Does the score-matched optimal convex estimator attain the full semiparametric efficiency bound?

Let $p\in\mathbb{N}$ be fixed. Observe i.i.d. pairs $(X_i,Y_i)_{i=1}^n$ from the linear model

where $\beta_0\in\mathbb{R}^p$, $X_i\in\mathbb{R}^p$, $\varepsilon_i\perp X_i$, and the conditional density is

for unknown error density $p_0$. Assume the regularity conditions in Feng et al. (2024) guaranteeing root-$n$ asymptotic normality for convex $M$-estimators.

For a convex loss $\rho$, write $\psi=\rho'$. Under this parameterization, convexity corresponds to monotonicity of $\psi$; equivalently, the source works with an antitone class $\Psi_\downarrow(p_0)$ after a sign convention. For $\psi\in\Psi_\downarrow(p_0)$, define

where $P_0$ is the law of $\varepsilon$ and $S_0=\{x:p_0(x)>0\}$. Then regular convex $M$-estimators satisfy

The score-matching objective is

and it is linked to asymptotic variance by

Hence minimizing $D_{p_0}$ is equivalent to minimizing $V_{p_0}$.

Population optimizer in the source: let $F_0$ be the CDF of $p_0$, define density-quantile $J_0:=p_0\circ F_0^{-1}$ on $(0,1)$, let $\widehat J_0$ be the least concave majorant of $J_0$, and set

where $\widehat J_0^{(R)}$ is the right derivative. This achieves

Sample score-matching constructor for $\hat\beta_{\mathrm{cvx}}$ in Feng et al. (2024):

1. Split the sample into folds and compute pilot estimator(s) on auxiliary folds.

2. Form residuals from the pilot fit and estimate $p_0$ on each fold (kernel density estimate with truncation used in the source).

3. Build a preliminary score estimate from the density estimate.

4. Project this preliminary score onto the monotone cone (implemented via isotonic regression/PAVA in the source), then antisymmetrize in the symmetric-error construction.

5. Define a convex loss by integrating the projected score:

1. Compute fold-wise convex $M$-estimators and cross-fit/average them to obtain $\hat\beta_{\mathrm{cvx}}$.

Precise meaning of "asymptotically optimal within convex $M$-estimators": for any regular convex $M$-estimator $\tilde\beta$ in the same model class,

in Loewner order; equivalently, $V_{p_0}(\psi_0^\star)\le V_{p_0}(\psi)$ for all admissible convex-score choices $\psi$, with equality only for positive scalar multiples of $\psi_0^\star$.