Testing for the rank of a covariance operator
نویسندگان
چکیده
How can we discern whether the covariance operator of a stochastic process is reduced rank, and if so, what its precise rank is? And how do so at given level confidence? This question central to great deal methods for functional data, which require low-dimensional representations by PCA or other methods. The difficulty that determination be made on basis i.i.d. replications observed discretely with measurement error contamination. adds ridge empirical covariance, obfuscating underlying dimension. We build matrix-completion inspired test statistic circumvents this issue measuring best possible least square fit covariance’s off-diagonal elements, optimised over covariances finite rank. For fixed grid sufficiently large size, determine statistic’s asymptotic null distribution as number grows. then use it construct bootstrap implementation stepwise testing procedure controlling familywise rate corresponding collection hypotheses formalising hand. Under minimal regularity assumptions, prove consistent valid. smoothing associated parameters, indifferent heteroskedasticity, does not assume low-noise regime. An extensive simulation study reveals an excellent practical performance, stably across wide range settings further illustrated means two data analyses.
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ژورنال
عنوان ژورنال: Annals of Statistics
سال: 2022
ISSN: ['0090-5364', '2168-8966']
DOI: https://doi.org/10.1214/22-aos2238