbert Spaces with Applications‎, ‎ Academic Press‎, ‎1990‎.
‎P.J‎. ‎Diggle‎, ‎P‎. ‎Heagerty‎, ‎K.Y‎. ‎Liang and S.L‎. ‎Zeger‎, ‎ Analysis of Longitudinal Data‎, ‎ Oxford Univ‎. ‎Press‎, ‎New York‎, ‎2002‎.
‎I.M‎. ‎Gelfand and N‎. ‎Ya‎. ‎Vilenkin‎, ‎ Generalized Functions‎, ‎ Applications of Harmonic Analysis‎, ‎Vol‎. ‎4‎, ‎Academic‎ ‎press,‎ ‎1964‎.
‎I.T‎. ‎Jolliffe and N‎. ‎Ya‎. ‎Vilenkin‎, ‎ Principal Component Analysis‎, ‎Springer‎, ‎2nd edition‎, ‎New York‎, ‎2002‎.
‎T‎. ‎Kato‎, ‎ Perturbation Theory for Linear Operators‎, ‎ Springer-verlag‎, ‎Berlin‎, ‎1970‎.
‎Y‎. ‎Li‎, ‎ A note on Hilbertian elliptically contoured‎ ‎distribution‎, ‎ Technical Report‎, ‎Iowa State University‎, ‎2007‎.
‎Y‎. ‎Li‎, ‎N‎. ‎Wang and R.J‎. ‎Caroll‎, ‎ Selecting the number of principal components in functional data‎, ‎ J‎. ‎Amer‎. ‎Statist‎. ‎Assoc.‎ 108 (2013)‎, ‎no‎. ‎504‎, ‎1284--1294‎.
‎J.O‎. ‎Ramsay and B.W‎. ‎Silverman‎, ‎ Applied‎ ‎Functional Data Analysis‎, ‎ Springer‎, ‎New York‎, ‎2002‎.
‎J.O‎. ‎Ramsay and B.W‎. ‎Silverman‎, ‎ Applied Functional Data Analysis‎, ‎ Springer‎, ‎New York‎, ‎2005‎.
‎J.O‎. ‎Ramsay‎, ‎G‎. ‎Hooker and S‎. ‎Graves‎, ‎ Functional Data Analysis with R and MATLAB‎, ‎ Springer‎, ‎New York‎, ‎2009‎.
‎J.R‎. ‎Schott‎, ‎ Matrix Analysis for Statistics‎, ‎ John Wiley‎, ‎2nd edition‎, ‎New York‎, ‎2005‎.
Volume 43, Issue 2
April 2017
Pages 467-475
  • Receive Date: 04 July 2015
  • Revise Date: 30 November 2015
  • Accept Date: 01 December 2015
  • First Publish Date: 01 April 2017