7 problems

Given observations $\mathbf x=(x_1,\dots,x_n)$, define … where the observations are independent e-values. Gaffke's conjecture. The quantity $K_n(\mathbf x)$ is a valid $p$-value wh…

Let $U_1,\dots,U_n$ be standard uniform random variables whose dependence is described by a Gaussian copula with positive correlation coefficients, and let sub-uniformity denote th…

Let $U_1,\dots,U_n$ be random variables with a dependence structure given by an elliptical copula or an Archimedean copula, and let sub-uniformity denote the property that the gene…

Let $K$ be a positive integer, let $f_1,\dots,f_K$ be admissible calibrators, and let $(\lambda_1,\dots,\lambda_K)\in\Delta_K$. Suppose that, for each $k=1,\dots,K$, $f_k$ is conve…

Conservativeness conjecture. The $p$-value will only be slightly conservative if the number of layers skipped is small relative to the size of the subsets considered.

Validity conjecture. The random variable $\hat{p}$ is a valid $p$-value: uniformly over the composite null hypothesis space, its distribution stochastically dominates the uniform d…

Multivariate extension conjecture. A version of Theorem 2.1 should hold for $d$-dimensional Gaussian vectors $X_j$ orthogonalized by the inverse square root of their covariance mat…