12 problems

Consider model-based clustering by maximum-likelihood estimation in mixture models, with $D_m$ the number of parameters in model $m$, $p_2(m)$ the corresponding empirical penalty q…

A model-selection procedure chooses among a collection of models $\mathcal{M}$, and a large collection of models is one whose richness can substantially increase the minimal penalt…

Let $\operatorname{pen}_{\mathrm{opt}}^{\mathrm{gal}}(m)$ be the optimal penalty and suppose that Algorithms 5–6 are first-order optimal for a model-selection problem. Let…

Let Algorithms 5–6 be the minimal-penalty procedures applied to a collection of candidate estimators, and consider estimator-selection problems beyond the few settings for which co…

Let $\mathcal{M}$ be a continuous collection of tuning parameters or estimators, and let Algorithm 5 denote the minimal-penalty procedure considered in the paper. Suppose that a co…

Consider variable-selection problems with estimator collections and minimal-penalty algorithms, including settings beyond orthonormal variable selection and change-point detection.…

Consider penalized least-squares change-point detection, with model dimension $D_m$, sample size $n$, and a minimal penalty and optimal penalty for selecting the change-point model…

Consider variable selection with an orthonormal design and Gaussian noise, with $D_m$ denoting model dimension and $\sigma^2$ the noise variance. In this setting, the minimal penal…

Consider an orthogonal-design linear regression with Gaussian noise of variance $\sigma^2$, and use the number of selected variables $k$ as the tuning parameter for the soft-thresh…

Consider statistical settings with an estimator collection $(\widehat{s}_m)_{m\in\mathcal{M}}$, a complexity measure $\mathcal{C}_m$, and minimal-penalty algorithms based on a pena…

Let $\widehat{C}_{\mathrm{jump}}$ be the jump-based estimator of the minimal-penalty constant, constructed from a collection of candidate estimators. Jump-estimator variance conjec…

Let $\mathcal{M}$ be a collection of models, let $\widehat{s}_m$ be the estimator associated with $m\in\mathcal{M}$, and let $\mathcal{R}$ and $\widehat{\mathcal{R}}_n$ denote the…