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Robert E. Gaunt

Publications and source records attributed to Robert E. Gaunt.

At least 73 records · Page 4Linked to original sources
arXiv2020

Bounds for modified Lommel functions of the first kind and their ratios

The modified Lommel function $t_{μ,ν}(x)$ is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for $t_{μ,ν}(x)$. In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio $t_{μ,ν}(x)/t_{μ-1,ν-1}(x)$ in terms of the ratio $I_ν(x)/I_{ν-1}(x)$ of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers $xt_{μ,ν}'(x)/t_{μ,ν}(x)$, the ratio $t_{μ,ν}(x)/t_{μ,ν}(y)$ and the modified Lommel function $t_{μ,ν}(x)$ itself that are given in terms of $xI_ν'(x)/I_ν(x)$, $I_ν(x)/I_ν(y)$ and $I_ν(x)$, respectively. The bounds obtained in this paper are quite accurate and often tight in certain limits. As an important special case we deduce bounds for modified Struve functions of the first kind and their ratios, some of which are new, whilst others extend the range of validity of some results given in the recent literature.

math.CA↗
arXiv2019

Inequalities for some integrals involving modified Lommel functions of the first kind

In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $t_{μ,ν}(x)$. In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function $t_{μ,ν}(x)$, for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.

math.CA↗
arXiv2019

Some new Stein operators for product distributions

We provide a general result for finding Stein operators for the product of two independent random variables whose Stein operators satisfy a certain assumption, extending a recent result of Gaunt, Mijoule and Swan \cite{gms18}. This framework applies to non-centered normal and non-centered gamma random variables, as well as a general sub-family of the variance-gamma distributions. Curiously, there is an increase in complexity in the Stein operators for products of independent normals as one moves, for example, from centered to non-centered normals. As applications, we give a simple derivation of the characteristic function of the product of independent normals, and provide insight into why the probability density function of this distribution is much more complicated in the non-centered case than the centered case.

math.PR↗
arXiv2019

Stein's Method for the Single Server Queue in Heavy Traffic

Following recent developments in the application of Stein's method in queueing theory, this paper is intended to be a short treatment showing how Stein's method can be developed and applied to the single server queue in heavy traffic. Here we provide two approaches to this approximation: one based on equilibrium couplings and another involving comparison of generators.

math.PR↗
arXiv2019

Stein's method for functions of multivariate normal random variables

By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate normal random variable $Σ^{1/2}\mathbf{Z}$, then the sequence of random variables $(g(\mathbf{W}_n))_{n\geq1}$ converges in distribution to $g(Σ^{1/2}\mathbf{Z})$ if $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. In this paper, we develop Stein's method for the problem of deriving explicit bounds on the distance between $g(\mathbf{W}_n)$ and $g(Σ^{1/2}\mathbf{Z})$ with respect to smooth probability metrics. We obtain several bounds for the case that the $j$-component of $\mathbf{W}_n$ is given by $W_{n,j}=\frac{1}{\sqrt{n}}\sum_{i=1}^nX_{ij}$, where the $X_{ij}$ are independent. In particular, provided $g$ satisfies certain differentiability and growth rate conditions, we obtain an order $n^{-(p-1)/2}$ bound, for smooth test functions, if the first $p$ moments of the $X_{ij}$ agree with those of the normal distribution. If $p$ is an even integer and $g$ is an even function, this convergence rate can be improved further to order $n^{-p/2}$. These convergence rates are shown to be of optimal order. We apply our general bounds to some examples, which include the distributional approximation of asymptotically chi-square distributed statistics; the approximation of expectations of smooth functions of binomial and Poisson random variables; rates of convergence in the delta method; and a quantitative variance-gamma approximation of the $D_2^*$ statistic for alignment-free sequence comparison in the case of binary sequences.

math.PR↗
arXiv2019

Stein's method and the distribution of the product of zero mean correlated normal random variables

Over the last 80 years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a recent technique from the Stein's method literature to obtain a simple new proof, which also serves as an exposition of a general method that may be useful in related problems.

math.ST↗
arXiv2018

Wasserstein and Kolmogorov error bounds for variance-gamma approximation via Stein's method I

The variance-gamma (VG) distributions form a four parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. Recently, Stein's method has been extended to the VG distribution. However, technical difficulties have meant that bounds for distributional approximations have only been given for smooth test functions (typically requiring at least two derivatives for the test function). In this paper, which deals with symmetric variance-gamma (SVG) distributions, and a companion paper \cite{gaunt vgii}, which deals with the whole family of VG distributions, we address this issue. In this paper, we obtain new bounds for the derivatives of the solution of the SVG Stein equation, which allow for approximations to be made in the Kolmogorov and Wasserstein metrics, and also introduce a distributional transformation that is natural in the context of SVG approximation. We apply this theory to obtain Wasserstein or Kolmogorov error bounds for SVG approximation in four settings: comparison of VG and SVG distributions, SVG approximation of functionals of isonormal Gaussian processes, SVG approximation of a statistic for binary sequence comparison, and Laplace approximation of a random sum of independent mean zero random variables.

