Principal components of random variables with values in a seperable hilbert space
- 1 January 1973
- journal article
- research article
- Published by Taylor & Francis in Mathematische Operationsforschung und Statistik
- Vol. 4 (5) , 391-406
- https://doi.org/10.1080/02331887308801137
Abstract
We extend the theory of principal components to random variabies X with values in a separable HILBERT space and prove optima! properties well-known for finite-dimen-sional spaces, Further we give an estimate of the variance operator D from a series of independent, observations and prove the strong consistency if the estimate for nuclear D.In the case D has single eigenvalues we find that with probability 1 the limiting properties of the sample principal components computed from such an estimate of D are those of the exact principal components.Keywords
This publication has 6 references indexed in Scilit:
- Introduction to Optimization Theory in a Hilbert SpacePublished by Springer Nature ,1971
- Estimation of Heteroscedastic Variances in Linear ModelsJournal of the American Statistical Association, 1970
- Minimization of Eigenvalues of a Matrix and Optimality of Principal ComponentsThe Annals of Mathematical Statistics, 1968
- Die Gesetze der Grossen ZahlenPublished by Springer Nature ,1968
- An Optimal Property of Principal ComponentsThe Annals of Mathematical Statistics, 1965
- A generalized inverse for matricesMathematical Proceedings of the Cambridge Philosophical Society, 1955