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 $\{\mathcal{X}_{n}(i)$ $:$ $1$ $\leq$ $i$ $\leq$ $\mathcal{L}\}_{n\geq 1}$, where each $\mathcal{X}_{n}(i)$ is assumed to converge in probability as $n$ $\rightarrow$ $\infty$. 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 $\max_{1\leq i\leq \mathcal{L}}|\mathcal{X}_{n}(i)|$ by restricting dependence across $i$. 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 $\mathcal{X}_{n}(i,\hat{\theta}_{n})$ is observed where $\hat{\theta}_{n}$ 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.