For a continuous one-parameter natural exponential family (NEF) with variance function V, Bar-Lev and Hoessly (arXiv:2608.28221) prove that coincidence of the equal-tail and likelihood-ratio two-sided p-values for one observation, throughout the family and…
For a radially symmetric Gaussian in n dimensions restricted to the shell R ≥ a, Petrella (arXiv:2511.11566) writes the variance of R at a fixed mean M as M² V(n,r) with r = a/σ, x = r²/2 and V(n,r) = Γ(frac n2,x)Γ(frac(n+2)2,x)/Γ(frac(n+1)2,x)² - 1, an exp…
Fix ℓ ≥ 2 ordered symbols and let all ℓⁿ words x₁x₂… xₙ be equally likely. The trend score S=S⁺-S⁻ is the number of increasing pairs of positions minus the number of decreasing ones: Kendall's score with repetitions allowed and — unlike in the classical the…
Sturma and Drton (arXiv:2507.18170) give a graphical criterion for the identifiability of parameters of a linear structural equation model with arbitrarily structured latent variables, together with a sound and complete algorithm for testing it. The algorit…
A tail dependence matrix is the matrix of pairwise tail dependence coefficients of a random vector; the set T_d of d × d tail dependence matrices is a polytope whose facets are known only for d ≤ 6. Inside it, Shyamalkumar and Tao single out the m-dependenc…
In partially rank-ordered set (PROS) sampling one fixes a set size S and a design parameter D={d₁,…,dₙ} splitting the ranks {1,…,S} into n consecutive blocks, and measures one unit drawn uniformly at random from each block; the measured value from block dᵣ…
Komaba, Johno and Nakamoto attach to a pair of distribution functions F,G on ℝ and an integer q ≥ 1 the overlap discrepancy D_q(F,G)=1-inf_(v₁ ≤ … ≤ v_q)Σᵢ₌₀^(q)min(F|_(vᵢ)^(vᵢ₊₁), G|_(vᵢ)^(vᵢ₊₁)), and build from it a one-sample and a two-sample goodness-of…
Kabata, Matsumoto and Okuno (arXiv:2608.01776) constructed an equally weighted three-component bivariate Gaussian mixture p_(r,M) with at least seven modes for r ∈ [97/100,99/100] and M ≥ 10⁴, refuting the conjectured maximum C(d+k-1, d)=6 recorded by Aménd…
Let X ∼ Nₚ(θ,Iₚ) be estimated under quadratic loss. Maruyama and Takemura build, from a positive non-decreasing function Φ, shrinkage estimators hat(θ)_φ that dominate the James–Stein estimator hat(θ)_JS provided they dominate it at the origin, i.e. provide…
Rossellini, Barber, Ren and Soloff [RBRS] bound the number of distinct fitted values of isotonic regression on a binary sample of size n by an explicit quantity f(n) built from the Euler totient, and write f(n)=c n^(2/3)+δₙ with c=3/(4π²)^(1/3). A propositi…
On the discrete cube X = {-1,+1}ᵏ let Bᵏ_q be the span of the Walsh functions of degree at most q and let (Bᵏ_q)₊ be its cone of non-negative members; Bᵏ₂ is the Ising model on the complete graph. A set U ⊆ X is a set of uniqueness for (Bᵏ_q)₊ if 0 is the o…
Let f(x)=Σⱼ₌₁³ wⱼexp(-norm(x-μⱼ)²/2) be a mixture of three Gaussian densities with a common covariance (normalised to the identity), nonnegative weights of positive sum, and centres μⱼ in a real inner product space. The largest possible number of modes (loc…
A one-dimensional discrete statistical model of maximum likelihood degree one is, after Bik and Marigliano, the image of a parameterisation t ↦ (cᵢ t^(νᵢ)(1-t)^(μᵢ))ᵢ₌₀ⁿ of the probability simplex Δₙ; it is fundamental when the exponent pairs (νᵢ,μᵢ) determ…
Baranwal (arXiv:2607.16676v2) certifies, for the pairwise message-passing classifier on the broadcast-labelled Poisson Galton–Watson tree with a two-point feature law at (Δ,γ,t₀)=(3,0.55,0.95), the first four values of the depth-error curve, E(0)=0.025, E(1…
Forbes and Lauritzen study the score matching estimator (SME) of a linear concentration model, a linear subspace L of the symmetric p × p matrices, and call L n-estimable when the SME exists with probability one, equivalently when det M(x)>0 for some sample…
