10 problems

Let $\theta=\sqrt{\alpha_n}{\bf 1}_n$ under the null hypothesis and $\sqrt{\alpha_n}\bigl({\bf 1}_n+c\cdot n^{-5/4}\alpha_n^{-3/4}\eta\bigr)$ under the alternative, where…

Let $\mathcal{S}$ be the operator associated with the goodness-of-fit problem for the Wishart distributions, and let $\mathcal{S}_0$ denote the corresponding null operator. Distinc…

Let $\rho_k$, $k\in\mathbb{N}_0$, denote the quantities appearing in the spectral decomposition of the covariance operator $\mathcal{S}$, and let $\delta$ be an eigenvalue of…

Consider a goodness-of-fit test for a semiparametric transformation model based on the marked empirical process, with the transformation estimated from the data. Transformation-est…

Let a semiparametric transformation model use an estimated transformation parameter, and consider a goodness-of-fit test based on an empirical characteristic function. Transformati…

Let $T_{n,\gamma}$ be the test statistic for testing multivariate normality, and let an alternative distribution mean any distribution other than the null multivariate normal distr…

Distinguishability conjecture. Similar results on the distinguishability of the null and alternative hypotheses by means of the test statistics $T_n^+(q,I)$, with…

Adaptivity conjecture. The ideas of adaptivity should be developable for the multivariate nonparametric regression models considered in this paper, although the exact details requi…

Unknown-design extension conjecture. Subject to additional constraints similar to those in the cited work, the main results established in Theorems … should extend to unknown produ…

For integers $k\leq p$, let $\mathcal{M}(k,p)$ be the class of all subsets of $\{1,\ldots,p\}$ of cardinality $k$, and let $\Theta[k,p]$ be the set of vectors in $\mathbb{R}^p$ wit…