17 problems

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…

Naturalness conjecture for Benford's law. Benford's law is an act of nature and can be used to indicate the randomness of our world.

Newcomb's first-digit conjecture. The first digit is $1$ about $30.1\%$ of the time, whereas it is $9$ only about $4.6\%$ of the time. This is the numerical form of the proposed lo…

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 $X$ and $Y$ be real-valued random variables with finite variance, let $\rho(X,Y)$ denote Pearson's correlation, and let $\mathop{\mathrm{eCor}}(X,Y)$ denote their earth mover's…

Let $n$ denote the dimension of a high-dimensional problem, and let $D$ be the degree of a polynomial algorithm. Write $n^{\widetilde{\Theta}(D)}$ for a running time up to factors…

Approximate message passing (AMP) is an algorithmic framework for high-dimensional statistical problems. In the settings under discussion, AMP may be information-theoretically opti…

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…

Let $X$, $Y$, and $Z$ be random variables, and let the null hypothesis consist of distributions under which $X$ and $Y$ are conditionally independent given $Z$. A test is required…

Carreira-Perpiñán–Williams conjecture. Among homoscedastic Gaussian mixtures in dimension $d$ with $k$ components, the maximum number of modes is

Let $U=(u_1,u_2,\dots)$ be a sequence of points in $[0,1]^m$, with each point $u_i=(x_1,x_2,\dots,x_m)$ chosen independently and uniformly from $[0,1]^m$, and let…

Let $X$ be a continuous real-valued random variable for which the first slide statistic $\rho_1(X)$ exists. The lower-bound conjecture. … This conjecture proposes a universal lower…

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…

The case of Lucia de Berk concerns the statistical analysis of reported medical incidents and the ensuing criminal investigation. The paper argues that the data were heterogeneous…

Quantile position symmetry conjecture. The symmetry property is off by at most one position in the dataset.

Quantile asymmetry conjecture. The asymmetric definition of the traditional quantile is the reason for the observed asymmetry when quantiles are compared across oppositely ordered…

Let $N\geq 1$ be fixed, and draw a sample of size $N$ from the population described in the paper's probability space. Let the sample quantities be estimators of the corresponding p…