42 problems

Let $n$ be the sample size, let $d$ be the dimension of the covariate, and let $s$ be the smoothness parameter. For mean integrated square error (MISE), consider estimation over an…

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…

Let $X_1,\ldots,X_n$ be independent and uniformly distributed on the sphere $\mathbb{S}^d$, let $\varepsilon_1,\ldots,\varepsilon_n$ be independent standard Gaussian variables inde…

Let $M$ be a compact Riemannian manifold, and consider non-parametric regression on $M$ together with the corresponding Gaussian white noise experiment. The manifold extension conj…

Consider the heteroscedastic nonparametric regression model … where $ta=V_0(X)^{1/2}ta^{\star}$, $ta^{\star}$ is bounded almost surely, and satisfies…

Let $\hat{f}$ be a local nonparametric estimator for a regression function, and suppose that it exhibits optimal global sup-norm convergence. For an adversarial perturbation set…

The estimated density flow map $t\mapsto \widehat R_t$ is compared with the ground-truth map $t\mapsto R_t^\ast$ through the integrated squared Hellinger distance … The preceding r…

Li–Racine conjecture. The associated optimal bandwidth should diverge to infinity.

Let $\hat{m}^{(\mathrm{LL})}_{n,b}$ and $\hat{m}^{(\mathrm{NW})}_{n,b}$ denote the local linear and Nadaraya–Watson estimators with Dirichlet kernel, respectively. Near the boundar…

$M^{-1}$ variance conjecture. If $\nu$ is dominated by $\lambda_\mathbb{X}$, then this variance attains the order of

$M^{-1}$ variance conjecture. If $\xi\sim\nu$, the variance could potentially attain the order of

Let $\Sigma$ be a covariance matrix, and let the rows of a design be of the form $\Sigma^{1/2}Z_i$, where the vectors $Z_i$ are isotropic and block-dependent sub-Gaussian vectors.…

Large-$N$ convergence-rate conjecture. The convergence rate should be slower than

The setting is local polynomial regression for spatial data on $\mathbb{R}^d$, with bandwidths chosen for estimating the regression function and its derivatives from spatially depe…

Laplace-privatization rate conjecture. On the basis of data privatized in this way, the same rates of convergence as those derived below should be attainable.

Slow-rate conjecture. If the SDA is not fulfilled, the rate of convergence can be arbitrarily slow.

General-input conjecture. Up to constants, the same lower bounds as in the paper can be derived for this model. The paper establishes the lower bounds for a more restricted input a…

Let $\mathcal{G}$ be the class of generalized additive models considered in the paper, and let $n^{-(2+d)/(4+4d)}$ be the lower-bound rate established for arbitrary mean-zero Gauss…

Shape-constraint performance conjecture. The performance of the estimators could be improved by incorporating additional shape constraints.

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…

Deep-network extension conjecture. Theorem 1 should extend to very deep fully connected neural network classes in which the number of neurons per layer is fixed and the number of h…

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…

Consider the bike-sharing data set described above, with rental-bike count as the response and attributes including season, hour, and working day. For each combination of season, h…

Let $f^\dagger$ range over functions in a bounded-variation unit ball, and let $\mathcal{GMS}_{f,\varepsilon,n}$ be the graph Mumford–Shah functional defined in the source's equati…

Consider the partially linear panel-data model … where $X_{it}$ is real-valued, $Z_{it}\in\mathbb{R}^d$, $u_{it}=\alpha_i+\gamma_t+\varepsilon_{it}$ with…