Robert E. Gaunt

Bio: Robert E. Gaunt is an academic researcher from University of Manchester. The author has contributed to research in topics: Stein's method & Random variable. The author has an hindex of 17, co-authored 86 publications receiving 898 citations. Previous affiliations of Robert E. Gaunt include University of Oxford.

Variance-Gamma approximation via Stein's method

Abstract: Variance-Gamma distributions are widely used in financial modelling and contain as special cases the normal, Gamma and Laplace distributions. In this paper we extend Stein's method to this class of distributions. In particular, we obtain a Stein equation and smoothness estimates for its solution. This Stein equation has the attractive property of reducing to the known normal and Gamma Stein equations for certain parameter values. We apply these results and local couplings to bound the distance between sums of the form $\sum_{i,j,k=1}^{m,n,r}X_{ik}Y_{jk}$, where the $X_{ik}$ and $Y_{jk}$ are independent and identically distributed random variables with zero mean, by their limiting Variance-Gamma distribution. Through the use of novel symmetry arguments, we obtain a bound on the distance that is of order $m^{-1}+n^{-1}$ for smooth test functions. We end with a simple application to binary sequence comparison.

Chi-square approximation by Stein's method with application to Pearson's statistic

Abstract: This paper concerns the development of Stein’s method for chi-square approximation and its application to problems in statistics. New bounds for the derivatives of the solution of the gamma Stein equation are obtained. These bounds involve both the shape parameter and the order of the derivative. Subsequently, Stein’s method for chi-square approximation is applied to bound the distributional distance between Pearson’s statistic and its limiting chi-square distribution, measured using smooth test functions. In combination with the use of symmetry arguments, Stein’s method yields explicit bounds on this distributional distance of order $n^{-1}$.

On Stein's Method for products of normal random variables and zero bias couplings

Abstract: In this paper, we extend Stein’s method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random variables. We end by proving some product normal approximation theorems.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Linear systems

Abstract: Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

British Society of Gastroenterology consensus guidelines on the management of inflammatory bowel disease in adults

Abstract: Ulcerative colitis and Crohn’s disease are the principal forms of inflammatory bowel disease. Both represent chronic inflammation of the gastrointestinal tract, which displays heterogeneity in inflammatory and symptomatic burden between patients and within individuals over time. Optimal management relies on understanding and tailoring evidence-based interventions by clinicians in partnership with patients. This guideline for management of inflammatory bowel disease in adults over 16 years of age was developed by Stakeholders representing UK physicians (British Society of Gastroenterology), surgeons (Association of Coloproctology of Great Britain and Ireland), specialist nurses (Royal College of Nursing), paediatricians (British Society of Paediatric Gastroenterology, Hepatology and Nutrition), dietitians (British Dietetic Association), radiologists (British Society of Gastrointestinal and Abdominal Radiology), general practitioners (Primary Care Society for Gastroenterology) and patients (Crohn’s and Colitis UK). A systematic review of 88 247 publications and a Delphi consensus process involving 81 multidisciplinary clinicians and patients was undertaken to develop 168 evidence- and expert opinion-based recommendations for pharmacological, non-pharmacological and surgical interventions, as well as optimal service delivery in the management of both ulcerative colitis and Crohn’s disease. Comprehensive up-to-date guidance is provided regarding indications for, initiation and monitoring of immunosuppressive therapies, nutrition interventions, pre-, peri- and postoperative management, as well as structure and function of the multidisciplinary team and integration between primary and secondary care. Twenty research priorities to inform future clinical management are presented, alongside objective measurement of priority importance, determined by 2379 electronic survey responses from individuals living with ulcerative colitis and Crohn’s disease, including patients, their families and friends.

Linear System Theory

Abstract: Chapter 8 establishes the relationship between the input and output power spectral densities of a linear system. Limitations on results are carefully detailed and the case of oscillator noise is considered. The theory is extended to multiple input-multiple output systems.