19 problems

The nonlinear regression model is approximated by a correlated Gaussian sequence experiment whose mean is a scalar multiple of $\mu_\ast$, and the empirical Bayes procedure uses on…

Rate-dependence conjecture. The optimal rate depends on $(L,U)$ analogously to how the optimal rate depends on the problem parameters in the Gaussian case.

In the normal mean model, let the prior be either compactly supported or subgaussian, and consider the minimax regret over the corresponding class of priors. The previously establi…

Let $p$ and $n$ denote the dimension and sample size, respectively, and let the limit referred to as the normal limit hold for the debiased estimator after correcting the likelihoo…

Let $Z(\lambda)$ denote the marginal likelihood for a regularized linear regression model, with ridge, lasso, and group lasso corresponding to their respective regularizers. Quasic…

Concentration conjecture. For high dimensions $p$ and design matrices $\mathbf{X}$ that have limited long-range dependence across variables, the estimate

Consistency conjecture. This approach would yield a consistent estimator of $\delta^*$, and its rate of convergence should be studied.

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…

Let $\Phi$ be the regression matrix, let $\hat{\eta}_{\operatorname*{EB}}$ be the empirical Bayes hyper-parameter estimator, and let $\eta_{\operatorname*{b}}^{*}$ denote its limit…

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

Let $(\bar{Q},Q)$ be a fixed point of the state-evolution equations, and suppose that this fixed point is not unique. Consider the fixed point reached in Proposition (a), and let i…

Sieve–Grenander conjecture. The sieve estimator above is the same as Grenander's estimator. Consequently, the clinical version of the empirical-prior model is covered by the stated…

Consider the priors … and … where $\{\lambda_i\}_{i\ge1}$ is a sequence of positive nondecreasing real numbers. The question concerns empirical- or full-Bayes estimation of the reg…

Consider Ma's empirical Bayes confidence interval, modified by replacing its empirical Bayes point estimator with a hierarchical Bayes-type point estimator. Let $\alpha$ denote the…

Let $S$ denote the model-support variable in the empirical Bayes prior, and consider priors on $S$ that yield the minimax posterior concentration rate under prediction error loss.…

Permutation-invariant extension conjecture. In Theorem 2, the condition that $Delta^n$ is simple symmetric for every $n$ may be replaced by the weaker condition that $Delta^n$ is p…

Consider the simple small-area model with area random effects having distribution $F$ and sampling errors having distribution $G$. Let…

Let $m$ denote the number of areas, let $boldsymbol{theta}{( ,)}$ be the ordered random effects, and let $boldsymbol{theta}^{[2]}{( ,)}(boldsymbol{ ,})$ denote the predictor indexe…