47 problems

Faster-rate conjecture. The nonparametric maximum likelihood estimator has a faster rate of convergence.

Cross-validation adaptation conjecture for compositions of moment estimators. In view of techniques from Richter and Dahlhaus (2018), a result similar to the cross-validation resul…

Let $F$ be an unknown distribution function on ${\mathbb R}^d$, with observations from the multivariate in-out interval-censoring model, and let $\hat{F}_n(\underline{t}_0)$ denote…

Let $W_\rho$ be the Wigner function of a quantum state, and let $\mathcal{A}(\alpha,r,L)$ be the smoothness class considered in the paper. Consider testing … against alternatives s…

MSE conjecture. The mean squared error of the improved estimator satisfies

Structural convergence-rate gap conjecture. The gap in convergence rates between the smooth regime $\beta>d/2$ and the rough regime $\beta\leq d/2$ is structural rather than an art…

Rate-dependence conjecture. The optimal rate depends on $(L,U)$ analogously to how the optimal rate depends on the problem parameters in the Gaussian case.

Let $\mathcal{B}$ denote the assignment boundary in a boundary discontinuity design. Consider new distance-based (isotropic) nonparametric smoothing estimators that either account…

Finite-component conjecture. For every $\alpha\in[0,1/2]$, the estimator $\widetilde{\pi}_{n,\alpha}$ has no more than $N=X_{(n)}+1$ components.

Beta-prior conjecture. The appropriate prior for mixtures of Geometric and Negative Binomial distributions is most likely a $\text{Beta}(a,b)$ prior, with $a,b otin (0,infty)$.

Let $r_{mu,n}$ denote the convergence rate of the outcome-regression estimator, let $gamma_0$ govern the simulated propensity-score distribution, and let $b_n$ be the clipping thre…

Let $e_e$ be the order of a local polynomial regression estimator for the propensity score, let $d$ be the covariate dimension, let $r_{e,n}$ denote its global consistency rate, an…

The A-OTF method uses training data consisting of input-output samples for learning an approximation to a transport map, and its training objective is the optimization problem in e…

Single-trajectory generalization conjecture. Similar arguments should establish the same lower bounds for arbitrary $d\in\mathbb{N}_{\geq 2}$ and $\beta\in\mathbb{R}_{\geq 1}$, wit…

Consider the Poisson mixture model with unknown mixing distribution $Q$ supported on $[0,\theta_\ast]$, and let $\hat Q_{\rm MLE}$ be its nonparametric maximum likelihood estimator…

Pointwise loss representation conjecture. The finite, high-dimensional optimization reduction described for the squared-error loss can be generalized to loss functions admitting th…

Let $p_{\nu,t,y}$ denote the density appearing in the spatially ergodic semilinear stochastic partial differential equation setting, and let $h$ and $\sigma$ be the localisation an…

The kernel-based density-weighted average derivative estimator is studied under small bandwidth asymptotics, with bootstrap-based inference procedures considered through the nonpar…

The paper considers estimation of the drift coefficient of a stochastic differential equation from discretely sampled trajectory data; this coefficient is related to the mean funct…

Donsker property conjecture. The analogous Donsker property should hold for the lower-dimensional classes A0a) and A0b) whenever

The paper considers average treatment effect estimation using deep neural networks, with regression functions having smoothness parameter $s$. Nearly parametric rate conjecture. Th…

Let $L_{T,n}(y)$ denote the Riemann estimator of the local time at $y\in\mathbb{R}$ for a symmetric stable process, and let $\mathbb{E}[L_T(y)\mid\mathcal{G}_n]$ be the correspondi…

Let $T$ denote the observation horizon, let $d$ be the spatial dimension, and consider the upper bounds for the squared relative risk of the estimator $\hat{\sigma}^2_*$ of…

Let $F(x_0)$ be the distribution function of particle displacement evaluated at $x_0$, and let $\hat{F}_{h_*}(x_0)$ be the estimator with the bandwidth $h_*$ used in the preceding…

Impossibility conjecture. There is a regime in which $f$ cannot be estimated consistently when $t>\frac{3}{2}d$.