121 problems

Let ${\mathbb D}_\beta$ denote the limiting distribution associated with the flatness parameter $\beta$, for $0<\beta<\infty$. Consider Brownian motion with no drift, corresponding…

Adaptive-rate conjecture. A result similar to the established bound

Let $f_0$ be an $b1$-Hölder continuous density, with $b1\in(0,1]$, and consider the Bayes histogram estimator and its convergence rate from Theorem. Sharpness conjecture. The rate…

Let $k\geq 1$ be an integer, and consider a $k$-monotone density whose $k$th derivative satisfies $(-1)^k g_0^{(k)}(x_0)>0$ at the point $x_0$. In the corresponding regression-spli…

General dominating-bias conjecture. This is a general property of nonparametric problems with dominating bias.

Let $(\beta)_{1-\tau_A}$ be the population version of the transformed generator, where $\tau_A$ is the Kendall-type parameter associated with the Pickands dependence function, and…

Let $A$ be a Pickands dependence function with discrete Pickands dependence measure $\vartheta$ having full support, and let $(A_\vartheta)_\alpha$ denote the associated transforma…

The transformer architecture is used to estimate a context function $\mathbf{h}$, with smoothness index $\beta_2\leq 1$, and can represent a Nadaraya--Watson estimator. If the data…

Let $k\ge1$. On a regular problem with an equispaced dense design, consider a degree-$k$ temporal local-polynomial forecaster on the tangent bundle, obtained by geodesic or barycen…

Consider the nonparametric density demixing model with $K$ component densities, sample-size parameters $N$ and $n$, and a plug-in estimator of the mixing matrix $\Pi$. Plug-in erro…

Combinatorial-term conjecture. The full minimax rate should include this combinatorial term as an additive contribution, in addition to the smoothness term.

Let $M_p(\epsilon,L)$ denote the constrained moment-matching quantity used in the paper, with tolerance parameter $\epsilon$ and moment parameter $L$. Constrained moment-matching c…

The interpolation regime concerns testing under non-smooth $\ell_p$ norms, with the rates given by the upper bounds for fixed odd $p\geq 1$. Tightness conjecture. The interpolation…

Let the MLE be the maximum likelihood estimator for multivariate mixtures of power series distributions with a conditional independence structure, and let $\ell_p$ denote the relev…

Sugiyama et al.'s tuning conjecture. Optimal estimation of the density difference may be difficult when the kernel order $\gamma$ is unknown.

Ye et al.'s rate conjecture. A better rate than $\mathcal{O}(h+1/\sqrt{t}+1/ht)$ should not be attainable even when the mixing distribution has higher-order smoothness.

Let $m$ be a density model indexed by $\theta\in\Theta$, with projections $Q_\theta:m\to m_\theta$ satisfying the regularity assumptions described above. Let $k$ denote the number…

Consider the SPDE approach with coefficient regularity parameter $\beta$ and the associated posterior contraction rate for the unknown function under the conditions described above…

Let $F_0$ have a continuous positive density $f_0$ on $(0,M)$, where $M>0$, and let $t_0\to(a,b)$ be an interior point. Let $\hat F_n$ be the nonparametric maximum likelihood estim…

Intrinsic-dimension conjecture. In general, the parameter $d$ can be replaced by $d_{\mathrm{int}}$ in all the results: tempered overfitting occurs whenever…

Let $x$ be the point at which the function is estimated, and let \underaccent{\bar}{x} and $\bar{x}$ be boundary points delimiting an interval on which the underlying function…

Let the truth and the prior both be $alpha$-smooth for some large value $alpha\bin (0,+infty]$. Suppose that the parent sets are norming sets with cardinality … where…

Combined margin-density conjecture. Under this condition, the combined margin and density condition holds.

Berrett–Györfi–Walk conjecture. Without the strong density assumption, the convergence rate in the privatized case could be arbitrarily slow.

Let $A$ be the observed adjacency matrix, let $M$ denote the graphon-generated mean adjacency matrix, and let $\ell$ be the estimation loss. The computational optimality conjecture…