15 problems

Lasso active-set constraints. The Lasso choice must satisfy ${\mathcal A}_1\subseteq {\mathcal A}$ and ${\mathcal A}\subseteq {\mathcal A}_{10}$; moreover, $w_{\mathcal A}$ cannot…

Consider the estimator $\widehat{\mathscr{E}}_{\texttt{test}}^{(t)}(X,Y)$ constructed by Algorithm; Theorem proves its consistency under the paper's single-index or multi-index reg…

Let $r({\mathbf{X}}, {\mathbf{Z}}, K_0,s_0)$ be the design-dependent quantity defined by … Assume that the design satisfies a restricted eigenvalue (RE) assumption and that the fea…

Consider fitting atmospheric concentration data with regression models after filtering and smoothing, while assessing the fit against raw noisy data. Second-order polynomial conjec…

Let $d$ denote the dimension and $n$ the sample size. Suppose the dimension grows at the scaling … The authors conjecture that this scaling cannot be weakened while retaining the p…

Consider a general regression model with design matrix $X$, using a polynomially decaying shrinkage prior with polynomial-order parameter $\alpha$. Nearly orthogonal design conject…

The cited approximation results concern neural networks approximating smooth functions, with one result requiring sparsity to obtain an $L_2$ approximation bound and another requir…

Hybrid screening conjecture. An appropriate hybrid screening procedure should make it possible to decrease this upper bound to $\O(\sqrt{n}\,\sigma_{(c\log n)})$. This proposed imp…

Consider convex M-estimators and convex procedures that select among members of a minimizing set. A convex M-estimator is said to return a singleton minimizing set when its minimiz…

Fix $\pi\in\mathcal P_2(\mathbb R)$, $\delta\in(0,\infty)$, and $\sigma>0$. Consider sequences of dimensions, Gaussian design matrices, coefficient vectors, and noise vectors satis…

Let $K$ be the number of convexity change points, let $p$ be the number of knots, and define the Piecewise Convex Information Criterion by … Here the default values are…

Consider the pointwise regression testing problem described above, with conditional variance function $sigma(x)$, covariate distribution for $X_i$, smoothness parameter $beta$, rad…

Let $\boldsymbol{H}$ be the hat matrix of a regression model, let $\boldsymbol{\boldsymbol{\epsilon}}=(\epsilon_1,\ldots,\epsilon_n)$ be a vector of independent identically distrib…

Let $(Y,X)$ denote an observation from a distribution $F$, let $\xi$ be the regression parameter, and consider monitoring schemes based on the residual $Y-X\xi$. Assume that the cd…

Let $m$ denote the number of components in a partial least squares regression fit, and let $\operatorname{DoF}(m)$ denote its degrees of freedom. Degrees-of-freedom lower-bound con…