\section{Extensions: unequal blocks and misspecification}\label{sec:robust}

Both extensions are known in the literature \citep{zhu2021least,gu2023heterogeneity,jordan2019communication} and are stated here only so that the frontier can be used in practice.

\begin{proposition}[Unequal block sizes and pseudo-likelihood]\label{prop:robust}
\begin{enumerate}[label=(\roman*)]
\item Let the blocks have sizes $m_1,\dots,m_K$, with $\sum_k m_k = N$, with $m_{\min} \to \infty$, and with each $s_j$ dividing every $m_k$. With the size weights $m_k/N$ in the pilot,
\begin{equation}\label{eq:unequal-pilot}
\pilot{r} - \thetaN = \op\Big(m_{\mathrm{eff}}^{-r} + \frac{\sqrt K}{N} + \frac1N\Big), \qquad m_{\mathrm{eff}} := \Big(\frac1N\sum_{k=1}^K m_k^{\,1-r}\Big)^{-1/r} \ \ge\ m_{\min},
\end{equation}
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:
\begin{equation}\label{eq:unequal-naive}
\frac1K\sum_{k=1}^K U_k - U_N = \sum_{i=1}^N c_i\,\psi(X_i), \qquad c_i = \frac{1}{K\,m_{k(i)}} - \frac1N, \qquad
\sum_{i=1}^N c_i^2 = \frac{1}{K^2}\sum_{k=1}^K\frac{1}{m_k} - \frac1N \ \ge\ 0,
\end{equation}
with equality only for equal sizes; the discrepancy is of exact order $N^{-1/2}$ when the sizes are bounded away from equality, so the equal-weight pilot is not first-order equivalent for $q = 0$, while for $q \ge 1$ the first Newton step squares it to $\op(N^{-1})$.
\item If the model is misspecified and $\theta_0$ is the pseudo-true value, all statements hold with
\begin{equation}\label{eq:sandwich}
\psi(x) = -\bar H(\theta_0)^{-1}\,\nabla_\theta\log f(x;\theta_0), \qquad \Sigma_\psi = \bar H(\theta_0)^{-1}\,\bar J(\theta_0)\,\bar H(\theta_0)^{-1},
\end{equation}
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}; observed-information intervals are not valid in general (they happen to be close to the sandwich in the misspecified logistic design of \cref{sec:mc}, and far from it in the contaminated exponential design).
\end{enumerate}
\end{proposition}

The 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}.