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Our starting point is recognition of some mixture relationships involving the (continuous) Gauss hypergeometric distribution. Our main emphasis is then to generalize these relationships to ones involving (discrete) generalized hypergeometric distributions and their rarely considered continuous counterparts. Two such sets of relationships are derived one involving beta distributions and the other gamma distributions. A wide variety of interesting special cases arise along the way: Poisson, binomial, negative binomial, logarithmic, Conway–Maxwell–Poisson and Libby–Novick distributions all appear. There are also comments on the wider context within which the relationships of interest in this article arise. [ABSTRACT FROM AUTHOR]

We propose a family of four-parameter distributions on the circle that contains the von Mises and wrapped Cauchy distributions as special cases. The family is derived by transforming the von Mises distribution via a Mobius transformation, which maps the unit circle onto itself. The densities in the family have a symmetric or asymmetric, unimodal or bimodal shape, depending on the values of the parameters. Conditions for unimodality are explored. Further properties of the proposed model are obtained, many by applying the theory of Mobius transformation. Properties of a three-parameter symmetric submodel are investigated as well; these include maximum likelihood estimation, its asymptotics, and a reparameterization that proves useful quite generally. A three-parameter asymmetric subfamily, which often proves to be an adequate model, is also discussed, with emphasis on its mean direction and circular skewness. The proposed family and subfamilies are used to model an asymmetrically distributed data set and are then adopted as the angular error distribution of a circular-circular regression model. Two applications of the latter are given. It is in this regression context that the Mobius transformation especially comes into its own. Comparisons with other families of circular distributions are made. Supplemental materials for this article are available online. KEY WORDS: Asymmetric distribution; Circular skewness; Circular-circular regression; Unimodality; Von Mises distribution; Wrapped Cauchy distribution.

We propose a new family of symmetric unimodal distributions on the circle that contains the uniform, von Mises, cardioid, and wrapped Cauchy distributions, among others, as special cases. The basic form of the densities of this family is very simple, although its normalization constant involves an associated Legendre function. The family of distributions can also be derived by conditioning and projecting certain bivariate spherically and elliptically symmetric distributions on to the circle. Trigonometric moments are available, and a measure of variation is discussed. Aspects of maximum likelihood estimation are considered, and likelihood is used to fit the family of distributions to an example set of data. Finally, extension to a family of rotationally symmetric distributions on the sphere is briefly made. KEY WORDS: Associated Legendre function; Cardioid distribution; Circular uniform distribution; Distributions on the sphere; von Mises distribution; Wrapped Cauchy distribution.

A phase-locking model is developed from circuit theory to qualitatively explain the various regimes observed in magnetron injection-locking experiments. The model recovers quantitatively many of the phase-locking frequency characteristics observed in the experiments.

The authors present a new procedure for discovering where stone artefacts come from without having to cut a slice through them. The method is tested on axes of spotted dolerite bluestone from Preseli in Wales, source of monoliths at Stonehenge. Keywords: stone axe-heads, provenancing, magnetic susceptibility, X-ray fluorescence See http://antiquity.ac.uk/ProjGall/williamsthorpe/henceforward 'web' for the tables containing the data which supports this paper., Introduction to non-destructive characterisation of axes Scientific provenancing of stone axe-heads has become one of the key elements in assessing trade and contact in prehistoric Britain (e.g. Bradley & Edmonds [...]

We consider estimation of mean and variance functions with kernel-weighted local polynomial fitting in a heteroscedastic nonparametric regression model. Our preferred estimators are based on a localized normal likelihood, using a standard local linear form for estimating the mean and a local log-linear form for estimating the variance. It is important to allow two bandwidths in this problem, separate ones for mean and variance estimation. We provide data-based methods for choosing the bandwidths. We also consider asymptotic results, and study and use them. The methodology is compared with a popular competitor and is seen to perform better for small and moderate sample sizes in simulations. A brief example is provided. KEY WORDS: Bandwidth selection; Heteroscedasticity; Kernel estimation; Nonparametric regression., 1. INTRODUCTION Let X and Y be random variables representing univariate covariate and response in a regression relationship that can be modeled as Y = m(X) + v[(X).sup.1/2][epsilon]. (1) Here [...]

We present a rather thorough investigation of the use of kernel-based nonparametric estimators of the binary regression function in the case of a single covariate. We consider various versions of Nadaraya-Watson and local linear estimators, some involving a single bandwidth and others involving two bandwidths. The locally linear logistic estimator proves to be a good single-bandwidth estimator, although the basic Nadaraya-Watson estimator also fares quite well. Two-bandwidth methods show great potential when bandwidths are selected with knowledge of the target function, but much of their potential vanishes when data-based bandwidths are used. Likelihood cross-validation and plug-in approaches are the data-based bandwidth selection methods tested; both prove quite useful, with a preference for the latter. Adaptive two-bandwidth methods retain particularly good performance only in certain special situations (and separate estimation of the two bandwidths as for optimal density estimation is never recommended). We therefore propose a hybrid estimation procedure in which the local linear logistic estimator is used unless the ratio of (robust) variances of the covariate in the success and failure groups is greater than 2, in which case we switch to a two-bandwidth Nadaraya-Watson-type estimator, each using plug-in bandwidth selection. KEY WORDS: Bandwidth selection; Cross-validation; Local linear; Logistic; Nadaraya-Watson; Nonparametric regression; Plug-in; Two bandwidths., 1. INTRODUCTION. This article presents a rather thorough investigation of the use of kernel-based nonparametric estimators of the binary regression function in the case of a single covariate. We consider [...]

