17 problems

Consider the nonparametric estimation problems for the space-dependent diffusivity, transport, and source coefficients in stochastic evolution equations described above, with the r…

The rodeo estimator is studied in a triangular-array setting: the regression function $m$, design density $f$, dimension $d$, and number $r$ of relevant variables may depend on the…

LCA extension conjecture. The LCA principle should extend to the problem of estimating optimal transport maps and couplings.

Under analogous small-noise and smoothness assumptions for dynamic treatment regimes with multiple categorical treatments per stage, the regret rate … should be minimax-optimal. La…

Let the minimax risk be considered in the circular-response nonparametric regression setting described above, with regression functions of smoothness $beta$ and estimators based on…

Let $n$ players be ranked under the Bradley–Terry–Luce model, with skill parameters $\theta^*$, sampling probability $p$, comparison count $L$, permutation space $\mathfrak{S}_n$,…

Let $A$ be an adjacency matrix from a sparse stochastic block model, with sparsity level tending to zero as the network size increases. The posterior contraction rate for estimatin…

Shamir's conjecture. The rate

Let $\expholder$ denote the smoothness parameter in the Hölder class of two-dimensional totally positive densities, and let $N$ be the sample size. The preceding bounds give a rate…

Let $F(x)=f(\langle v,x\rangle)$ be a single-index regression function on $\mathbb{R}^d$, where $v\in\mathbb{R}^d$ is an unknown unit index vector and $f:\mathbb{R}\to\mathbb{R}$ i…

Let $n_1=n_2=n$, let $\theta$ be the worst-case anti-Monge matrix constructed above, and observe $y=\theta+\varepsilon$, where the entries of $\varepsilon$ are independent and dist…

Let y=\theta^*(\boldsymbol{[?])+[?] be observed under the latent-permutation anti-Monge model, and let $\bar{\mathcal{M}}$ denote the unrestricted anti-Monge parameter space. The…

Let $d$ be bounded, let $0<\lambda\leq 1$, and let $\mathcal{F}_{d}(\lambda,\kappa)$ be the function class of $d$-index models. A function $f_{\lambda}$ is $d$-variate when it is d…

Non-minimax rate conjecture. The rate achieved by this estimator is not minimax optimal, not even in the exponent.

High-dimensional C2-minimax PRF conjecture. PRF models reaching $C^2$-minimax rates of convergence could also be derived in dimension $d$.

Proportional-size partitioning conjecture. Such a partitioning scheme should perform better than the BPRF scheme and could reach the minimax rate for $C^2$ functions.

Consider pointwise estimation of a cumulative distribution function when the unknown density is $m$ times differentiable and the error distribution is ordinary smooth. Fan's conjec…