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**The delta method for analytic functions of random operators with application to functional data.**
*(English)*
Zbl 1129.62011

Summary: The asymptotic distributions of estimators for the regularized functional canonical correlation and variates of the population are derived. The method is based on the possibility of expressing these regularized quantities as the maximum eigenvalue and the corresponding eigenfunctions of an associated pair of regularized operators, similar to the Euclidean case. The known weak convergence of the sample covariance operator, coupled with a delta-method for analytic functions of covariance operators, yields the weak convergence of the pair of associated operators. From the latter weak convergence, the limiting distributions of the canonical quantities of interest can be derived with the help of some further perturbation theory.

### MSC:

62E20 | Asymptotic distribution theory in statistics |

62H20 | Measures of association (correlation, canonical correlation, etc.) |

46N30 | Applications of functional analysis in probability theory and statistics |

### Keywords:

delta-method for analytic functions of covariance operators; perturbation theory; regularization of operators; regularized functional canonical correlation and variates; weak convergence
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\textit{J. Cupidon} et al., Bernoulli 13, No. 4, 1179--1194 (2007; Zbl 1129.62011)

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