66 problems

Faster-rate conjecture. The nonparametric maximum likelihood estimator has a faster rate of convergence.

Huh–Sturmfels' conjecture. The maximum likelihood degree satisfies

Collapsing conjecture. Suppose that

Ma et al.'s conjecture. If the MLE satisfies the asymptotic normality condition, then the estimation problem is super-teachable.

Let $u(k,q)$ denote the size of the smallest set of uniqueness for the positive function class $\left(\mathcal{B}^{k}_{q}\right)_+$. In particular, consider the case $q=2$. Minimum…

Let $\hat{\theta}_n$ be the maximum likelihood estimator of the location parameter $\theta$ in the Pearson Type VII model, and let $m$ denote its shape parameter. Mean-squared-erro…

Let $m>1/2$, and let $D_n(t)$ denote the likelihood-equation function for the Pearson Type VII location model. Write $c_m$ for the constant appearing in the limiting root-count law…

Let (\hat\gamma_1\hat\pmb v_1,\dots,\hat\gamma_r\hat\pmb v_r) be critical points of the likelihood, with summary statistics consisting of…

Let $Y_1,\ldots,Y_n$ be the observations, let $\bar{Y}_n$ be their sample mean, and define the sample variance by … The function $g(\nu)$ is the efficient score function, up to the…

Stabilisation conjecture. For star-shaped graphs, the $\Lambda$-MLE given any $f$-stabilisation of a sample $f$ is a $\Lambda$-MLE given $f$ if either a $\Lambda$-MLE given $f$ exi…

Entropic characterization conjecture. The entropic characterization of DBMR can be shown to provide a theoretical foundation for the observed size-sensitive properties of DBMR.

A determinantal point process (DPP) assigns probabilities to subsets through a positive semidefinite ensemble matrix, and learning its likelihood means estimating the model paramet…

MLE extension conjecture. The result that, with probability $1$, the MLE exists and is unique when the sample size is greater than the size of the largest clique should hold for ev…

Consider a hidden Markov model with trends, with true transition matrix $Q^*$ and maximum likelihood estimator $Q^{\hat{\theta}_n}$ based on the first $n$ observations. Let…

Worst-case exponent conjecture. At the worst case, $c$ should be strictly smaller than $1/2$. The paper gives explicit attainable exponents for Poisson and negative-binomial mixtur…

Let $Q_n(x)$ be the undersmoothed HAL-MLE at $x$, let $Q_0(x)$ be the true target function, and let $c3_n(x)$ denote the scale in the conjectured normalization. Pointwise asymptoti…

Let $n$ be a positive integer, let $(x_1,\dots,x_n)$ be a sample from the Cauchy distribution, and let the maximum likelihood estimates refer to the joint estimates of the location…

Let $n$ be the size of a sample from the Cauchy distribution, and let the maximum likelihood estimator be the joint estimator of the location and scale parameters. Nonexistence con…

Let $\hat{\theta}_n$ be the maximum likelihood estimator of the Cauchy parameter $\theta \in \mathbb{H}$, and let $E$ denote expectation under the Cauchy model. MLE mean-square exp…

Let $\hat{\theta}_n$ be the maximum likelihood estimator in the Cauchy location-scale model, and let $b(\epsilon,\theta)$ denote the rate function used for the Bahadur-efficiency r…

Let $K$ be the Gaussian kernel, let $T=[0,1]$, and take uniform points $t_i=i/N$ for $i=1,\ldots,N$. For $f(t)=t^p$, let $\hat{\sigma}_{f,N}^2$ be the maximum likelihood variance e…

Let $K$ be the Gaussian kernel, let $T=[0,1]$, and take uniform points $t_i=i/N$ for $i=1,\ldots,N$. For a monomial $f(t)=t^p$, let $\hat{\sigma}_{f,N}^2$ be the corresponding maxi…

Let $X$ be a Gaussian process with reproducing-kernel Hilbert spaces $H(K_0)$, $H(K_1)$, and $H(K_2)$ associated with kernels $K_0$, $K_1$, and $K_2$, respectively. Let…

Consider the sparse Bernoulli tensor-PCA model … with $\mathbf{x}\in\mathcal{C}_{p,k}=\{0,1/\sqrt{k}\}^p\cap\mathcal{S}_{p-1}$, and let $\operatorname{MLE}$ denote the maximum-like…

Let $H(\theta^*)$ be the Hessian matrix of the objective at the true skill parameter $\theta^*$, and let $e_j$ denote the $j$th canonical vector. For players with true ranks…