\section{The general local correction}\label{sec:construction}

\subsection{Weights}
Let $V$ be the $r\times r$ Vandermonde matrix with entries $V_{aj} = s_j^{a}$, $a = 0,\dots,r-1$. The weights are the unique solution of
\begin{equation}\label{eq:vandermonde}
\sum_{j=1}^r w_j = 1, \qquad \sum_{j=1}^r w_j\, s_j^{a} = 0, \quad a = 1,\dots,r-1,
\end{equation}
i.e.\ $w = V^{-1}e_1$. For $s = (1,2)$, $w = (2,-1)$; for $(1,2,4)$, $w = (8/3,-2,1/3)$; for $(1,2,4,8)$, $w = (64/21,-8/3,2/3,-1/21)$; for $(1,2,3)$, $w = (3,-3,1)$. Two constants summarize the cost of a pattern: the \emph{leading bias coefficient} $\betar := \sum_j w_j s_j^{\,r}$, which multiplies the first uncancelled bias term, and the \emph{remainder amplification} $\Amp := \sum_j \abs{w_j}\, s_j$, which multiplies the $L_2$ bound of the stochastic remainder. \Cref{tab:weights} lists them; the weights are computed exactly in rational arithmetic in the replication code and the moment constraints are verified to machine precision.

\begin{table}[t]
\centering\small
\begin{threeparttable}
\caption{Split patterns, weights and constants.}\label{tab:weights}
\begin{tabular}{@{}lclrrrrr@{}}
\toprule
Pattern $s$ & $r$ & Weights $w$ & $\betar$ & $\Amp$ & $\sum_j\abs{w_j}$ & $\kappa(V)$ & Local fits \\
\midrule
$(1)$ & 1 & $1$ & $1$ & $1$ & $1$ & $1$ & 1 \\
$(1,2)$ & 2 & $2,\,-1$ & $-2$ & $4$ & $3$ & $6.9$ & 3 \\
$(1,3)$ & 2 & $3/2,\,-1/2$ & $-3$ & $3$ & $2$ & $5.8$ & 4 \\
$(1,2,4)$ & 3 & $8/3,\,-2,\,1/3$ & $8$ & $8$ & $5$ & $80$ & 7 \\
$(1,2,3)$ & 3 & $3,\,-3,\,1$ & $6$ & $12$ & $7$ & $71$ & 6 \\
$(1,2,4,8)$ & 4 & $64/21,\,-8/3,\,2/3,\,-1/21$ & $-64$ & $80/7$ & $6.4$ & $3284$ & 15 \\
$(1,2,3,4)$ & 4 & $4,\,-6,\,4,\,-1$ & $-24$ & $32$ & $15$ & $1171$ & 10 \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item $\kappa(V)$ is the spectral condition number of the Vandermonde matrix; ``local fits'' is $\sum_j s_j$ (each level costs one pass over the block's data). Source: \texttt{results/tables/weights\_analysis.csv}.
\end{tablenotes}
\end{threeparttable}
\end{table}

\subsection{Influence preservation, bias cancellation and the random remainder}
Write $U_S := \abs{S}^{-1}\sum_{i\in S}\psi(X_i)$ and $U_k := U_{D_k}$. The sub-blocks of level $j$ partition $D_k$ exactly, so the average of their influence terms is $U_k$ whatever $j$; since the weights sum to one, the influence term of $\Jr{r,k}$ is $U_k$ (\cref{lem:influence}). Nothing else about $\psi$ is used, which is why the construction applies verbatim to pseudo-MLEs. Collecting the remainders of the $\sum_j s_j$ fits into
\begin{equation}\label{eq:RJ}
R^{J}_{k} := \sum_{j=1}^r w_j\, s_j^{-1}\sum_{\ell=1}^{s_j} R_{S_{j\ell,k}},
\end{equation}
we have $\Jr{r,k} - \theta_0 = U_k + R^J_k$. The mean of $R^J_k$ is deterministic and, by \eqref{eq:bias} and \eqref{eq:vandermonde}, equals $\sum_{a<r} b_a m^{-a}\sum_j w_j s_j^a + \sum_j w_j\rho_{m/s_j} = O(m^{-r})$ (\cref{lem:bias}). Its $L_2$ norm is at most $\sqrt{C}\Amp/m$ by the triangle inequality (\cref{lem:remainder}); the nested fits of one worker are dependent, and no independence is used at this step. Across workers the blocks are independent, so the centered average has second moment at most $C\Amp^2/(Km^2)$ (\cref{lem:average}). Finally the full-sample expansion $\thetaN - \theta_0 = U_N + R_N$ with $U_N = K^{-1}\sum_k U_k$ cancels the common influence term exactly:
\begin{equation}\label{eq:pilot}
\pilot{r} - \thetaN = \E R^J_k + K^{-1}\sum_k (R^J_k - \E R^J_k) - R_N = \op\!\Big(m^{-r} + \frac{1}{m\sqrt K} + \frac{1}{N}\Big),
\end{equation}
(\cref{lem:pilot}). The middle term is the centered random part of the local corrections. Its mean is zero but its magnitude, $\sqrt K/N$, exceeds $N^{-1}$ by the factor $\sqrt K$ and must be carried through every subsequent step; a proof that drops it is incomplete even where the final exponent happens to be unaffected.

\subsection{Conditioning and the choice of pattern}
Higher order costs conditioning. The weights grow geometrically with $r$ for the doubling pattern ($\max_j\abs{w_j} \approx 3.0, 3.2$ for $r = 4, 5$, with Vandermonde condition numbers $3\times10^3$ and $5\times10^5$), and the finest sub-block has $m/2^{r-1}$ observations, which for $p > 1$ quickly reaches the regime where local MLEs fail to exist (separation in logistic regression) or have no moments (sub-blocks of two or three exponential observations). The patterns of \citet{dhaene2015split}, $(1,2,3)$ and $(1,2,3,4)$, use coarser finest blocks and smaller $\abs{\betar}$ but larger $\Amp$ ($12$ against $8$ for $r = 3$): they trade bias constants against remainder amplification. Which trade is better is a finite-sample question answered in \cref{sec:mc}. Adaptive selection of $r$ from the data is not covered by the theory: a data-driven order changes the estimator and its limit, and the theorems below are for fixed $(r,q)$; see \cref{sec:discussion}.