88 problems

Consider the spiked Wigner model with the two-point prior … where $p\in(0,1)$, and let $\lambda_c$ be the information-theoretic threshold. In the region $\lambda_c<\lambda<1$, non-…

Let $X(1),\ldots,X(N)$ be an i.i.d. sample, and let $\widehat H_i$, $i=1,\ldots,M$, be a permutation bootstrap sample of the estimator $\widehat H(j_1,j_2)$. Let the scales $j_1$,…

Let $n$ be the number of vertices in a uniform attachment tree, and let the noisy edge probability be $o(\log n/n)$. A confidence set for the root vertex is optimal if it has the b…

In the binary case, let $t(\bx,\by)$ be a statistic that depends only on the response vector $\by$. Write $\hthe_a$ and $\hthe_b$ for the corresponding labeled-data and prediction-…

Let $t(\bx,\by)$ be a statistic that depends only on the response vector $\by$. Write $\hthe_a$ and $\hthe_b$ for the corresponding labeled-data and prediction-powered estimates, r…

Let Holm's multiple-testing procedure be considered under the assumptions in force for the procedure, with admissibility meaning that it cannot be uniformly improved without invali…

Let $C_\lambda$ be the matrix whose kernel determines the true eigenvalue set, and suppose … For $i\geq 1$, let $\chi_i^2$ denote a chi-square random variable with $i$ degrees of f…

Extension conjecture. The approach based on weak convergence of $sigma$-fields and strong maximal nuisance-ancillarity should extend, mutatis mutandis, to situations in which the l…

Let a heterogeneous hypergraph be generated under the model considered in the paper, and let the maximal clique estimator be the estimator defined by Algorithm 1. The recovery thre…

Consider rectangular spiked matrix models with general rotationally invariant noise, and let the optimal orthogonal approximate message passing (OAMP) algorithm be the variant that…

Let $\beta\geq -1$ and $\gamma>0$, and for $n\in\mathbb{N}$ set $N=\lceil n/\gamma\rceil$. Under $\mu_0$, draw $\mathbf{z}_1,\ldots,\mathbf{z}_N\sim N(0,I_n)$ independently. Under…

Incompatibility conjecture. Probabilism and calibration are fundamentally incompatible: there is no construction of a data-dependent probability distribution quantifying uncertaint…

Let the conditional distribution of the observed responses in a dependent mixture model belong to an exponential family, and suppose that its partition function is nontrivial and e…

Consider signal plus noise models beyond the four models analyzed in the paper, where the empirical eigenvalue distribution may differ from the specific parametric forms in the pap…

Consider signal plus noise models with a signal parameter and confidence intervals constructed by the procedure described in the paper. In particular, distinguish delocalized signa…

Let $H$ be a hypothesis concerning the values of a non-linear function of the model parameters, such as a hypothesis about a ratio. False-confidence conjecture. False confidence te…

LCA extension conjecture. The LCA principle should extend to the problem of estimating optimal transport maps and couplings.

Let $\mathbb{X}_{n, 0}^*$ and $\mathbb{X}_{n, \lambda}^*$ be samples produced by the subsampling procedure with threshold values $0$ and $\lambda$, respectively, and let…

Drton–Xiao's conjecture. Based on simulations, the distribution of $T_{\boldsymbol{w}}$ is $\operatorname{Cauchy}(0,1)$, the Cauchy distribution with center $0$ and scale $1$.

Nonlinearity conjecture. All non-linear hypotheses about $\Theta$ have at least a mild case of false confidence.

Let $Y_1,\ldots,Y_n$ be the observations, let $\bar{Y}_n$ be their sample mean, and define the sample variance by … The function $g(\nu)$ is the efficient score function, up to the…

Let $M=M_N=N^\alpha$ with $\alpha<1$, and consider the growing-rank spiked Wigner model with a fully factorized centered prior distribution having $D$-bounded support, under the hy…

Let $sigma_{delta}^2(\cdot,\widehat{\alpha})$ be the variance function of the model, with $sigma_{Y,\delta}^2(\cdot,\widehat{\alpha})$ denoting the contribution associated with…

Leighton–Samaniego's conjecture. The deterministic minimax risk is of this order:

Let $d$ be a degree parameter. A degree-$d$ polynomial algorithm is an algorithm represented by a polynomial of degree at most $d$, and a “robust” algorithm is an algorithm whose r…