It is well known that if X and Y are uniformly distributed over the region between the horizontal axis and a density function f, then X is distributed according to density f. The algorithm 'generate Y from its marginal distribution, then X from its uniform conditional distribution given Y = y' follows. The main point made in this article is that for monotone and unimodal distributions, this construction reduces to Khintchine's theorem, thereby yielding a simple explication thereof. This observation is followed up with further consideration of the general, nonunimodal, case for both univariate and multivariate distributions, and parallels are drawn with an alternative random variate generation method called vertical density representation. KEY WORDS: Monotone density; Uniform distribution; Unimodal density; Vertical density representation., 1. INTRODUCTION Suppose one wishes to generate an observation on a univariate random variable X from a given absolutely continuous distribution with density f. Fundamental to this task is the [...]

A study was conducted to examine nonparametric regression quantile estimation by kernel weighted local linear fitting. Several conditions for the mean estimation case were considered, including partial derivatives, conditional density, population conditional quantiles and second-order symmetric. Results indicated the feasibility and practicality of local linear conditional quantile estimation and showed rules for bandwidth selection that gave reasonable results., 1. INTRODUCTION Although most regression investigations are concerned with the regression mean function m(x), the conditional mean of a response Y given values x of a predictor X, other aspects [...]

Kernel-based density estimators such as variable kernel methods, higher-order kernels, and transformation and multiplicative correction methods are all believed to improve bias reduction. This may be attributed to similarities the mean squared errors of these approaches. However, questions are raised regarding the capability of these approaches in improving statistical analysis for small-to-moderate exploratory purposes., 1. INTRODUCTION Kernel density estimation is by now a well-established technique (see, e.g., Silverman 1986, Scott 1992, and Wand and Jones 1995). Recently, a wide variety of 'sophistications' of the [...]

There has been major progress in recent years in data-based bandwidth selection for kernel density estimation. Some 'second generation' methods, including plug-in and smoothed bootstrap techniques, have been developed that are far superior to well-known 'first generation' methods, such as rules of thumb, least squares cross-validation, and biased cross-validation. We recommend a 'solve-the-equation' plug-in bandwidth selector as being most reliable in terms of overall performance. This article is intended to provide easy accessibility to the main ideas for nonexperts. KEY WORDS: Bandwidth selection; Kernel density estimation; Nonparametric curve estimation; Smoothing parameter selection., There has been several significant developments in data-based bandwidth selection for kernel density functions. New methods such as plug-in and smoothed bootstrap methods are more reliable than previous methods such as rules of thumb, least squares cross-validation and biased cross-validation. The 'solve-the-equation' plug-in bandwidth selector is the most dependable in terms of overall performance., 1. INTRODUCTION Smoothing methods provide a powerful methodology for gaining insights into data. Many examples of this may be found in the monographs of Eubank (1988), Hardle (1990), Muller (1988), [...]

We explore the aims of, and relationships between, various kernel-type regression estimators. To do so, we identify two general types of (direct) kernel estimators differing in their treatment of the nuisance density function associated with regressor variable design. We took at the well-known Gasser-Muller, Nadaraya-Watson, and Priestley-Chao methods in this light. In the random design case, none of these methods is totally adequate, and we mention a novel (direct) kernel method with appropriate properties. Disadvantages of even the latter idea are remedied by kernel-weighted local linear fitting, a well-known technique that is currently enjoying renewed popularity. We see how to fit this approach into our general framework, and hence form a unified understanding of how these kernel-type smoothers interrelate. Though the mission of this article is unificatory (and even pedagogical), the desire for better understanding of superficially different approaches is motivated by the need to improve practical estimators. In the end, we concur with other authors that kernel-weighted local linear fitting deserves much further attention for applications. KEY WORDS: Design density; Gasser-Muller estimator; Local linear fitting; Nadaraya-Watson estimator; Priestley-Chao estimator; Smoothing., Applications of various kernel-type regression estimators are evaluated to determine optimal applications. The study identifies two general kernel types that differ in simulating the nuisance density functions for regression analysis. It is shown that while all direct kernel estimators fail to exhibit complete adequacy, the kernel-weighted local linear fitting technique offers the best possible applications., 1. INTRODUCTION Given data ([t.sub.i], [Y.sub.i]), i = 1, 2,..., n, there exists a host of different formulas going by the name of the kernel estimator, m, of the regression [...]

An investigation of the effect of the various possible parameterizations of the bivariate kernal density estimator compares the performances of different possible smoothing parameterizations. The use of smoothing strategies based on the covariance matrix is examined.

Naively displaying a large collection of curves by super-imposing them one on another all on the same graph is largely uninformative and aesthetically unappealing. We propose that a simple principal component analysis be used to identify important modes of variation among the curves and that principal component scores be used to identify particular curves which clearly demonstrate the form and extent of that variation. As a result, we obtain a small number of figures on which are plotted a very few 'representative' curves from the original collection; these successfully convey the major information present in sets of 'similar' curves in a clear and attractive manner. Useful adjunct displays, including the plotting of principal component scores against covariates, are also described. Two examples - one concerning a data-based bandwidth selection procedure for kernel density estimation, the other involving ozone level curve data - illustrate the ideas. KEY WORDS: Graphics; Principal component.