Abstract.
In this talk we give theoretical accuracy for asymptotic expansions of the distributions of two MANOVA test statistics:
$$
T_{LH}=T_0^2=n \tr S_hS_e^{-1}, \quad T_{LR}=-n \log (|S_e|/|S_e+S_h|),
$$
where $S_h$ and $S_e$ are independently distributed as Wishart distributions $W_p(q, \Sigma)$ and $W_p(n, \Sigma)$, respectively. The first statistic is called Lawey-Hotelling trace criterion or Hotelling's generalized $T_0^2$ statistic. The second statistic is a likelihood ratio statistic for multivariate linear hypothesis. Our main purpose is to derive error bounds for the asymptotic expansions. It may be noted that our error bounds are given in explicit and computable forms. Further, their orders are the same as ones of the usual order estimates, and hence by products, the result provides a proof for the validity of asymptotic expansions in a strong sense. We also discuss about theoretical accuracy for their Cornish-Fisher expansion. This talk is based on a joint work of V.V. Ulyanov, etc.

Резюме.
В докладе приводятся новые результаты о поведении спектра (плотность состояний, максимальное собственное значение, локализация) для матриц случайных графов и их приложения в математической биологии. Будет рассказано о многих открытых проблемах в этой области.