Deviance Residuals for an Angular Response

This paper discusses deviance residual approximations in von Mises regression models. By using a relationship between the von Mises and the wrapped normal distributions, the paper shows that the deviance component of the von Mises distribution is approximately a linear function of the standard normal distribution. Two standardized forms are proposed for the deviance residual, and a simulation study is performed to compare the approximation of the proposed residuals to the standard normal distribution. An illustrative example is given.     

Statistics for a New Test of Discordance in Circular Data

This article considers the derivation of approximate distributions for two types of statistics that can be used in developing new tests of discordance in circular samples from the von Mises distribution. An alternative test of discordance is proposed based on the circular distance between sample points. The advantage of the test is that it allows users to detect possible outliers in both univariate and bivariate circular data. For illustration, the test is applied to two real circular data sets.     

Cusum control for data following the von Mises distribution

The von Mises distribution is widely used for modeling angular data. When such data are seen in a quality control setting, there may be interest in checking whether the values are in statistical control or have gone out of control. A cumulative sum (cusum) control chart has desirable properties for checking whether the distribution has changed from an in-control to an out-of-control setting. This paper develops cusums for a change in the mean direction and concentration of angular data and illustrates some of their properties.     

On an extension of the von Mises distribution due to Batschelet

This paper considers the three-parameter family of symmetric unimodal circular distributions proposed by Batschelet in [1 Batschelet, E. 1981. Circular Statistics in Biology, London: Academic Press.  [Google Scholar]], an extension of the von Mises distribution containing distributional forms ranging from the highly leptokurtic to the very platykurtic. The family's fundamental properties are given, and likelihood-based techniques described which can be used to perform estimation and hypothesis testing. Analyses are presented of two data sets which illustrate how the family and three of its most direct competitors can be applied in the search for parsimonious models for circular data.     

Some asymptotic properties of the circular median

In this article, the asymptotic distribution of the circular median is derived for symmetric distributions on the circle. Its asymptotic relative efficienty with respect to the mean direction and to an estimator proposed by Watson (1983) is then examined. Special attention is given to the cases where the underlying distribution is von Mises and contaminated von Mises. It is seen that the circular median can perform more efficiently than both estimators in presence of outliers.     

A random‐effects model for clustered circular data

This article considers a circular regression model for clustered data, where both the cluster effects and the regression errors have von Mises distributions. It involves β, a vector of parameters for the fixed effects, and two concentration parameters for the error distribution. A measure of intra‐cluster circular correlation and a predictor for an unobserved cluster random effect are studied. Preliminary estimators for the vector β and the two concentration parameters are proposed, and their performance is compared with that of the maximum likelihood estimators in a simulation study. A numerical example investigating the factors impacting the orientation taken by a sand hopper when released is presented. The Canadian Journal of Statistics 47: 712–728; 2019 © 2019 Statistical Society of Canada     

Multimodal exponential families of circular distributions with application to daily peak hours of PM2.5 level in a large city

In this paper, we propose two multimodal circular distributions which are suitable for modeling circular data sets with two or more modes. Both distributions belong to the regular exponential family of distributions and are considered as extensions of the von Mises distribution. Hence, they possess the highly desirable properties, such as the existence of non-trivial sufficient statistics and optimal inferences for their parameters. Fine particulates (PM2.5) are generally emitted from activities such as industrial and residential combustion and from vehicle exhaust. We illustrate the utility of our proposed models using a real data set consisting of fine particulates (PM2.5) pollutant levels in Houston region during Fall season in 2019. Our results provide a strong evidence that its diurnal pattern exhibits four modes; two peaks during morning and evening rush hours and two peaks in between.     

Testing homogeneity in a mixture of von mises distributions with a structural parameter

The modified likelihood ratio statistic can be used to test the homogeneity in a variety of mixture models. Here, the authors propose the use of the modified and the iterative modified likelihood ratio for testing homogeneity against a two‐component von Mises mixture with a structural parameter. They derive the limiting distributions of the test statistics and propose methods to improve the accuracy of the asymptotic approximation in finite samples. Their simulations show that the tests maintain their nominal level and that they have adequate power. Data on movements of turtles are used as an illustration     

Goodness-of-Fit Tests for the Skew-Normal Distribution When the Parameters Are Estimated from the Data

In this article, tests are developed which can be used to investigate the goodness-of-fit of the skew-normal distribution in the context most relevant to the data analyst, namely that in which the parameter values are unknown and are estimated from the data. We consider five test statistics chosen from the broad Cramér–von Mises and Kolmogorov–Smirnov families, based on measures of disparity between the distribution function of a fitted skew-normal population and the empirical distribution function. The sampling distributions of the proposed test statistics are approximated using Monte Carlo techniques and summarized in easy to use tabular form. We also present results obtained from simulation studies designed to explore the true size of the tests and their power against various asymmetric alternative distributions.     

A full Bayesian analysis of circular data using the von Mises distribution

This paper presents a full Bayesian analysis of circular data, paying special attention to the von Mises distribution. We obtain samples from the posterior distribution using the Gibbs sampler which, after the introduction of strategic latent variables, has all full conditional distributions of known type.     

A multivariate von mises distribution with applications to bioinformatics

Motivated by problems of modelling torsional angles in molecules, Singh, Hnizdo & Demchuk (2002) proposed a bivariate circular model which is a natural torus analogue of the bivariate normal distribution and a natural extension of the univariate von Mises distribution to the bivariate case. The authors present here a multivariate extension of the bivariate model of Singh, Hnizdo & Demchuk (2002). They study the conditional distributions and investigate the shapes of marginal distributions for a special case. The methods of moments and pseudo‐likelihood are considered for the estimation of parameters of the new distribution. The authors investigate the efficiency of the pseudo‐likelihood approach in three dimensions. They illustrate their methods with protein data of conformational angles     

Directional mixture models and optimal estimation of the mixing density

The authors develop consistent nonparametric estimation techniques for the directional mixing density. Classical spherical harmonics are used to adapt Euclidean techniques to this directional environment. Minimax rates of convergence are obtained for rotation ally invariant densities verifying various smoothness conditions. It is found that the differences in smoothness between the Laplace, the Gaussian and the von Mises‐Fisher distributions lead to contrasting inferential conclusions.     

