Related papers: Power Law in a Bounded Range: Estimating the Lower…
Skewness and kurtosis are fundamental statistical moments commonly used to quantify asymmetry and tail behavior in probability distributions. Despite their widespread application in statistical mechanics, condensed matter physics, and…
We provide an upper bound as a random variable for the functions of estimators in high dimensions. This upper bound may help establish the rate of convergence of functions in high dimensions. The upper bound random variable may converge…
Power laws arise in a variety of phenomena ranging from matter undergoing phase transition to the distribution of word frequencies in the English language. Usually, their presence is only apparent when data is abundant, and accurately…
Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with the finite Choquet expectation, upper mean…
We consider the sum of power weighted nearest neighbor distances in a sample of size n from a multivariate density f of possibly unbounded support. We give various criteria guaranteeing that this sum satisfies a law of large numbers for…
The laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of unknown parameters of a quantum system. In this work, we prove lower bounds on the size of confidence regions reported…
Measuring mutual information from finite data is difficult. Recent work has considered variational methods maximizing a lower bound. In this paper, we prove that serious statistical limitations are inherent to any method of measuring mutual…
Let $X$ denote a nonnegative random variable with $\mathsf{E} X<\infty$. Upper and lower bounds on $\mathsf{E} X-\exp\mathsf{E}\ln X$ are obtained, which are exact, in terms of $V_X$ and $E_X$ for the upper bound and in terms of $V_X$ and…
We give the first nonconstant lower bounds for the approximability of the Independent Set Problem on the Power Law Graphs. These bounds are of the form $n^{\epsilon}$ in the case when the power law exponent satisfies $\beta <1$. In the case…
In this paper, we consider the sublinear expectation on bounded random variables. With the notion of uncorrelatedness for random variables under the sublinear expectation, a weak law of large numbers is obtained. With the notion of…
We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying…
Analysis of experimental data must sometimes deal with abrupt changes in the distribution of measured values. Setting upper limits on signals usually involves a veto procedure that excludes data not described by an assumed statistical…
Given a non-negative $n \times n$ matrix viewed as a set of distances between $n$ points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree…
We consider the problem of distinguishing between two arbitrary black-box distributions defined over the domain [n], given access to $s$ samples from both. It is known that in the worst case O(n^{2/3}) samples is both necessary and…
This paper presents a systematic study on gap-dependent sample complexity in offline reinforcement learning. Prior work showed when the density ratio between an optimal policy and the behavior policy is upper bounded (the optimal policy…
We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…
Given $n$ samples of a regular discrete distribution $\pi$, we prove in this article first a serial of SLLNs results (of Dvoretzky and Erd\"{o}s' type) which implies a typical power law when $\pi$ is heavy-tailed. Constructing a (random)…
For a large class of statistical systems a geometric mean value of the observables is constrained. These observables are characterized by a power-law statistical distribution.
Extreme events can come either from point processes, when the size or energy of the events is above a certain threshold, or from time series, when the intensity of a signal surpasses a threshold value. We are particularly concerned by the…
Performance of optimization on quadratic problems sensitively depends on the low-lying part of the spectrum. For large (effectively infinite-dimensional) problems, this part of the spectrum can often be naturally represented or approximated…