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In this paper we work under the setting of data with high dimension d greater than the sample size n (HDLSS). We study asymptotics of the rst p 2 sample eigenvalues and their corresponding eigenvectors under a spiked covariance model for which its rst p largest population eigenvalues have the same asymptotic order of magnitude as d tends to innity and the rest are constant or bounded as d increases. We get the asymptotic joint distribution of the nonzero sample eigenvalues when d! 1 and the sample size n xed. We then prove that the p largest sample eigenvalues increase jointly at the same speed as their population counterpart, in the sense that the vector of ratios of the sample and population eigenvalues converges to a multivariate distribution when d!1 and n is xed, and to the vector of ones when both d;n!1 and d n. We also show the subspace consistency of the corresponding sample eigenvectors when d goes to innity and n is xed. Furthermore, using the asymptotic joint distribution of the sample eigenvalues we study some inference problems for the spiked covariance model and propose hypothesis tests for a particular case of this model and condence intervals for the p largest eigenvalues. A simulation is performed to assess the behavior of the proposed statistical methodologies.

PCA and eigen-inference for a spiked covariance model with largest eigenvalues of same asymptotic order