发布时间：2014-12-01 来源： [打印] 字号： T T T
Time: 1:30pm, Dec.3rd, 2014
Place: ISEM Conference Room (3rd floor, Chengming Building)
On the Invariant Distribution of the Extreme Values of Strongly Correlated Random Variables
Abstract: This paper shows that the limiting distribution of the extreme value of a large class of "strongly correlated" random variables is invariant. The condition the term "strongly correlated" imposed is that all of the random variables in the set whose extreme value to be taken on assume the form of projections of a multi-dimensional random vector and its stochastic limit, a scaled, or "deformed" Brownian motion, to a class of strongly correlated vectors in the same metric space. Our results can be applied to a broad range of test statistics generated from maximization of some objective function, including but not limited to Chi-square tests, F-tests, and sup-Wald tests. The random vector being projected can take a great variety of forms as long as its primitive shocks follows a FCLT, although cases of I(1) and fractionally integrated errors can also be discussed in a similar manner. Our results are also relevant to the literature of extreme value theory (EVT) in the sense that our key assumption, that the maximization has to be taken over a set of "strongly correlated" random variables, is parallel to the key assumption in EVT that the maximization is taken over a set of independent random variables. Moreover, we show that our result still holds if allowing for statistics before rescaling to have infinite distribution, hence includes EVT on normal variables as special case. We also point out that our results can be extended from scaling transformations to any invertible linear transformations. Simulation results on a special case of "joint segmented trend break test under heterogeneous innovations" are also reported to illustrate the finite sample performance of the pivotal statistic.