Hill estimator of projections of functional data on principal components

ABSTRACT

Functional principal component scores are commonly used to reduce mathematically infinitely dimensional functional data to finite dimensional vectors. In certain applications, most notably in finance, these scores exhibit tail behaviour consistent with the assumption of regular variation. Knowledge of the index of the regular variation, α, is needed to apply methods of extreme value theory. The most commonly used method of the estimation of α is the Hill estimator. We derive conditions under which the Hill estimator computed from the sample scores is consistent for the tail index of the unobservable population scores.

KEYWORDS:

• Functional data
• Hill estimator
• principal components

AMS CLASSIFICATIONS:

• 62G32
• 62H25

Acknowledgments

We thank the Associate Editor and two referees for reading the paper carefully and pointing out ways of improving the exposition.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research has been partially supported by National Science Foundation (NSF) grants DMS-1462067 and DMS-1737795.