arXiv subjects

Robert E. Gaunt

Publications and source records attributed to Robert E. Gaunt.

At least 37 records · Page 2Linked to original sources

arXiv2025

Inequalities for an integral involving the modified Bessel function of the first kind

Simple bounds are obtained for the integral $\int_0^x\mathrm{e}^{-γt}t^νI_ν(t)\,\mathrm{d}t$, $x>0$, $ν>-1/2$, $0\leqγ<1$, together with a natural generalisation of this integral. In particular, we obtain an upper bound that holds for all $x>0$, $ν>-1/2$, $0\leqγ<1$, is of the correct asymptotic order as $x\rightarrow0$ and $x\rightarrow\infty$, and possesses a constant factor that is optimal for $ν\geq0$ and close to optimal for $ν>-1/2$. We complement this upper bound with several other upper and lower bounds that are tight as $x\rightarrow0$ or as $x\rightarrow\infty$, and apply our results to derive sharper bounds for some expressions that appear in Stein's method for variance-gamma approximation.

math.CA↗

arXiv2025

Normal approximation for the posterior in exponential families

In this paper, we obtain quantitative, non-asymptotic, and data-dependent \textit{Bernstein-von Mises type} bounds on the normal approximation of the posterior distribution in exponential family models with arbitrary centring and scaling. Our bounds, stated in the total variation and Wasserstein distances, are valid for univariate and multivariate posteriors alike, and do not require a conjugate prior setting. They are obtained through a refined version of Stein's method of comparison of operators that allows for improved dimensional dependence in high-dimensional settings and may also be of interest in other problems. Our approach is rather flexible and, in certain settings, allows for the derivation of bounds with rates of convergence faster than the usual $ O(n^{-1/2}) $ rate (when $ n $ is the sample size). We illustrate our findings on a variety of exponential family distributions, including the Weibull, multinomial, and linear regression with unknown variance. The resulting bounds have an explicit dependence on the prior distribution and on sufficient statistics of the data from the sample, and thus provide insight into how these factors affect the quality of the normal approximation. Insights from our examples include identification of conditions under which faster $ O(n^{-1}) $ convergence rates occur for Bernoulli data, illustrations of how the quality of the normal approximation is influenced by the choice of standardisation, and dimensional dependence in high-dimensional settings.

math.ST↗

arXiv2024

On Stein factors in Stein's method for normal approximation

Building on the rather large literature concerning the regularity of the solution of the standard normal Stein equation, we provide a complete description of the best possible uniform bounds for the derivatives of the solution of the standard normal Stein equation when the test functions belong to the class of real-valued functions whose $k$-th order derivative is Lipschitz.

math.PR↗

arXiv2024

Stein's Method of Moments

Stein operators allow to characterise probability distributions via differential operators. Based on these characterisations, we develop a new method of point estimation for marginal parameters of strictly stationary and ergodic processes, which we call \emph{Stein's Method of Moments} (SMOM). These SMOM estimators satisfy the desirable classical properties such as consistency and asymptotic normality. As a consequence of the usually simple form of the operator, we obtain explicit estimators in cases where standard methods such as (pseudo-) maximum likelihood estimation require a numerical procedure to calculate the estimate. In addition, with our approach, one can choose from a large class of test functions which allows to improve significantly on the moment estimator. Moreover, for i.i.d.\ observations, we retrieve data-dependent functions that result in asymptotically efficient estimators and give a sequence of explicit SMOM estimators that converge to the maximum likelihood estimator. Our simulation study demonstrates that for a number of important univariate continuous probability distributions our SMOM estimators possess excellent small sample behaviour, often outperforming the maximum likelihood estimator and other widely-used methods in terms of lower bias and mean squared error. We also illustrate the pertinence of our approach on a real data set related to rainfall modelisation.

math.ST↗

arXiv2024

Asymptotic approximations for the distribution of the product of correlated normal random variables

We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the tail probabilities and quantile functions of this distribution, as well as an asymptotic approximation for the widely used risk measures value at risk and tail value at risk.

math.PR↗

arXiv2024

A Stein characterisation of the distribution of the product of correlated normal random variables

We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein characterisation is shown to naturally generalise a number of other Stein characterisations in the literature. From our Stein characterisation we derive recursive formulas for the moments of the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, which allows for efficient computation of higher order moments.

math.PR↗

arXiv2024

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

math.PR↗

arXiv2024

Stein's method of moments for truncated multivariate distributions

We use Stein characterisations to derive new moment-type estimators for the parameters of several truncated multivariate distributions in the i.i.d. case; we also derive the asymptotic properties of these estimators. Our examples include the truncated multivariate normal distribution and truncated products of independent univariate distributions. The estimators are explicit and therefore provide an interesting alternative to the maximum-likelihood estimator (MLE). The quality of these estimators is assessed through competitive simulation studies, in which we compare their behaviour to the performance of the MLE and the score matching approach.

