A0591
Title: An approximation for the quantiles of the maxima
Authors: Daniel Peer - University of Vienna (Austria) [presenting]
Moritz Jirak - University of Vienna (Austria)
Abstract: Let $X_1,\ldots, X_n \in \mathbb{R}^d$ be a sequence of i.i.d. random vectors, where $n\ll d$. A fundamental problem in high-dimensional statistics concerns normal approximations and convergence properties of the maximum statistic $M_n=\max_{1\leq k\leq d} \frac{1}{\sqrt{n}}\sum_{i=1}^n X_{i,k},$ whose study was initiated in a seminal study. The next step in understanding the asymptotic properties of $M_n$ and accompanying quantile approximations is the development of Edgeworth-type expansions and corresponding bootstrap methods. A very recent result in this direction was established by Koike, developing an Edgeworth expansion for $\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$ based on Stein kernels, subject to some regularity conditions. The focus is on the problem through the lens of Poisson-approximations to directly construct an Edgeworth expansion for $M_n$. The main assumptions are a Cram\'{e}r-type condition for all pairs of components of the $X_i$ and a notion of weak dependence across the dimension. Inverting the Edgeworth expansions, a Cornish-Fisher-type expansion is obtained for the quantiles of $M_n$, which is also second-order accurate. Furthermore, the results are extended to the studentized case, i.e., to the statistic $\max_{1\leq k\leq d} T_{n,k}$, where $T_{n,k}$ is the Student-t statistic of the $k$-th components of the $X_i$.
