14 problems

Let $X$ and $Y$ be independent random variables, both with continuous distributions, and let $\nu_{n}^{\text{1-dim}}(Y,X)$ denote the one-dimensional estimator defined by ordering…

Let $X$ be an $n$-Cartesian product and let $Y$ be an $m$-Cartesian product. Consider the continuous analogue of the generalized independence-test setting of Theorem 3.??, involvin…

Unit-efficiency conjecture. The Pitman asymptotic relative efficiency of $T$ with respect to $S$ for the independence test will usually be

Under the hypothesis of independence, the trivial partition is the partition with one cell containing the entire space. An adaptive data-driven partition is a partition whose cells…

Shi et al.'s consistency conjecture. The test based on this statistic with the van der Waerden score function is consistent. The conjecture concerns multivariate independence testi…

Let $\hat{C}_n$ and $\hat{r}_n$ be the statistics discussed above, and consider tests of independence that are consistent against all alternatives. A statistic has the stated extre…

Let $X$ and $Y$ be high-dimensional random vectors, and consider tests for independence based on a sample whose size is fixed. An omnibus-test conjecture is that an omnibus test, c…

Let $(X_1,X_2)$ be a random vector with an arbitrary distribution on $\mathbb{R}^2$, and let $h_{\tau^*}$ be the symmetric kernel defining Bergsma–Dassios–Yanagimoto's rank correla…

Let $\tau_P^*(X,Y)$ denote the multivariate partial $\tau^*$, obtained from the rank indicator $I_P$ and the alternating kernel $H_{\tau^*}$. A dependence measure is D-consistent w…

For bivariate random variables $X$ and $Y$, let $\tau^*(X,Y)$ denote the Bergsma–Dassios sign-covariance defined through concordant and discordant quadruples. Bergsma–Dassios' cons…

Let $T_\tau$ denote the unbiased estimator of signal strength formed from the kernel associated with Kendall's tau. The alternatives are high-dimensional dependence alternatives fr…

Curse-of-dimensionality conjecture. These tests suffer from the well-known curse of dimensionality as the dimension increases.

Let $X$ and $Y$ be real random variables, and define … where $(X_i,Y_i)$ are iid copies of $(X,Y)$ and … The paper proves the claim when $X$ and $Y$ are real continuous random vari…

Let $T_n$ be the infeasible test statistic, and let $T_n(\hat{\pmb{\theta}})$ and $T_n(\hat{\pmb{\gamma}})$ be the two feasible test statistics, with bandwidth parameter $B_n$. Div…