math.PR↗
arXiv2018

Inequalities for some integrals involving modified Bessel functions

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double inequality, involving the modified Bessel function of the first kind, for a generalized hypergeometric function. We also present some open problems that arise from this research.

math.CA↗
arXiv2018

Inequalities for integrals of the modified Struve function of the first kind II

Simple inequalities are established for integrals of the type $\int_0^x \mathrm{e}^{-γt} t^{-ν} \mathbf{L}_ν(t)\,\mathrm{d}t$, where $x>0$, $0\leqγ<1$, $ν>-\frac{3}{2}$ and $\mathbf{L}_ν(x)$ is the modified Struve function of the first kind. In most cases, these inequalities are tight in certain limits. As a consequence we deduce a tight double inequality, involving the modified Struve function $\mathbf{L}_ν(x)$, for a generalized hypergeometric function.

math.CA↗
arXiv2018

An algebra of Stein operators

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are $k$th degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of $k$ independent symmetric variance-gamma distributed random variables.

math.PR↗
arXiv2018

Stein operators for variables form the third and fourth Wiener chaoses

Let $Z$ be a standard normal random variable and let $H_n$ denote the $n$-th Hermite polynomial. In this note, we obtain Stein equations for the random variables $H_3(Z)$ and $H_4(Z)$, which represents a first step towards developing Stein's method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable $aZ^2+bZ+c$, $a,b,c\in\mathbb{R}$, which leads us to a Stein equation for the non-central chi-square distribution. We also provide a discussion as to why obtaining Stein equations for $H_n(Z)$, $n\geq5$, is more challenging.

math.PR↗
arXiv2018

Multivariate normal approximation of the maximum likelihood estimator via the delta method

We use the delta method and Stein's method to derive, under regularity conditions, explicit upper bounds for the distributional distance between the distribution of the maximum likelihood estimator (MLE) of a $d$-dimensional parameter and its asymptotic multivariate normal distribution. Our bounds apply in situations in which the MLE can be written as a function of a sum of i.i.d. $t$-dimensional random vectors. We apply our general bound to establish a bound for the multivariate normal approximation of the MLE of the normal distribution with unknown mean and variance.

math.ST↗
arXiv2018

Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

We use the Stein-Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. Both the cases that the fixed graph is a simple graph and that it has multiple edges are treated. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman \cite{kn11}.

math.PR↗
arXiv2018

A note on the distribution of the product of zero mean correlated normal random variables

The problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables dates back to 1936. Perhaps surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero mean correlated normal random variables as a variance-gamma random variable, from which an explicit formula for probability density function is immediate.

math.ST↗
arXiv2018

Bounds for modified Struve functions of the first kind and their ratios

We obtain a simple two-sided inequality for the ratio $\mathbf{L}_ν(x)/\mathbf{L}_{ν-1}(x)$ in terms of the ratio $I_ν(x)/I_{ν-1}(x)$, where $\mathbf{L}_ν(x)$ is the modified Struve function of the first kind and $I_ν(x)$ is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for $I_ν(x)/I_{ν-1}(x)$ to immediately deduce bounds for $\mathbf{L}_ν(x)/\mathbf{L}_{ν-1}(x)$. We note some consequences and obtain further bounds for $\mathbf{L}_ν(x)/\mathbf{L}_{ν-1}(x)$ by adapting techniques used to bound the ratio $I_ν(x)/I_{ν-1}(x)$. We apply these results to obtain new bounds for the condition numbers $x\mathbf{L}_ν'(x)/\mathbf{L}_ν(x)$, the ratio $\mathbf{L}_ν(x)/\mathbf{L}_ν(y)$ and the modified Struve function $\mathbf{L}_ν(x)$ itself. Amongst other results, we obtain two-sided inequalities for $x\mathbf{L}_ν'(x)/\mathbf{L}_ν(x)$ and $\mathbf{L}_ν(x)/\mathbf{L}_ν(y)$ that are given in terms of $xI_ν'(x)/I_ν(x)$ and $I_ν(x)/I_ν(y)$, respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature.

math.CA↗
arXiv2018

Inequalities for integrals of the modified Struve function of the first kind

Simple inequalities for some integrals involving the modified Struve function of the first kind $\mathbf{L}_ν(x)$ are established. In most cases, these inequalities have best possible constant. We also deduce a tight double inequality, involving the modified Struve function $\mathbf{L}_ν(x)$, for a generalized hypergeometric function.

math.CA↗
arXiv2018

Inequalities for integrals of modified Bessel functions and expressions involving them

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.

math.CA↗