Asymptotic Theory for the Maximum of an Increasing Sequence of Parametric Functions

\cite{HillMotegi2017} present a new general asymptotic theory for the maximum of a random array $\{{\mathcal{X}}_n(i)$ $:$ $1$ $\le$ $i$ $\le$ $\mathcal{L}{\}}_{n\ge 1}$, where each ${\mathcal{X}}_n(i)$ is assumed to converge in probability as $n$ $\to$ $\mathrm{\infty }$. The array dimension $\mathcal{L}$ is allowed to increase with the sample size $n$. Existing extreme value theory arguments focus on observed data ${\mathcal{X}}_n(i)$, and require a well defined limit law for $\max_{1\leq i\leq $\mathcal{L}$}|$\mathcal{X}${n}(i)|$byrestrictingdependenceacross$i$. The high dimensional central limit theory literature presumes approximability by a Gaussian law, and also restricts attention to observed data. \cite{HillMotegi2017} do not require $\max{1\leq i\leq $\mathcal{L}${n}}|$\mathcal{X}${n}(i)|$ to have a well defined limit nor be approximable by a Gaussian random variable, and we do not make any assumptions about dependence across $i$. We apply the theory to filtered data when the variable of interest ${\mathcal{X}}_n(i,{\theta }_0)$ is not observed, but its sample counterpart ${\mathcal{X}}_n(i,{\widehat{\theta }}_n)$ is observed where ${\widehat{\theta }}_n$ estimates ${\theta }_0$. The main results are illustrated by looking at unit root tests for a high dimensional random variable, and a residuals white noise test.