Editing Openai/6922876a-7988-8007-9c62-5f71772af6aa (section)

===== Insert after (2.1) defining ψτ\psi_\tauψτ: =====

<syntaxhighlight lang="latex">\medskip\noindent\textbf{Clipped score.}
For $B\ge 1$, define the clipped score
\[
\tilde\psi_{\tau,B}(u)\ :=\ \mathrm{clip}\big(\psi_\tau(u),\,[-B/\sqrt{\tau},\,B/\sqrt{\tau}]\big),
\]
and let $\tilde\psi'_{\tau,B}$ be any selection of its (a.e.) derivative; then $\|\tilde\psi_{\tau,B}\|_\infty\le B/\sqrt{\tau}$ and $\|\tilde\psi'_{\tau,B}\|_\infty\le C/\tau$ uniformly in $B$.
We henceforth use $\tilde\psi_{\tau_\star,B}$ in the estimating equation \eqref{eq:M-est}, and write $\hat\theta_\ep$ for any solution. The clipping level $B$ is fixed (e.g. $B=10$); on the $W_2$-ball all bounds below are uniform in $B$ and coincide with the unclipped estimator whenever no sample point falls in the extreme tails.

</syntaxhighlight>

Replace Lemma \ref{lem:score-moments} by the following (stated and proved for the clipped score):

<syntaxhighlight lang="latex">\begin{lemma}[Uniform control of clipped score moments]\label{lem:score-moments-clipped}
Fix $\tau\in(0,\tfrac12]$ and let $\tilde\psi_\tau:=\tilde\psi_{\tau,B}$ be the clipped score defined above. There exist constants $C_1,C_2,C_3<\infty$ (independent of $B$) such that for all $Q$ with $W_2(Q,\mu_\theta)\le \ep$,
\begin{align}
\left|\E_Q[\tilde\psi_\tau]-\E_{\nu_{\theta,\tau}}[\tilde\psi_\tau]\right| &\le C_1\,\frac{\ep}{\sqrt{\tau}},\label{eq:bias-score-clipped}\\
\left|\E_Q[\tilde\psi'_\tau]-\E_{\nu_{\theta,\tau}}[\tilde\psi'_\tau]\right| &\le C_2\,\frac{\ep}{\sqrt{\tau}},\label{eq:curv-score-clipped}\\
\left|\E_Q[\tilde\psi_\tau^2]-\E_{\nu_{\theta,\tau}}[\tilde\psi_\tau^2]\right| &\le C_3\,\frac{\ep}{\tau^{3/2}}.\label{eq:var-score-clipped}
\end{align}
\end{lemma}

\begin{proof}
Apply Lemma~\ref{lem:dyn-W2} with $P=\nu_{\theta,\tau}$, $\varphi=\tilde\psi_\tau$, $\varphi=\tilde\psi'_\tau$, and $\varphi=\tilde\psi_\tau^2$, respectively. Since $\|\tilde\psi'_\tau\|_\infty\lesssim \tau^{-1}$ and $\|(\tilde\psi_\tau^2)'\|_\infty\lesssim \tau^{-3/2}$, we have
\[
\int_0^1 \E_{P_t}[(\tilde\psi'_\tau(X_t-\theta))^2]\,dt \lesssim \tau^{-1},\quad
\int_0^1 \E_{P_t}[((\tilde\psi_\tau^2)'(X_t-\theta))^2]\,dt \lesssim \tau^{-3},
\]
uniformly over $Q$ with $W_2(Q,\mu_\theta)\le \ep$. The claim follows using $W_2(Q,\nu_{\theta,\tau})\le \ep + W_2(\mu_\theta,\nu_{\theta,\tau})$ and Lemma~\ref{lem:info-W2}.
\end{proof}

</syntaxhighlight>

Adjust the M‑estimation remainder bounds (replace all occurrences of ψ,ψ′,ψ′′\psi,\psi',\psi''ψ,ψ′,ψ′′ by ψ~τ,ψ~τ′\tilde\psi_\tau,\tilde\psi'_\tauψ~τ,ψ~τ′, and remove every use of ∥ψ′′∥∞\|\psi''\|_\infty∥ψ′′∥∞. Where you currently use Lipschitz in ttt via ∥ψ′′∥∞\|\psi''\|_\infty∥ψ′′∥∞, replace it by the uniform bound ∥ψ~τ′∥∞≲τ−1\|\tilde\psi'_\tau\|_\infty\lesssim\tau^{-1}∥ψ~τ′∥∞≲τ−1 and the simple inequality

∣mn(tˉ)−mn(θ)∣≤1n∑i=1n∣ψ~τ′(Xi′−tˉ)−ψ~τ′(Xi′−θ)∣≤∥ψ~τ′∥Lip ∣tˉ−θ∣≲τ−1 ∣tˉ−θ∣.|m_n(\bar t)-m_n(\theta)| \le \frac{1}{n}\sum_{i=1}^n\big|\tilde\psi'_\tau(X_i'-\bar t)-\tilde\psi'_\tau(X_i'-\theta)\big|
\le \|\tilde\psi'_\tau\|_\mathrm{Lip}\,|\bar t-\theta|
\lesssim \tau^{-1}\,|\bar t-\theta|.∣mn(tˉ)−mn(θ)∣≤n1i=1∑nψ~τ′(Xi′−tˉ)−ψ~τ′(Xi′−θ)≤∥ψ~τ′∥Lip∣tˉ−θ∣≲τ−1∣tˉ−θ∣.
Finally, every appearance of \EQ[ψ4]\E_Q[\psi^4]\EQ[ψ4] can be replaced by ∥ψ~τ∥∞4≲τ−2\|\tilde\psi_\tau\|_\infty^4\lesssim \tau^{-2}∥ψ~τ∥∞4≲τ−2, uniformly over the W2W_2W2-ball.