31 problems

Let $\bm{S}$ be the sample scatter matrix and let $\widehat{\bm{\Sigma}}^U=n^{-1}\bm{S}$ be the unbiased estimator of the covariance matrix $\bm{\Sigma}$. For an orthogonally equiv…

Let $\mathcal C(\sigma_{\min},\sigma_{\max})$ be the bounded-spectrum correlation class, and let $\widehat{\bm{\Sigma}}$ estimate the latent correlation matrix $\bm{\Sigma}$. Sharp…

Let $\mathrm{F}_{\alpha}\rbra{P}$ and $\mathrm{F}_{\alpha}\rbra{\rho}$ denote the high-order functionals of a discrete distribution $P$ and a quantum state $\rho$, respectively, an…

Let $\mathcal{F}_{K,B}$ denote the class of monotone functions considered in the paper, with influence budget $K$ and range or norm bound $B$. Under the paper's noisy-observation m…

Let $p,q>0$ be distinct exponents, and for each $r>0$ let $\mathcal{L}_r$ be the class of losses whose local behavior near $a=\theta$ is $c(\theta)|\theta-a|^r+o(|\theta-a|^r)$ wit…

Let $p=p_{i\in\mathbb{N}}$ be any distribution on $\mathbb{N}$, let $\tilde p$ be an arbitrary estimator based on $n$ observations, and let $V^*(p)$ denote the population variance-…

Let $X$ be a sample from the uniform location estimation problem, let $\theta$ denote the location parameter, and let $\hat\theta_k(X)$ be an estimator indexed by $k$. Write…

Let a multiple-index model have a central space, and let the generalized signal-to-noise ratio be denoted by $\operatorname{gSNR}$. Lin et al.'s conjecture. The minimax optimal rat…

Minimax linear estimation conjecture. The minimax linear estimation strategy should yield an optimal estimator for any such function class $\mathcal{F}$.

Variance-knowledge conjecture. Knowledge of $\sigma^2$ is not necessary to construct an estimator achieving the minimax rate for unbounded constraint sets $K$.

Optimal-error order conjecture. The optimal error has true order of magnitude

Feasibility conjecture. Estimating $U_{n,\lambda n}$ under such weaker, non-parametric power-law assumptions is possible only if $\lambda$ is small and/or the loss function is suff…

Let $q=p$, with $p\notin\mathbb{N}^*$, $p>1$, and let $\tau(p)<0$. Consider estimation of the $\mathbb{L}_p$-norm over the anisotropic class…

Let $K$ be a convex cone, and consider minimax lower bounds for estimating a signal over $K$ obtained from packing sets. Cone-specific packing conjecture. For specific convex cones…

In the slow regime, namely when $q>1+2/(d+2\beta)$, consider estimators for the linear inverse problem of estimating functions over the class $BV_L$, with singular values behaving…

Logarithmic-factor conjecture. The logarithmic terms appearing in the upper bound are superfluous and arise as an artifact of the proof techniques.

Let $n\in\mathbb{N}$ be the sample size, let $\alpha>0$ be the privacy parameter, and let $\omega_{TV}$ and $\omega_H$ denote the total-variation and Hellinger moduli appearing in…

For $\lambda>0$, consider estimating the qubit-state parameter in the Hellinger-distance asymptotic regime, with the corresponding Poisson limit parameter $\mu=n\lambda$. Minimax-c…

The estimator $chat{G}_{\textrm{s}}(chat{\theta}_n)$ is constructed by splitting the samples into independent estimators $chat{\theta}_n^{(1)}$ and $chat{\theta}_n^{(2)}$, and appl…

Let a minimax estimator be associated with a convex loss function, and let its penalty function have the equimax property, meaning that the relevant maxima on the successive subint…

Let $S\subset\mathbb{R}^d$ satisfy the assumptions above, and let $C_\alpha(\mathcal{X}_n)$ be the plug-in estimator of $\mu(S)$ constructed from a sample of size $n$, with known r…

Ordinal minimax-rate conjecture. The dependence of the squared $\ell_2$ minimax risk on the problem parameters $n$, $d$, and the graph topology is identical to that for the Paired…

For a generalized linear model with weakly collinear design, let model selection use a single selected estimator and let averaging use estimators from all models with properly chos…

Let $d$ be the dimension and $r$ a positive integer. Let $\mathcal P_r^{all}$ denote the class of all convex polytopes with at most $r$ vertices, without requiring them to be conta…

Let $d$ be the dimension, let $r\geq d+1$, and let $\mathcal P_r$ denote the class of convex polytopes with at most $r$ vertices contained in the unit cube. Let…