167 problems

Simultaneous-limit conjecture. We conjecture that

Let $\bmu$ and $\bDelta$ be the preliminary estimators, let $\hn$ be the preliminary density estimator, and let the untruncated one-step estimator be the estimator obtained from th…

Let $\widehat{\sigma}_{1,n}^2$ be the maximum likelihood estimator under a misspecified decay parameter, and let $C_{\nu,\alpha_i}$ denote the Matérn normalization constants for pa…

Let $\mathbf{a}(\bm{\theta})$ denote the asymptotic bias function, and let Theorem … remains true for a larger class of asymptotic bias functions. The conjecture concerns extending…

Weak-dependence extension conjecture. An extension of the multivariate results to the other type of weak dependence discussed in the paper should also be possible.

Arnold's density-profile conjecture. Asymptotically for large ${\bf d}^m$, the density at $s$ is

Arnold's conductor conjecture. The conductor is weakly asymptotic to

Consider the spin-half quantum model in which the state is parametrized by a two-dimensional equatorial-plane parameter, with rotationally symmetric priors and sample size $N$. A n…

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

Innovation-variance scaling conjecture. The same scaling of the estimator's asymptotic distribution by $\sigma_{\varepsilon}$ should hold when $\kappa<1$, in which case standard in…

Let the observations form a stationary $\alpha$-mixing sequence, rather than an i.i.d. sample, and let the estimators and theoretical results considered in the paper be applied to…

Asymptotic linear-form conjecture. Under the setting of Theorem InfluenceQuantiles-main,

Let $W$ be a two-sided standard Brownian motion with $W(0)=0$, and let … Write $f$ for the density of $Z$. Balabdaoui–Wellner conjecture. The density $f$ is strongly log-concave, m…

Let $B\geq 1$ be fixed, let $\nu$ be a positive integer, and let $\mathsf q\in\mathbb R$. Let $\Psi_B$ and $\psi_B$ denote the distribution function and density function, respectiv…

The nonparametric Kiefer--Weiss problem considers sequential testing with error probabilities tending to zero, equivalently as $c\to\infty$, where $c$ is the parameter governing th…

Let $\widehat{\beta}$ be the estimator based on the quantized data and let $\widehat{\beta}^{o}$ be the ordinary least squares estimator based on the uncompressed data. Write $R_n$…

Dense-limit conjecture. Asymptotic distributions can be derived in the dense setting as well, which would further strengthen the results.

Let $\theta$ denote the parameter of the skewed stable Ornstein–Uhlenbeck process, with stability index $\alpha$ and skewness parameter $\beta$. Let $\ell_n(\theta)$ be the true lo…

Let $\mathcal{F}_k$ denote sieve spaces used for boosting, and let the associated hyperparameters be selected from the data, for example by sample-splitting or cross-validation. Da…

Let $p$ and $n$ denote the dimension and sample size, respectively, and let the limit referred to as the normal limit hold for the debiased estimator after correcting the likelihoo…

Let $F$ be a distribution function and let $\mathbb G_F$ denote the Gaussian process introduced earlier in the paper. The paper proves that, under the stated assumptions—including…

Let $\alpha$ denote the local regularity parameter governing the nonsmoothness of the density at its mode, and consider the asymptotic distribution of the kernel mode estimator for…

The averaged Zaremba conjecture. For $N\in\mathbb{N}$,

Let $\widehat{\lambda}_{0,\delta_n}$ denote the AR(2)-based estimator of the spectral peak location for the stationary time series $\{X_{t,\delta_n},t\in\mathbb{Z}\}$. The estimato…

An incomplete U-statistic is formed from a kernel whose dependence structure is represented by a deterministic design and whose asymptotic normality is established under the assump…