1\section{Extensions: unequal blocks and misspecification}\label{sec:robust}23Both extensions are known in the literature \citep{zhu2021least,gu2023heterogeneity,jordan2019communication} and are stated here only so that the frontier can be used in practice.45\begin{proposition}[Unequal block sizes and pseudo-likelihood]\label{prop:robust}6\begin{enumerate}[label=(\roman*)]7\item Let the blocks have sizes $m_1,\dots,m_K$ with $\sum_k m_k = N$, $m_{\min} \to \infty$ and each $s_j \mid m_k$. With the size weights $m_k/N$ in the pilot, $\pilot{r} - \thetaN = \op(m_{\mathrm{eff}}^{-r} + \sqrt K/N + N^{-1})$ with $m_{\mathrm{eff}} := (N^{-1}\sum_k m_k^{1-r})^{-1/r} \ge m_{\min}$, and \cref{thm:frontier} holds with $m$ replaced by $m_{\mathrm{eff}}$ (the aggregated derivatives are size-weighted and equal the full-sample derivatives, so the Newton steps are unchanged). With equal weights $1/K$ the influence term is not preserved: $K^{-1}\sum_k U_k - U_N = \sum_i c_i\psi(X_i)$ with $c_i = 1/(Km_{k(i)}) - 1/N$, a centered term of order $(\sum_i c_i^2)^{1/2}$ that is of exact order $N^{-1/2}$ unless the sizes are asymptotically equal; the equal-weight pilot is therefore not first-order equivalent for $q = 0$, while for $q \ge 1$ the discrepancy is of order $N^{-1/2}$ and is absorbed by the first Newton step.8\item If the model is misspecified and $\theta_0$ is the pseudo-true value, all statements hold with $\psi = -\bar H(\theta_0)^{-1}\nabla\log f(X;\theta_0)$ and $\Sigma_\psi = \bar H(\theta_0)^{-1}\bar J(\theta_0)\bar H(\theta_0)^{-1}$, provided \cref{ass:lin,ass:bias,ass:newton} are read for the pseudo-MLE. The estimator is then first-order equivalent to the centralized pseudo-MLE and Wald intervals require the sandwich of \cref{cor:inference}; the observed-information intervals are not valid.9\end{enumerate}10\end{proposition}1112The proof is in \cref{app:proofs}; the Monte Carlo evidence is in \cref{sec:mc} (skewed partitions with size ratios 4 and 16, and an omitted-nonlinearity logistic design). Heterogeneous distributions across blocks, in which the local pseudo-true values differ, are outside the scope: the common-influence identity of \cref{lem:influence} fails, and the appropriate target and weighting are those of \citet{gu2023heterogeneity} and \citet{duan2022heterogeneity}.13