8 problems

Piecewise Convex Information Criterion conjecture. PCIC consistently selects the number of convexity change points and is asymptotically efficient within the class of methods that…

Let $f_0$ be the true regression function in a convex regression model, and let $x_0$ be the point at which the function value is evaluated. The likelihood-ratio-test-based procedu…

Let $N$ denote the total sample size and let $K$ be the number of bins used in a convexified regressogram estimator for distributed convex regression. Convexified regressogram conj…

Let $Y_i=m(x_i)+\varepsilon_i$, where the $\varepsilon_i$ are mean-zero independent and identically distributed observations, the $x_i$ are fixed real numbers, and $m$ is convex. L…

Negative adaptation conjecture. Adaptation does not occur when the support $\Omega$ is a general convex body and the regression function is polyhedral, within the current methods b…

Boundary-smoothness conjecture. The smoothness of the boundary of the domain $\Omega$ plays a significant role in determining the degree of difficulty in estimating a convex functi…

Let the true function be non-convex, and let its projection onto the class of convex functions have linear components, called misspecified regions. Let the LSE denote the convex le…

The paper studies risk bounds for the least-squares estimator of a univariate convex regression function, including Theorems and. In the adaptive bound, the relevant logarithmic fa…