18 problems

Brown's conjecture. The prior $g(\theta)$ leads to an admissible estimator, regardless of the density $p$ and the loss function $W$.

Let $\psi$ denote the mean contrast and let $\xi$ denote the variance ratio in the Behrens--Fisher problem. Consider prior-free procedures that are exactly and uniformly valid for…

Let $\mathscr{L}$ be a rich loss space, for example the space of all continuous losses satisfying mild growth conditions, and let optimality be defined by a natural optimality noti…

Let $p,q>0$ be distinct exponents, and for each $r>0$ let $\mathcal{L}_r$ be the class of losses whose local behavior near $a=\theta$ is $c(\theta)|\theta-a|^r+o(|\theta-a|^r)$ wit…

Donoho's conjecture. A different approach is necessary to develop a sequence of non-asymptotic minimax lower bounds that converges to the correct asymptotic limit.

Conjecture 8.1. Proposition 4.2, namely the EPV criterion for the corresponding single-observation problem, remains valid for repeated observations.

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…

Consider the proposed sequential change-diagnosis scheme, operating over possible change-points. Uniform error control means that its error probability is bounded by the prescribed…

Variance-knowledge conjecture. Knowledge of $\sigma^2$ is not necessary to construct an estimator achieving the minimax rate for unbounded constraint sets $K$.

Singh's conjecture. For any exponential family, bounded total regret is not possible even when the priors are compactly supported.

Consider estimating the mean vector of a $p$-variate Normal distribution when the covariance matrix is an unknown multiple of the identity. For the class of generalized prior distr…

Approximate complete-class simplification conjecture. Further simplifications can be obtained by allowing approximate versions of complete class theorems, Bayesian estimators, opti…

Let $\mathcal{F}$ be a class of distributions for which the assertion of Osband's principle holds when $\mathcal{F}$ contains all distributions with finite support. In particular,…

The setting is a multidimensional exponential family, with MAP estimators of the form described in the preceding discussion and with KL divergence as the loss. Multidimensional suf…

Let $L$ be the Frobenius loss, and let $L_\infty(\eta\mid\mathbf{x})$ denote the asymptotic loss of a continuous singular-value shrinker $\eta$ at any signal singular-value vector…

Herschkorn's conjecture. The inequality

Let $s$ be a proper local scoring rule, and suppose that $s$ can be generated from some concave gauge choice. The standard gauge choice is $\phi=q_0s$, and write $u_i=q_i/q_0$ for…

Let $\delta^H$ and $\delta_U$ denote the generalized Bayes estimators corresponding to $a=2$ and $a=p$, respectively, and let $\delta^{GB}_a$ be the generalized Bayes estimator ind…