Let $X = (X_1, \ldots, X_n)$ be a random vector from any Borel probability law on $\mathbb{R}_+^n$. We revisit the problem of deriving a lower confidence bound (LCB) on a scalar parameter of that law. We recast classical work, beginning with Buehler, in purely probabilistic terms to form a more accessible and extensible framework. We then specialize the framework to the case where the components of $X$ are independent. In this context, we prove that Gaffke's bound is Buehler optimal for the order that it induces with respect to the maximum marginal mean parameter: $max_{i \in [n]} E_Q[X_i]$, which reduces to the common mean when the $X_i$ are independent and identically distributed. That is to say, no other valid LCB that orders samples in the same way as Gaffke's bound can improve on it with respect to this parameter.
Given observations $\mathbf x=(x_1,\dots,x_n)$, Gaffke (2005) defined \[ K_n(\mathbf x)=\mathbb{P}_{\mathbf D}\!\left\{\sum_{i=1}^n x_iD_i\le 1\right\}, \qquad (D_0,D_1,\ldots,D_n)\sim\mathrm{Dirichlet}(1,\ldots,1), \] and conjectured that it is a $p$-value whenever the inputs are independent e-values. Recently, Vlassi...
Jiahao Ming, Aaditya Ramdas, Yi Shen et al.· arXiv.org· 4 citations· ⚡1
We give upper and lower bounds for the number of solutions of the equation $e_n(x,y) = g$ in the group $W_k=(C_p\wr C_{p^k})^2$, where $e_n(x,y)$ is the $n$-th Engel word and $g\in W_k$. We obtain several corollaries from this. First, we prove a stronger version of the Amit-Ashurst conjecture for Engel words in $W_k$....
Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This proves the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Mat...
Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we cons...
Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as...
Andrii Arman, A. Bondarenko, A. Prymak et al.· 0 citations
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every...
Christopher D. Long· 2 citations· ⚡1
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