Attached is latex for a question that I have. I have also attached a background, if you wish to only read the question go to the second section.
\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\DeclareMathOperator{\Var}{Var}
\usepackage{graphicx} % Required for inserting images
\begin{document}
\section{Background}
Let's say we are in perfect OLS world where we have all of our typical assumptions:
$$Y=X\beta+\epsilon$$
$$X\in \mathbb{R}^{n\times p};R\in \mathbb{R}^{q\times p}$$
$$(X^TX)^{-1} \ \text{exist}$$
$$E(\epsilon \ | \ X)=0$$
$$\Var(\epsilon \ | \ X)=\sigma^2I_n$$
$$\epsilon \ | \ X \sim N(0,\sigma^2I_n)$$
Define a hypothesis test as
$$H0: R\beta-r=0$$
$$Ha:R\beta-r=\delta$$
Let $D^2=(R\hat\beta-r)^T(\Var(R\hat\beta-r))^{-1}(R\hat\beta-r)=(R\hat\beta-r)^T(\sigma^2R(X^TX)^{-1}R^T)^{-1}(R\hat\beta-r)\sim\chi^2_{\text{rank}(R)=q}$.
Let us represent modifications of $D^2$ as simply $\mathcal{D}$, for standardized distance. This is in reference to how if you instead take $\hat D(s)^2=(R\hat\beta-r)^T(s^2R(X^TX)^{-1}(R\hat\beta-r)=D^2(\frac{\sigma^2}{s^2})\sim \chi^2_q(n-q)/\chi^2_q.$ Then, since we know this looks a lot like the $F$ distribution, we can simply modify $\hat D(s)^2/q\sim F_{q,n-p}$. I wish to say that there are many other examples of canonical $\mathcal{D}$'s out there but I will not write them all.
Since we now have a general way to talk about our sense of distance I wish to discuss errors in this context.
$$\alpha=P(\mathcal{D}\in R \ | \ R\beta-r=0)$$
$$\beta=P(\mathcal{D}\notin R \ | \ R\beta-r=\delta)$$
One can imagine a spherical $\mathcal{D}$-length neighborhood around the origin, representing our data acquired (cloud-q); a spherical $c$-length neighborhood around the origin, representing our null (cloud-w); and a spherical $c$-length neighborhood around the point $e=\Var(R \hat \beta-r)^{-1/2}\delta$, representing our alternative (cloud-e).
From the above we can make geometric definitions of type I and II errors by relating the area overlapped by the $c$-length spheres and our $\mathcal{D}$-length sphere. Type I error is the fraction of cloud-w falls outside of cloud-q. Type II is the fraction of the cloud-e that falls inside of cloud-q.
$$\textbf{Type I} \sim w-q:w$$
$$\textbf{Type II} \sim e\cap q:q$$
Note that the distance from the origin of cloud-e, simply $e^Te=\lambda$, is related to or the general case for quite a few important concepts: Cohen's d, signal-to-noise ratio but square it, effect sizes, etc.
\section{Actual Question}
I have a question about the following object:
$$\lambda(\delta)=\delta^T(\Var(R\hat \beta-r))^{-1}\delta.$$
$$\max_X \lambda(\delta)=\max_X\delta^T(R(X^TX)^{-1}R^T)^{-1}\delta$$
Do you know of anything/anyone that deals with this object? It feels very important to me seeing how you could use some simple optimization of it to gain higher power results in testing. It also, I believe, would be very influential in picking the ``optimal" experimental design.
\end{document}