Dependent models for observations which include angular ones

This paper discusses some stochastic models for dependence of observations which include angular ones. First, we provide a theorem which constructs four-dimensional distributions with specified bivariate marginals on certain manifolds such as two tori, cylinders or discs. Some properties of the submodel of the proposed models are investigated. The theorem is also applicable to the construction of a related Markov process, models for incomplete observations, and distributions with specified marginals on the disc. Second, two maximum entropy distributions on the cylinder are discussed. The circular marginal of each model is distributed as the generalized von Mises distribution which represents a symmetric or asymmetric, unimodal or bimodal shape. The proposed cylindrical model is applied to two data sets.     

The Consistency and Robustness of Modified Cramér–Von Mises and Kolmogorov–Cramér Estimators

This article focuses on the minimum distance estimators under two newly introduced modifications of Cramér–von Mises distance. The generalized power form of Cramér–von Mises distance is defined together with the so-called Kolmogorov–Cramér distance which includes both standard Kolmogorov and Cramér–von Mises distances as limiting special cases. We prove the consistency of Kolmogorov-Cramér estimators in the (expected) L₁-norm by direct technique employing domination relations between statistical distances. In our numerical simulation we illustrate the quality of consistency property for sample sizes of the most practical range from n = 10 to n = 500. We study dependence of consistency in L₁-norm on ?-contamination neighborhood of the true model and further the robustness of these two newly defined estimators for normal families and contaminated samples. Numerical simulations are used to compare statistical properties of the minimum Kolmogorov–Cramér, generalized Cramér–von Mises, standard Kolmogorov, and Cramér–von Mises distance estimators of the normal family scale parameter. We deal with the corresponding order of consistency and robustness. The resulting graphs are presented and discussed for the cases of the contaminated and uncontaminated pseudo-random samples.     

An integral formula for the distribution of self-normalized Gaussian random samples

Given i.i.d. Gaussian random variables and after standardizing the sample by subtracting the sample mean and dividing it by the sample deviation, we obtain an integral formula for the distribution of these self-normalized variables. Using geometrical arguments, we obtain the distribution of each and the joint distribution of two of them. These formulas can be used to calculate the expected value of the particular type of Cramér von Mises statistic to test normality.     

Mixtures of concentrated multivariate sine distributions with applications to bioinformatics

Motivated by examples in protein bioinformatics, we study a mixture model of multivariate angular distributions. The distribution treated here (multivariate sine distribution) is a multivariate extension of the well-known von Mises distribution on the circle. The density of the sine distribution has an intractable normalizing constant and here we propose to replace it in the concentrated case by a simple approximation. We study the EM algorithm for this distribution and apply it to a practical example from protein bioinformatics.     

Segmentation of circular data

Circular data – data whose values lie in the interval [0,2π) – are important in a number of application areas. In some, there is a suspicion that a sequence of circular readings may contain two or more segments following different models. An analysis may then seek to decide whether there are multiple segments, and if so, to estimate the changepoints separating them. This paper presents an optimal method for segmenting sequences of data following the von Mises distribution. It is shown by example that the method is also successful in data following a distribution with much heavier tails.     

A Bayesian analysis of the change-point problem for directional data

In this paper, we discuss a simple fully Bayesian analysis of the change-point problem for the directional data in the parametric framework with von Mises or circular normal distribution as the underlying distribution. We first discuss the problem of detecting change in the mean direction of the circular normal distribution using a latent variable approach when the concentration parameter is unknown. Then, a simpler approach, beginning with proper priors for all the unknown parameters – the sampling importance resampling technique – is used to obtain the posterior marginal distribution of the change-point. The method is illustrated using the wind data [E.P. Weijers, A. Van Delden, H.F. Vugts and A.G.C.A. Meesters, The composite horizontal wind field within convective structures of the atmospheric surface layer, J. Atmos. Sci. 52 (1995. 3866–3878]. The method can be adapted for a variety of situations involving both angular and linear data and can be used with profit in the context of statistical process control in Phase I of control charting and also in Phase II in conjunction with control charts.     

Testing fit for the grouped exponential distribution

Grouped data can often arise due to the lack of resolution of the measurement instruments; they also arise when data are deliberately rounded to a certain accuracy and are presented, say, in the form of a histogram. The author uses statistics of the Cramér‐von Mises type to test for the exponential distribution when data are grouped.     

A Modular CDF Approach for the Approximation of Percentiles

This article describes a method for computing approximate statistics for large data sets, when exact computations may not be feasible. Such situations arise in applications such as climatology, data mining, and information retrieval (search engines). The key to our approach is a modular approximation to the cumulative distribution function (cdf) of the data. Approximate percentiles (as well as many other statistics) can be computed from this approximate cdf. This enables the reduction of a potentially overwhelming computational exercise into smaller, manageable modules. We illustrate the properties of this algorithm using a simulated data set. We also examine the approximation characteristics of the approximate percentiles, using a von Mises functional type approach. In particular, it is shown that the maximum error between the approximate cdf and the actual cdf of the data is never more than 1% (or any other preset level). We also show that under assumptions of underlying smoothness of the cdf, the approximation error is much lower in an expected sense. Finally, we derive bounds for the approximation error of the percentiles themselves. Simulation experiments show that these bounds can be quite tight in certain circumstances.