Estimation of the number of principal components in high‐dimensional multivariate extremes

For multivariate regularly random vectors of dimension d$$ d $$, the dependence structure of the extremes is modeled by the so‐called angular measure. When the dimension d$$ d $$ is high, estimating the angular measure is challenging because of its complexity. In this paper, we use Principal Component Analysis (PCA) as a method for dimension reduction and estimate the number of significant principal components of the empirical covariance matrix of the angular measure under the assumption of a spiked covariance structure. Therefore, we develop Akaike Information Criteria (AIC$$ \mathrm{AIC} $$) and Bayesian Information Criteria (BIC$$ \mathrm{BIC} $$) to estimate the location of the spiked eigenvalue of the covariance matrix, reflecting the number of significant components, and explore these information criteria on consistency. On the one hand, we investigate the case where the dimension d$$ d $$ is fixed, and on the other hand, where the dimension d$$ d $$ converges to ∞$$ \infty $$ under different high‐dimensional scenarios. When the dimension d$$ d $$ is fixed, we establish that the AIC$$ \mathrm{AIC} $$ is not consistent, whereas the BIC$$ \mathrm{BIC} $$ is weakly consistent. In high‐dimensional contexts, we utilize methods from random matrix theory to establish sufficient conditions ensuring the consistency of the AIC and BIC. Finally, the performance of the different information criteria is compared in a simulation study and applied to high‐dimensional precipitation data.