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Jan Vrbik

Publications and source records attributed to Jan Vrbik.

14 recordsLinked to original sources
arXiv2022

Fisher transformation via Edgeworth expansion

We show how to calculate individual terms of the Edgeworth series to approximate the distribution of the Pearson correlation coefficient with the help of a simple Mathematica program. We also demonstrate how to eliminate the corresponding skewness, thus making the approximation substantially more accurate. This leads, in a rather natural way, to deriving a superior (in terms of its accuracy) version of Fisher's z transformation. The code can be easily modified to deal with any sample statistics defined as a function of several sample means, based on a random independent sample from a multivariate distribution.

math.ST↗
arXiv2021

Numerical instability in B&D models

When computing the expected value of time till extinction of a Birth and Death process, the usual textbook approach results in an extreme case of numerical ill-conditioning, which prevents us from getting accurate answers beyond the first few low-lying states; in this brief note we present a potential solution, together with a novel derivation of related formulas.

math.PR↗
arXiv2019

General proof of a limit related to AR(k) model of Statistics

Computing moments of various parameter estimators related to an autoregressive model of Statistics, one needs to evaluate several non-trivial limits. This was done by arXiv:1506.03131 for the case of two, three and four dimensions; in this article, we present a proof of a fully general formula, based on an ingenious solution of https://mathoverflow.net/users/4312/fedor-petrov.

math.ST↗
arXiv2018

Asymptotic Distribution of Centralized $r$ When Sampling from Cauchy

Assume that $X$ and $Y$ are independent random variables, each having a Cauchy distribution with a known median. Taking a random independent sample of size $n$ of each $X$ and $Y$, one can then compute their centralized empirical correlation coefficient $r$. Analytically investigating the sampling distribution of this $r$ appears possible only in the large $n$ limit; this is what we have done in this article, deriving several new and interesting results.

math.ST↗
arXiv2017

Molecular geometry and vibrational frequencies by parallel sampling

Quantum Monte Carlo is an efficient technique for finding the ground-state energy and related properties of small molecules. A major challenge remains in accurate determination of a molecule's geometry, i.e. the optimal location of its individual nuclei and the frequencies of their vibration. The aim of this article is to describe a simple technique to accurately establish such properties. This is achieved by varying the trial function to accommodate changing geometry, thereby removing a source of rather unpleasant singularities which arise when the trial function is fixed (the traditional approach).

physics.comp-ph↗
arXiv2015

Yet Another Proof of Sylvester's Determinant Identity

In 1857 Sylvester stated a result on determinants without proof that was recognized as important over the subsequent century. Thus it was a surprise to Akritas, Akritas and Malaschonok when they found only one English proof - given by Bareiss 111 years later! To rectify the gap in the literature these authors collected and translated six additional proofs: four from German and two from Russian. These proofs range from long and "readily understood by high school students" to elegant but high level. We add our own proof to this collection which exploits the product rule and the fact that taking a derivative of a determinant with respect to one of its elements yields its cofactor. A differential operator can then be used to replace one row with another.

math.HO↗
arXiv2015

Playing Several Patterns Against One Another

We revisit the game in which each of several players chooses a pattern and then a coin is flipped repeatedly until one of these patterns is generated. In particular, we demonstrate how to compute the probability of any one player winning this game, and find the distribution of the game's duration. Our presentation is an extension (and perhaps a simplification) of the results of Blom and Thornburn.

math.PR↗
arXiv2015

Cumulants of products of Normally distributed random variables

To find moments of various estimators related to Autoregressive models of Statistics, one first needs the cumulants of products of two Normally distributed random variables. The purpose of this article is to derive the corresponding formulas, and extend them to products of three or more such variables.

math.ST↗
arXiv2014

Three competing patterns

Assuming repeated independent sampling from a Bernoulli distribution with two possible outcomes S and F, there are formulas for computing the probability of one specific pattern of consecutive outcomes (such as SSFFSS) winning (i.e. being generated first) over another such pattern (e.g. SFSSFS). In this article we will extend the theory to three competing patterns.

math.PR↗
arXiv2014

Accurate distribution of X^{T}X with singular, idempotent variance-covariance matrix

Assume that X is a set of sample statistics which follow a special case Central Limit Theorem, namely: as the sample size n increases the corresponding distribution becomes multivariate Normal with the mean (of each X) equal to zero and with an idempotent variance-covariance matrix V. It is well known that X^{T}X has (in the same limit), a chi-squared distribution with degrees of freedom equal to the trace of V. In this article we extend the above result to include the corresponding (1/n)-proportional corrections, making the new approximation substantially more accurate and extending its range of applicability to small-size samples.

math.ST↗
arXiv2014

Improving Accuracy of Goodness-of-fit Test

It is well known that the approximate distribution of the usual test statistic of a goodness-of-fit test is chi-square, with degrees of freedom equal to the number of categories minus 1 (assuming that no parameters are to be estimated -- something we do throughout this article). Here we show how to improve this approximation by including two correction terms, each of them inversely proportional to the total number of observations.

math.ST↗
arXiv2014

Finding an ARMA(p,q) model given its spectral density or its correlogram

An ARMA model can be fully determined based on either its spectral density, or its correlogram, i.e. a formula for computing the corresponding k th serial correlation for any integer k. In this article we describe how to find, given one of these three ways of specifying the model, the other two.

math.ST↗