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Solutions to Open Problems in Mathematical Sciences

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2Verified Solutions
2Counterexamples
120Open Problems
SolvedCounterexampleMathematical Statistics
solveall.org — Score-Matched Optimal Convex Estimator Efficiency Bounds

The Score-Matched Optimal Convex Estimator Does Not Attain the Full Semiparametric Efficiency Bound

We show that the score-matched optimal convex -estimator of Feng et al. attains the full semiparametric efficiency bound if and only if the error density is log-concave. For non-log-concave errors, we construct an explicit counterexample—Rademacher design with a bimodal Gaussian mixture error—in which a nonconvex kernel-density-based estimator achieves strictly smaller asymptotic variance.

Pythagorean Identity for the Efficiency Gap

The efficiency gap is exactly the squared -distance from the true score to the cone of non-increasing functions. Equality holds iff is log-concave.

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SolvedCounterexampleProbability Theory
solveall.org — Gaussian Correlation Inequality: Optimal Extensions

A Note on Non-Centered and Log-Concave Extensions of the Gaussian Correlation Inequality

We prove that two natural extensions of the Gaussian correlation inequality fail in the strongest possible sense. The naive analogue for arbitrary non-centered Gaussian measures and symmetric convex sets is false already in , even for positive-definite covariance matrices and strips. For log-concave measures, the optimal correlation constant collapses to zero in all dimensions .

Optimal Log-Concave Correlation Constants

Where is the largest constant such that holds for every log-concave probability measure on and all symmetric convex sets . The proof is elementary and geometric.

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