17 problems

Complex fractional Ornstein–Uhlenbeck estimation conjecture. The strong consistency and the asymptotic normality of the estimator $\hat{\gamma}_T$ hold for the complex-valued fract…

Parameter-estimation rate conjecture. While the mixture of experts model is more complicated, a similar rate for $\eta_{{\sf est}}$ should carry over to this setting; in particular…

EGD statistical-radius conjecture. There exist universal constants $C_{3},C_{4},C_{5}$ such that, whenever $n\geq C_{3}(d\log(d/\delta))^{4}$ and…

Let $(\gamma_i)_{i\in\mathbb{N}}$ be the sequence of mixing parameters, and let ${\bm{\mu}}_1,\ldots,{\bm{\mu}}_k$ be the mixture component means. A consistent estimator is an esti…

Consider the generalized modulated Ornstein--Uhlenbeck kernel … The estimation procedure uses a weight function $w$ and a minimization algorithm. Performance conjecture. The subopt…

The estimators are considered with $m=1$ and $m=2$, where $m$ is the dimension used in the integral approximation, and their asymptotic variances are given in Theorem MCE-conv, ite…

Joint-sampling extension conjecture. Similar to the results of Cialenco, Delgado-Vences, and Kim and of Bibinger and Trabs, the theorems on these estimators should remain true when…

Minimax lower-bound conjecture. The lower-bound conclusion stated for the regime $M/\sqrt N\gtrsim 1$ should also hold when $M/\sqrt N\to 0$; in particular, the minimax estimation…

Square-root rate conjecture. All estimators in the preceding display should exhibit a rate of convergence equal to $\sqrt{nm}$. The consistency of these estimators follows from the…

In the nonlinear one-bit observation model, let $\bm{y}=\operatorname{sgn}(\bm{X}\bm{\theta}^*)$, where $\bm{\theta}^*$ is a sparse vector with $s$ nonzero entries. Let $n_0$ and…

Generalized singularity-structure conjecture. The same result holds for general $k_0$; namely,

Let an orbit lie on a Moser invariant curve, so that its Lyapunov exponents are exactly zero, and let $n$ denote the number of iterations. Consider the standard deviations of all t…

Analytic-coefficient moment parametrization conjecture. This parametrization holds when $a(x,\boldsymbol\theta)$ and $b(x,\boldsymbol\theta)$ are analytic in $x$ and…

Analytic moment parametrization conjecture. If $a(x,\boldsymbol\theta)$ and $b(x,\boldsymbol\theta)$ are analytic in $x$ and $\boldsymbol\theta$, then $\boldsymbol\theta$ is a gene…

Let $H$ be the Hurst parameter of the long-memory noise, and let $\widehat{\theta}_T$ and $\widehat{\rho}_T$ denote the estimators of the model parameters $\theta^*$ and $\rho^*$ b…

Optimality conjecture. The above-mentioned modulation schemes are essentially optimal, and the gap between the lower bound $e^{-CT/2}$ and the upper bound $e^{-CT/3}$ is due to the…

Let $\mathcal{G}(n,a/n,b/n)$ be the sparse planted partition model with two equal-sized clusters and parameters $a>b>0$. A consistent estimator of $a$ and $b$ is an estimator whose…