32 problems

Let model selection mean adaptive selection of nonzero means, and let the underlying estimand be $bc$, belonging to one of the parameter spaces and considered under one of the loss…

Linear worst-case FDR conjecture. There exists a universal constant $C>0$ such that, for every nominal level $q$, the worst-case FDR is at most $Cq$.

BH FDR control conjecture. For every $q\in(0,1)$, every $N\in\mathbb N$, every $\theta\in\mathbb R^N$, and every correlation matrix $\Sigma$, the Benjamini--Hochberg procedure sati…

Extension conjecture. All the results of the paper should remain valid for the Spike and Slab LASSO prior.

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…

Optimal augmentation-size conjecture. Under this regime,

Consider the oracle procedure In-sample BH and the non-oracle procedure CRT + BH in the multivariate normal means testing problem. Both procedures produce $p$-values, and their exp…

Let $\tilde m_{\textnormal{opt}}$ be the choice of augmentation size that maximizes the power of BONuS for a point-mass prior, with $m$ fixed. Optimal augmentation-size conjecture.…

Extension conjecture. The theory can extend to scenarios where $n$ increases slowly as $alpha$ decreases, similar to results shown for special cases.

Equivalence conjecture. The compound $\overline{\operatorname{eBH}}$ procedure and the standard compound eBH procedure are equivalent, so that

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…

Let $\overline{H}$ be the matrix appearing in the maxT procedure, let $\iota$ denote the all-ones vector, and define … Suppose that $E$ has independent and identically distributed…

The two-group mixture model has independently drawn hypothesis labels, with each hypothesis non-null with probability $pi_1$. In the fixed non-null model, the number of non-null hy…

Let $n,d\to\infty$ with $n/d\to\delta$, let $K=K_n\to\infty$ and $\varepsilon=\varepsilon_n\to0$ slowly enough. Suppose that, for every $m\in[M]$, $\widehat\Pi_{-m}$ converges weak…

Let $n,d\to\infty$ with $n/d\to\delta$, let $K=K_n\to\infty$ and $\varepsilon=\varepsilon_n\to0$, and assume the Bayesian linear model of Assumption. Let $\boldsymbol T_\star$ be e…

Limiting FDP and TPP conjecture. The limiting false discovery proportion and true positive proportion of TPoP satisfy

Consider the multiple testing setting of Theorem, with Subbotin noise and risk bound $\Lambda_\infty+o(1)$. The empirical Bayes $\ell$-value procedure is the procedure defined in.…

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.

Conjecture on faster dimensional growth. Similar theoretical results for the RESS and refined RESS methods should also hold when

Let $\mathbf{X}=(X_1,\ldots,X_d)$ and $\mathbf{Y}=(Y_1,\ldots,Y_d)$ be $d$-dimensional Gaussian vectors with common mean $\bm{\mu}=(\mu_1,\ldots,\mu_d)$ and covariance matrices…

Let $N$ be the total number of Monte Carlo samples, and consider the instance-wise upper bound on $\mathbb{E}[N\mid\{P_i^{\mathrm{fMC}}\}=\{p_i\}]$ given for the confidence bounds…

Closed-testing optimality conjecture. The optimality of closed testing will translate to FDP control in countable and even uncountable multiple testing problems, when competing met…

Let $F_X$ and $F_Y$ be distribution functions for two samples, and let $alpha$ be the target familywise error rate. Under assumptions (a:iid) and (a:F), consider the Dirichlet two-…

Let $n$ denote the sample size and let $alpha$ denote the familywise error rate, with approximation formulas relating the familywise error rate to the pointwise type I error rate.…

The paper considers asymptotic uniformity when the population distribution is allowed to vary with the sample size. Uniformity conjecture. The basic multiple-testing methods are co…