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An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential…
State-space models are pivotal for dynamic system analysis but often struggle with outlier data that deviates from Gaussian distributions, frequently exhibiting skewness and heavy tails. This paper introduces a robust extension utilizing…
We prove the existence of secondary terms of order $X^{5/6}$ in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error…
Importance sampling is often used in machine learning when training and testing data come from different distributions. In this paper we propose a new variant of importance sampling that can reduce the variance of importance sampling-based…
Motivated by real-world machine learning applications, we analyze approximations to the non-asymptotic fundamental limits of statistical classification. In the binary version of this problem, given two training sequences generated according…
For a given parametric probability model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback--Leibler divergence, the Hellinger distance and $\chi^2$…
We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal distribution are provided, one of which takes the…
We prove that the classical Laplace asymptotic expansion (AE) of $\int_{\mathbb R^d} g(x)e^{-nu(x)}dx$, $n\gg1$ extends to the high-dimensional regime in which $d$ may grow large with $n$. More specifically, we use new techniques suitable…
A two-stage adaptive optimal design is an attractive option for increasing the efficiency of clinical trials. In these designs, based on interim data, the locally optimal dose is chosen for further exploration, which induces dependencies…
Learning joint probability distributions on n random variables requires exponential sample size in the generic case. Here we consider the case that a temporal (or causal) order of the variables is known and that the (unknown) graph of…
Laplace approximations are commonly used to approximate high-dimensional integrals in statistical applications, but the quality of such approximations as the dimension of the integral grows is not well understood. In this paper, we prove a…
Laplace's method approximates a target density with a Gaussian distribution at its mode. It is computationally efficient and asymptotically exact for Bayesian inference due to the Bernstein-von Mises theorem, but for complex targets and…
A basic result is that the sample variance for i.i.d. observations is an unbiased estimator of the variance of the underlying distribution (see for instance Casella and Berger (2002)). But what happens if the observations are neither…
Considering two random variables with different laws to which we only have access through finite size iid samples, we address how to reweight the first sample so that its empirical distribution converges towards the true law of the second…
Fr\'echet mean and variance provide a way of obtaining mean and variance for general metric space valued random variables and can be used for statistical analysis of data objects that lie in abstract spaces devoid of algebraic structure and…
We introduce methods to bound the mean of a discrete distribution (or finite population) based on sample data, for random variables with a known set of possible values. In particular, the methods can be applied to categorical data with…
Rejective sampling improves design and estimation efficiency of single-phase sampling when auxiliary information in a finite population is available. When such auxiliary information is unavailable, we propose to use two-phase rejective…
In this paper exponential ratio and exponential product type estimators using two auxiliary variables are proposed for estimating unknown population variance $S_y^2$. Problem is extended to the case of two-phase sampling. Theoretical…
The recent seminal work of Chernozhukov, Chetverikov and Kato has shown that bootstrap approximation for the maximum of a sum of independent random vectors is justified even when the dimension is much larger than the sample size. In this…
Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games,…