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The difficulties of estimating and representing the distributions offunctional data mean that principal component methods play a substantiallygreater role in functional data analysis than in more conventionalfinite-dimensional settings. Local maxima and minima in principal componentfunctions are of direct importance; they indicate places in the domain of arandom function where influence on the function value tends to be relativelystrong but of opposite sign. We explore statistical properties of therelationship between extrema of empirical principal component functions, andtheir counterparts for the true principal component functions. It is shown thatempirical principal component funcions have relatively little trouble capturingconventional extrema, but can experience difficulty distinguishing a``shoulder'' in a curve from a small bump. For example, when the true principalcomponent function has a shoulder, the probability that the empirical principalcomponent function has instead a bump is approximately equal to 1/2. We suggestand describe the performance of bootstrap methods for assessing the strength ofextrema. It is shown that the subsample bootstrap is more effective than thestandard bootstrap in this regard. A ``bootstrap likelihood'' is proposed formeasuring extremum strength. Exploratory numerical methods are suggested.

Assessing extrema of empirical principal component functions