Optimality of Monotonized Asymptotic Risk

Definitions and mathematical statement

Let $\widehat f$ denote the estimator under consideration, and write $R_{\mathrm{det}}(\gamma;\widehat f)$ for its deterministic asymptotic risk. The supplied excerpt uses positive parameter values $\gamma>0$, but does not specify the statistical model, the meaning of $\gamma$, the asymptotic regime, or the definition of this risk. The clean statement suppresses the dependence on $\widehat f$.

The monotonized risk is given formally by \[ M_{\widehat f}(\gamma) :=\inf_{\zeta\geq\gamma}R_{\mathrm{det}}(\zeta;\widehat f), \qquad \gamma>0. \] This formula presupposes that the risk is defined at the parameter values over which the infimum is taken. The supplied record does not specify further domain restrictions, finiteness assumptions, or attainment of the infimum. Its clean statement uses an infimum, whereas the transcribed original excerpt uses a minimum; no attainment hypothesis is supplied.

Open-problem request as supplied. Establish some form of optimality principle for $M_{\widehat f}$. The source describes this function as nondecreasing in $\gamma$, even when $R_{\mathrm{det}}(\gamma;\widehat f)$ is nonmonotonic, and records the comparison \[ M_{\widehat f}(\gamma)\leq R_{\mathrm{det}}(\gamma;\widehat f) \qquad (\gamma>0), \] where these expressions are defined. These properties provide context for the request; the supplied record does not identify the optimality criterion, comparison class, admissible procedures, or statistical hypotheses needed to turn the request into a fully specified optimization statement. Those choices remain unresolved.

Short English statement

For a fixed estimator, take the infimum of its deterministic asymptotic risk over parameter values $\zeta\geq\gamma$. The supplied problem asks for an optimality principle for this monotonized risk. The statistical setting and the intended meaning of optimality are unspecified, so a precise optimality theorem cannot yet be stated.

Sources

• Local UnsolvedMath problem registry, permanent ID $30005253$, source problem code catalogue entry; supplied complete source record, titled "Optimality of Monotonized Asymptotic Risk."
• Supplied report citation: Mini-Workshop: Mathematical Foundations of Robust and Generalizable Learning (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/46, pp. 2659--2686. Supplied source URL and literature reference "Oberwolfach Report 46/2022": https://doi.org/10.4171/owr/2022/46. The record gives proposed year 2022 and names no individual proposer. The relationship between that year, the report identifier, and the citation's parenthetical 2023 is not explained.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the report explicitly presents establishing an optimality principle for the monotonized risk as an open problem. This is supplied dated curation, not independently verified literature assessment.
• The registry labels the statement corrected_verified and reports checking it against the official Oberwolfach report. This is a supplied verification claim; the report has not been independently consulted for this draft.