Related papers: On the Chi square and higher-order Chi distances f…
We study the accuracy of a scaled Poisson approximation to the weighted sum of independent Poisson random variables, focusing on in particular the relative error of the tail distribution. A bound on the relative approximation error is…
Using the technique developed in approximation theory, we construct examples of exponential families of infinitely divisible laws which can be viewed as deformations of the normal, gamma, and Poisson exponential families. Replacing the…
Approximating complex probability distributions, such as Bayesian posterior distributions, is of central interest in many applications. We study the expressivity of geometric Gaussian approximations. These consist of approximations by…
In this paper, we state and prove a simple geometric interpretation of the degrees of freedom of a chi-square distribution. The general geometric idea goes back at least to Fisher in the 1920s, but the exact result does not appear to have…
A new large deviation results for the Pearson chi-square and Log-likelihood ratio statistics are obtained. Here attention is focused on the case when the number of groups increases to infinity and the probabilities of groups decreases to…
There are many applications that benefit from computing the exact divergence between 2 discrete probability measures, including machine learning. Unfortunately, in the absence of any assumptions on the structure or independencies within…
I present here a generalization of the maximum likelihood method and the $\chi^2$ method to the cases in which the data are {\it not} assumed to be Gaussian distributed. The method, based on the multivariate Edgeworth expansion, can find…
In this paper we show that the conditional distribution of perturbed chi-quare risks can be approximated by certain distributions including the Gaussian ones. Our results are of interest for conditional extreme value models and multivariate…
$f$-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics and information theory such as Kullback-Leibler…
Nearest neighbor cells in $R^d,d\in\mathbb{N}$, are used to define coefficients of divergence ($\phi$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a…
Computing the Fr\'{e}chet distance for surfaces is a surprisingly hard problem and the only known algorithm is limited to computing it between flat surfaces. We adapt this algorithm to create one for computing the Fr\'{e}chet distance for a…
Polynomial series approximations are a central theme in approximation theory due to their utility in an abundance of numerical applications. The two types of series, which are featured most prominently, are Taylor series expansions and…
The binary divergences that are divergences between probability measures defined on the same 2-point set have an interesting property. For the chi-squared divergence and the relative entropy, it is known that their binary divergence attain…
We consider distributional limit of the Pearson chi-square statistic when the number of classes m increases with the sample size n in such way that $n/\sqrt{m} \to {\lambda}$. Under mild moment conditions, the limit is Gaussian for…
This paper presents a new method to estimate systematic errors in the maximum-likelihood regression of count data. The method is applicable in particular to X-ray spectra in situations where the Poisson log-likelihood, or the Cash…
For some smooth special case of generalized $\varphi-$divergences as well as of new divergences (called scaled shift divergences), we derive approximations of the omnipresent (weighted) $\ell_{1}-$distance and (weighted) $\ell_{1}-$norm.
We introduce generalized notions of a divergence function and a Fisher information matrix. We propose to generalize the notion of an exponential family of models by reformulating it in terms of the Fisher information matrix. Our methods are…
In a thought-provoking paper, Efron (2011) investigated the merit and limitation of an empirical Bayes method to correct selection bias based on Tweedie's formula first reported by \cite{Robbins:1956}. The exceptional virtue of Tweedie's…
Approximate distributions for sum and difference of linearly correlated $\chi^{2}$ distributed random variables are derived. It is shown that they can be reduced to conveniently parametrized gamma and Variance-Gamma distributions,…
Bayesian Poisson probability distributions for the average n can be analytically converted into equivalent chi-squared distributions. These can then be combined with other Gaussian or Bayesian Poisson distributions to make a total…