15 problems

Let ${\mathbb D}_\beta$ denote the limiting distribution associated with the flatness parameter $\beta$, for $0<\beta<\infty$. Consider Brownian motion with no drift, corresponding…

Let $F$ be a distribution function and let $\mathbb G_F$ denote the Gaussian process introduced earlier in the paper. The paper proves that, under the stated assumptions—including…

Let $n$ denote the sample size, and let the convex/concave component be estimated under the $L_{2}(P)$ norm, where $P$ is the relevant covariate distribution. Minimax lower-bound c…

Let $H$ be a non-negative strongly convex drift function such that $H(0)=0$, and let the $\mathbf{slGCM}$ distribution denote the limiting distribution associated with the least co…

Statistical-dimension conjecture. For every fixed $k\geq 0$, the correct order of $\delta_{K_{\uparrow,k}}$ should be

General k-monotone-cone conjecture. Results analogous to those of Theorem 1 should hold for the general $k$-monotone cone $K_{\uparrow,k}$.

Gaussian-limit conjecture. The same result should hold for any concave $F$; that is, generally

Non-normality conjecture. In this scenario, the sampling distribution of $\sqrt{n}\{\tau(g,\hat f_n)-\tau(g,f)\}$ does not converge to a normal limit with mean $0$ and the efficien…

Let $2\log\lambda_n$ denote the likelihood ratio statistic for testing the value of a concave regression function, based on a null-constrained concave least-squares estimator. Doss…

Linear-functional extension conjecture. Statements such as $$ should continue to hold for other one-dimensional shape constraints.

Curvature-effect conjecture. The observed difference in the convergence behavior is caused by the presence or absence of a strictly curved region of $f_0$.

Parametric-rate conjecture. The $n^{1/2}$ convergence behavior of linear functionals and the entropy functional should continue to hold for other shape constraints.

Local-rate conjecture. In misspecified regions, the rate of convergence should be $n^{1/2}$, whereas in well-specified regions it should depend on whether the density lies locally…

Shape-constraint conjecture. Phenomena similar to those revealed in this paper should appear under more complex shape constraints, such as convexity, although both the boundary val…

Competitive convergence-rate conjecture. The rates of convergence for the shape-constrained estimator are competitive with those of the adaptive kernel and logspline estimators.