Related papers: A note on the maximum probability of ultra log-con…

Suppose the edges of the complete graph on $n$ vertices are assigned a uniformly chosen random ordering. Let $X$ denote the corresponding number of Hamiltonian paths that are increasing in this ordering. It was shown in a recent paper by…

Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $\mu$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15],…

Let $M_n=\max \left(X_1, X_2, \ldots, X_n \right)$ denote the partial maximum of an independent and identically distributed skew-normal random sequence. In this paper, the rate of uniform convergence of skew-normal extremes is derived. It…

Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence…

In the present paper, we show that under the Riemann hypothesis, and for fixed $h, \epsilon > 0$, the supremum of the real and the imaginary parts of $\log \zeta (1/2 + it)$ for $t \in [UT -h, UT + h]$ are in the interval $[(1-\epsilon)…

We study asymptotic probabilities of attaining the maximum in heterogeneous Gaussian samples. In the two-group setting, the first sample has variance $1$ and size $n_1$, while the second has variance $\sigma^2>1$ and size $n_2$. We…

We propose a unified framework for likelihood-based regression modeling when the response variable has finite support. Our work is motivated by the fact that, in practice, observed data are discrete and bounded. The proposed methods assume…

We study the smoothed log-concave maximum likelihood estimator of a probability distribution on $\mathbb{R}^d$. This is a fully automatic nonparametric density estimator, obtained as a canonical smoothing of the log-concave maximum…

The partition function $p(n)$ and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: $p(n)^2 \geq p(n-1)p(n+1)$ for $n\geq 26$, and satisfy the inequality: $p(n)p(m) \geq…

Let $ X_1, \ldots, X_n $ be independent random variables taking values in the alphabet $ \{0, 1, \ldots, r\} $, and $ S_n = \sum_{i = 1}^n X_i $. The Shepp--Olkin theorem states that, in the binary case ($ r = 1 $), the Shannon entropy of $…

In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the…

We study the {\em robust proper learning} of univariate log-concave distributions (over continuous and discrete domains). Given a set of samples drawn from an unknown target distribution, we want to compute a log-concave hypothesis…

Markov Chain Monte Carlo (MCMC) requires to evaluate the full data likelihood at different parameter values iteratively and is often computationally infeasible for large data sets. In this paper, we propose to approximate the log-likelihood…

Let $X_1,X_2,...$ be independent identically distributed random variables with $\mathbb E X_k=0$, $\mathrm{Var} X_k=1$. Suppose that $\varphi(t):=\log \mathbb E e^{t X_k}<\infty$ for all $t>-\sigma_0$ and some $\sigma_0>0$. Let…

Let $S$ and $X$ be independent random variables, assuming values in the set of non-negative integers, and suppose further that both $\mathbb{E}(S)$ and $\mathbb{E}(X)$ are integers satisfying $\mathbb{E}(S)\ge \mathbb{E}(X)$. We establish a…

Let $\xi_i$, $i=1,...,n$, and $\eta_j$, $j=1,...,m$ be iid p-stable respectively q-stable random variables, $1<p<q<2$. We prove estimates for $\Ex_{\Omega_1} \Ex_{\Omega_2}\max_{i,j}\abs{a_{ij}\xi_i(\omega_1)\eta_j(\omega_2)}$ in terms of…

Variable selection in linear regression has been a central topic in statistical research for decades. Bayesian variable selection methods, which account for uncertainty in both the regression coefficients and the noise variance, have…

Maximum-likelihood estimation (MLE) is arguably the most important tool for statisticians, and many methods have been developed to find the MLE. We present a new inequality involving posterior distributions of a latent variable that holds…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. The pointwise…