lin-0205

16.5 Statistical admission

For estimated log odds \(\widehat\ell _s\) with covariance \(\widehat\Sigma _s\), set

\[ \widehat\Omega _s=Z_s^\top \widehat\ell _s, \qquad \widehat V_s=Z_s^\top \widehat\Sigma _sZ_s. \]

Under the scalarizable null and the usual fixed-graph asymptotics,

\[ Q_s = \widehat\Omega _s^\top \widehat V_s^\dagger \widehat\Omega _s \ \Longrightarrow \ \chi _d^2, \]

where \(d=\operatorname {rank}V_s\). This yields a three-way decision:

\[ \mathcal D(G_s,\widehat\ell _s)= \begin{cases} \mathsf U, & d=0,\\ \mathsf R, & d{\gt}0\text{ and }p_Q\leq \alpha ,\\ \mathsf A_G, & d{\gt}0\text{ and }p_Q{\gt}\alpha . \end{cases} \]

Here \(\mathsf U\) requests more comparisons or a weaker target, \(\mathsf R\) rejects scalarization, and \(\mathsf A_G\) conditionally admits it on observed support. Non-rejection is not proof of a scalar population reward.