math.ST↗

arXiv2024

Stein's Method of Moments on the Sphere

We use Stein characterizations to obtain new moment-type estimators for the parameters of three classical spherical distributions (namely the Fisher-Bingham, the von Mises-Fisher, and the Watson distributions) in the i.i.d. case. This leads to explicit estimators which have good asymptotic properties (close to efficiency) and therefore lead to interesting alternatives to classical maximum likelihood methods or more recent score matching estimators. We perform competitive simulation studies to assess the quality of the new estimators. Finally, the practical relevance of our estimators is illustrated on a real data application in spherical latent representations of handwritten numbers.

math.ST↗

arXiv2024

A probabilistic proof of some integral formulas involving incomplete gamma functions

The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals $\int_0^\infty x^{-2ν}\cos(bx)γ(ν,αx^2)\,\mathrm{d}x$ (for $ν>1/2$, $b>0$ $α>0$) and $\int_0^\infty x^{2ν-1}\cos(bx)Γ(-ν,αx^2)\,\mathrm{d}x$ (for $ν>0$, $b>0$ $α>0$), where $γ(a,x)$ and $Γ(a,x)$ are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas.

math.PR↗

arXiv2024

The distribution of the product of independent variance-gamma random variables

Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the product $XY$ is derived. Some basic distributional properties are also derived, including formulas for the cumulative distribution function and the characteristic function, as well as asymptotic approximations for the density, tail probabilities and the quantile function. As special cases, we deduce some key distributional properties for the product of two independent asymmetric Laplace random variables as well as the product of four jointly correlated zero mean normal random variables with a particular block diagonal covariance matrix. As a by-product of our analysis, we deduce some new reduction formulas for the Meijer $G$-function.

math.PR↗

arXiv2024

On Pólya's random walk constants

A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms of the Lauricella function $F_C$, for the return probability for all $d\geq3$. Previously, a closed-form formula had only been available for $d=3$.

math.PR↗

arXiv2023

Modified Method of Moments for Generalized Laplace Distribution

In this note, we consider the performance of the classic method of moments for parameter estimation of symmetric variance-gamma (generalized Laplace) distributions. We do this through both theoretical analysis (multivariate delta method) and a comprehensive simulation study with comparison to maximum likelihood estimation, finding performance is often unsatisfactory. In addition, we modify the method of moments by taking absolute moments to improve efficiency; in particular, our simulation studies demonstrate that our modified estimators have significantly improved performance for parameter values typically encountered in financial modelling, and is also competitive with maximum likelihood estimation.

stat.ME↗

arXiv2023

On the cumulative distribution function of the variance-gamma distribution

We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero mean normal random variables.

math.PR↗

arXiv2023

On the moments of the variance-gamma distribution

We obtain new closed-form formulas for the moments and absolute moments of the variance-gamma distribution. We thus deduce new formulas for the moments and absolute moments of the product of two correlated zero mean normal random variables.

math.PR↗

arXiv2023

The rate of convergence of some asymptotically chi-square distributed statistics by Stein's method

We build on recent works on Stein's method for functions of multivariate normal random variables to derive bounds for the rate of convergence of some asymptotically chi-square distributed statistics. We obtain some general bounds and establish some simple sufficient conditions for convergence rates of order $n^{-1}$ for smooth test functions. These general bounds are applied to Friedman's statistic for comparing $r$ treatments across $n$ trials and the family of power divergence statistics for goodness-of-fit across $n$ trials and $r$ classifications, with index parameter $λ\in\mathbb{R}$ (Pearson's statistic corresponds to $λ=1$). We obtain a $O(n^{-1})$ bound for the rate of convergence of Friedman's statistic for any number of treatments $r\geq2$. We also obtain a $O(n^{-1})$ bound on the rate of convergence of the power divergence statistics for any $r\geq2$ when $λ$ is a positive integer or any real number greater than 5. We conjecture that the $O(n^{-1})$ rate holds for any $λ\in\mathbb{R}$.

math.PR↗

arXiv2023

Bounds for distributional approximation in the multivariate delta method by Stein's method

We obtain bounds to quantify the distributional approximation in the delta method for vector statistics (the sample mean of $n$ independent random vectors) for normal and non-normal limits, measured using smooth test functions. For normal limits, we obtain bounds of the optimal order $n^{-1/2}$ rate of convergence, but for a wide class of non-normal limits, which includes quadratic forms amongst others, we achieve bounds with a faster order $n^{-1}$ convergence rate. We apply our general bounds to derive explicit bounds to quantify distributional approximations of an estimator for Bernoulli variance, several statistics of sample moments, order $n^{-1}$ bounds for the chi-square approximation of a family of rank-based statistics, and we also provide an efficient independent derivation of an order $n^{-1}$ bound for the chi-square approximation of Pearson's statistic. In establishing our general results, we generalise recent results on Stein's method for functions of multivariate normal random vectors to vector-valued functions and sums of independent random vectors whose components may be dependent. These bounds are widely applicable and are of independent interest.

math.ST↗

arXiv2023

The Variance-Gamma Distribution: A Review

The variance-gamma (VG) distributions form a four-parameter family which includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and distributional approximation on Wiener space. In this review, we provide an up-to-date account of the basic distributional theory of the VG distribution. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review methods for parameter estimation and some applications of the VG distribution, including the aforementioned applications to financial modelling and distributional approximation on Wiener space.

math.ST↗