8 problems

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 $Z_n$ be the one-step estimator considered in the paper for the Cauchy parameter $\theta \in \mathbb{H}$, where $\mathbb{H}$ is the upper half-plane, and let $E$ denote expecta…

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 $X_1,X_2,\ldots$ be independent observations from the Cauchy model, with parameter $\theta$, and let $f$ be a generator of a quasi-arithmetic mean satisfying the residue and ex…

Drton–Xiao conjecture. The random variable

Drton–Xiao's Cauchy conjecture. For $m>2$ and arbitrary $\Sigma$, $Z$ has the standard Cauchy distribution $\operatorname{Cauchy}(0,1)$. The result is already known when…