39 problems

Let $C$ be a bivariate copula, that is, a distribution function on $[0,1]^2$ with uniform marginals. A copula is stochastically increasing when it satisfies the stochastic increasi…

Let the observations form a stationary $\alpha$-mixing sequence, rather than an i.i.d. sample, and let the estimators and theoretical results considered in the paper be applied to…

Let $(\beta)_{1-\tau_A}$ be the population version of the transformed generator, where $\tau_A$ is the Kendall-type parameter associated with the Pickands dependence function, and…

Let $A$ be a Pickands dependence function with discrete Pickands dependence measure $\vartheta$ having full support, and let $(A_\vartheta)_\alpha$ denote the associated transforma…

Nešlehová's conjecture. Multidimensional directional $\rho$-coefficients can be expressed as a linear combination of lower-dimensional directional coefficients.

For a copula whose operator tail density is defined using a diagonal norming matrix, suppose that the entries of the matrix are not all equal. Hidden regular variation conjecture.…

For $\lambda \leq -1$, let $\phi_{\lambda}^{-1}$ denote the strict inverse of the power-divergence copula generator $\phi_{\lambda}$, defined on $t \in [0,\infty)$. Complete monoto…

Let $F$ be the distribution function of the transformed observations, let $\hat F_n$ be the empirical distribution function based on the rank-based pseudo-observations, and define…

Independence conjecture. The random intervals $\mathcal{I}_s(u_1,u_2)$ and $\mathcal{I}_z(v_1,v_2)$ are independent; equivalently, the copula density satisfies

Fix $d\ge 2$. For a $d$-dimensional quasi-copula $Q$ and a $d$-box $\mathcal B$, let $V_Q(\mathcal B)$ denote the associated $Q$-volume. The conjecture concerns the minimum of…

Let $\mathbb{I}=[0,1]$, let $\mathcal{C}$ denote the family of bivariate copulas, and let $\mathcal{C}_{mcd}$ be the subclass of mutually completely dependent copulas. Let…

For a copula $C$, write $\partial_1C(x,y)$ for the partial derivative with respect to the first variable, when it exists. A subclass of copulas is called typical when it contains a…

The tau–rho region conjecture.

Let $U_1,\dots,U_n$ be random variables with a dependence structure given by an elliptical copula or an Archimedean copula, and let sub-uniformity denote the property that the gene…

Let $s:[-\frac{1}{2},1]\to[-1,1]$ be the upper bound constructed in the preceding theorem, and let $\rho$ and $\phi$ denote Spearman's rho and Spearman's footrule for copulas. Non-…

Quasi-copula patching conjecture. Under the condition that the volume $V$ of $R$ is nonzero, the desired patch is equal to

Let $(X_1,Y_1),(X_2,Y_2),\ldots$ be a random sample from $(X,Y)$, let $E_n$ be the empirical object used in the estimator, let $N(n)$ be the resolution chosen as in the paper, and…

Let $C_{\mathcal{G},\bm{\psi},\bm{\ell},Q}$ be a clustered Archimax copula satisfying the extended max-domain-of-attraction assumption and the standard stable tail dependence funct…

Let $C_{\mathcal{G},\bm{\psi},\bm{\ell},Q}$ be a clustered Archimax copula, and let $\mathcal{D}_2$ and $\mathcal{D}_3$ be the classes of clusters specified by the extended max-dom…

Let a random variable have a bounded copula density, and let a Lipschitz-continuous diffeomorphism be applied to it. Bounded-copula-density conjecture. The transformed random varia…

Generalized energy-distance copula conjecture. The solution is attained by the circular bivariate $L_p$-norm spherical copula with $p=3-\beta$ for arbitrary $\beta\in(0,2)$. For…

Let $d\geq 2$, let $f$ be a suitable density on $[0,1]$, and for each $\bm{i}=(i_1,\ldots,i_d)\in\{0,1\}^d$ define … Let the generalized family consist of all copulas whose densiti…

Consistency-range conjecture. The permissible range of the parameter $s$ according to the main consistency theorem can be enlarged to

Marginal-incorporation conjecture. When working with multilinear interpolations, the marginal distribution of $Y$ has to be incorporated into the definition of $\zeta^1$, particula…

Optimal-parameter conjecture. The optimal choice of $s$, in the sense that the estimator performs well independently of the underlying dependence structure, is