20 problems

Let $\lambda^2_c$ denote the correctly calibrated version of the likelihood-ratio test statistic, and let $\lambda^2_u$ and $\lambda^1$ denote the alternative test statistics consi…

Consider the two-mean Gaussian mixture model … with homogeneity null hypothesis $H_0:p=0$ or $t_1=t_2$ and alternative $H_1:p\in(0,1)$, $t_1\ne t_2$. Let the likelihood-ratio stati…

Let $X_1,\dots,X_n$ be i.i.d. standard normal random variables under the homogeneity null hypothesis $H_0:p=0$ or $t=0$ in the contaminated Gaussian mixture model … where…

Consider the contaminated Gaussian mixture model … with unknown mixture weight $p$ and mean parameter $t$, and let the quadratic approximation and the likelihood-ratio statistic be…

Alternative-hypothesis intrinsic Bayes factor conjecture. If $k_1-k_0>r$, then

Let $T(x_1,\ldots,x_n):=2[h_1(x_1,\ldots,x_n)-h_0(x_1,\ldots,x_n)]$ be the nonnegative likelihood-ratio test statistic, where $h_0$ and $h_1$ are the hypotheses' maximized log-like…

Consider the simple linear regression change-point model with independent, zero-mean normal errors, and let $W$ be the maximum likelihood ratio statistic … where … with…

Consider spiked Wigner models with different priors on the spike, including the Rademacher, spherical, and sparse Rademacher priors. Let ${\mathcal L}(Y;\lambda)$ denote the likeli…

General k-monotone-cone conjecture. Results analogous to those of Theorem 1 should hold for the general $k$-monotone cone $K_{\uparrow,k}$.

Optimal normal-approximation rate. The error rate

Let $f_0$ be the true regression function in a convex regression model, and let $x_0$ be the point at which the function value is evaluated. The likelihood-ratio-test-based procedu…

Doss's localization conjecture. The quantity $E_{n,b}$ is asymptotically negligible for sufficiently large $n$ and $b$. This is intended to justify localization of the likelihood r…

Assume the regression model holds with $Ee^{t\epsilon_{n,i}^2}<\infty$ for some $t>0$. Let $r_0$ be concave, satisfy $r_0(x_0)=y_0$, be twice continuously differentiable in a neigh…

Let $2\log\lambda_n$ denote the likelihood ratio statistic for testing the value of a concave regression function, based on a null-constrained concave least-squares estimator. Doss…

Let $Gamma$ be the set of parameter pairs $(\alpha,\beta)$ for which the replica-symmetric formula vanishes, and let $Gamma'$ be its interior. For the likelihood ratio…

Let $LR_n$ be the likelihood ratio, let $\widetilde{LR}_n^{-}$ and $\widetilde{LR}_n^{+}$ be the proxy likelihood ratios associated with the lower and upper proxy covariance models…

Let $LR_n$ be the likelihood ratio and let $\widetilde{LR}_n$ be the proxy likelihood ratio used in the paper; let $E_0$ denote expectation under the null model. Likelihood-ratio a…

Let a likelihood ratio test concern a parametric family of probability distributions, and suppose the alternative is restricted by replacing that family with a subfamily of lower d…

Let a likelihood ratio test be constructed for a statistical hypothesis concerning a parametric family of probability distributions, and let the alternative hypothesis be restricte…

High-dimensional alternative-distribution conjecture. In the high-dimensional case, there could be new results regarding the limiting distributions of likelihood ratio test statist…