9 problems

Let $\Pi$ be the set of all bivariate copulas. For $C\in\Pi$, define Spearman's rho and Gini's gamma by $\rho(C)=12\int_0^1\int_0^1 C(u,v)\,du\,dv-3$ and…

Fix $\alpha\in(0,1)$. Let $\mathcal{P}$ be a class of distributions of $(X,Y,Z)$ on Euclidean spaces having densities bounded on compact subsets, and let…

Let $n\ge 2$ independent arms have distributions $X_i\sim\mathcal{N}(\mu_i,1)$, with $\mu_i\in[0,1]$ and a unique optimal arm $*$. Write $\Delta_i=\mu_*-\mu_i$ for $i\ne *$,…

Let $X_1,\ldots,X_N$ be observations from a centered stationary Gaussian process with spectral density $f$ belonging to a Sobolev-type class $\mathcal F_s$, and let the observation…

Let $\mathcal{C}$ be the set of bivariate copulas, equivalently the set of joint laws of random variables $U,V\sim\mathcal{U}(0,1)$. Define Spearman's rho and Spearman's footrule b…

For each $n,k$, consider left Gog and GOGAm trapezoids of shape $(n,k)$, and let $X_{1,1}=l$ denote their bottom entry. Refined left Gog–GOGAm conjecture. For each $n,k,l$, the…

Let $(X,Y)$ be bivariate normal random variables, and write $\varrho(X,Y)=\varrho$ for their correlation. The bivariate normal earth mover's correlation conjecture. One has … The p…

Let $X$ be a random variable on $\mathbb{S}^m$ whose local Fréchet mean is smeary, as in the curse of dimensionality theorem, and let the corresponding local Fréchet means be those…

For Gog, Magog and GOGAm triangles, define the statistics $\alpha$, $\beta$, and $\gamma$ as in the paper; in particular, $\alpha_{\mathrm{Gog}}(X)$ counts indices $k$ with…