Change-point problems for the von Mises distribution

A generalized likelihood ratio procedure and a Bayes procedure are considered for change-point problems for the mean direction of the von Mises distribution, both when the concentration parameter is known and when it is unknown. These tests are based on sample resultant lengths. Tables that list critical values of these test statistics are provided. These tests are shown to be valid even when the data come from other similar unimodal circular distributions. Some empirical studies of powers of these test procedures are also incorporated.     

A fast algorithm for sampling from the posterior of a von Mises distribution

Motivated by molecular biology, there has been an upsurge of research activities in directional statistics in general and its Bayesian aspect in particular. The central distribution for the circular case is von Mises distribution which has two parameters (mean and concentration) akin to the univariate normal distribution. However, there has been a challenge to sample efficiently from the posterior distribution of the concentration parameter. We describe a novel, highly efficient algorithm to sample from the posterior distribution and fill this long-standing gap.     

Qualitative robustness of von Mises statistics based on strongly mixing data

In this article, the property of qualitative robustness is studied for von Mises statistics in the situation where the observations are not necessarily independent but are drawn from a strongly mixing sequence of identically distributed random variables. The notion of qualitative robustness is taken from “Zähle (2012, submitted)” where Huber’s version of Hampel’s original definition was adapted to the case of dependent observations. The main result is illustrated by means of several examples including the sample variance, the sample Gini’s mean difference and the Cramér–von Mises statistic.     

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.     

A general censoring scheme for circular data

A generalized censoring scheme in the survival analysis context was introduced by the authors in Jammalamadaka and Mangalam [S. Rao Jammalamadaka, V. Mangalam, Nonparametric estimation for middle censored data, J. Nonparametr. Stat. 15 (2003) 253–265]. In this article we discuss how such a censoring scheme applies to circular data and in particular when the original data is assumed to come from a parametric model such as the von Mises. Maximum likelihood estimation of the parameters as well as their large-sample properties are considered under this censoring scheme. We also consider nonparametric estimation of the circular probability distribution under such a general censoring scheme and use Monte Carlo methods to investigate its effects on the estimation of the mean direction and concentration.     

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.     

A modified likelihood ratio test for the mean direction in the von mises distribution

The likelihood ratio test (LRT) for the mean direction in the von Mises distribution is modified for possessing a common asymptotic distribution both for large sample size and for large concentration parameter. The test statistic of the modified LRT is compared with the F distribution but not with the chi-square distribution usually employed, Good performances of the modified LRT are shown by analytical studies and Monte Carlo simulation studies, A notable advantage of the test is that it takes part in the unified likelihood inference procedures including both the marginal MLE and the marginal LRT for the concentration parameter.     

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.     

Goodness-of-fit tests based on the distance between the Dirichlet process and its base measure

The Dirichlet process is a fundamental tool in studying Bayesian nonparametric inference. The Dirichlet process has several sum representations, where each one of these representations highlights some aspects of this important process. In this paper, we use the sum representations of the Dirichlet process to derive explicit expressions that are used to calculate Kolmogorov, Lévy, and Cramér–von Mises distances between the Dirichlet process and its base measure. The derived expressions of the distance are used to select a proper value for the concentration parameter of the Dirichlet process. These tools are also used in a goodness-of-fit test. Illustrative examples and simulation results are included.     

Tables for the Lilliefors and Modified Cramer–von Mises Tests of Normality

We present new tabulations of the Lilliefors distribution and the modified Cramer–von Mises distribution, which are used to test for normality when the population mean and variance are unknown. Some practical remarks and an example are given.     

A Bayesian Analysis of Directional Data Using the von Mises–Fisher Distribution

ABSTRACT

This article presents a Bayesian analysis of the von Mises–Fisher distribution, which is the most important distribution in the analysis of directional data. We obtain samples from the posterior distribution using a sampling-importance-resampling method. The procedure is illustrated using simulated data as well as real data sets previously analyzed in the literature.     

A bayesian approach to prediction in the presence of spurious observations for several models

The problem of spuriousity has been dealt with from a Bayesian perspective by, among others, Box and Taio (1968) and in several papers by Guttman with various co-authors, beginning with Guttman (1973), The main objective of these papers has been to obtain posterior distributions of parameters, and to base inference on these distributions. In the current paper, the Bayesian argument is carried one step further by deriving predictive distributions of future observations. Inferences are then based on these distributions. We will obtain predictive results for several models, First, we consider the univariate normal case with one spurious observation, This is then generalized to several spurious observations. The multivariate normal situation is studied next. Finally, we consider the general linear model with normal errors.     

Analysis of variance for circular data

Over the last 25 years, increasing attention has been given to the problem of analysing data arising from circular distributions. The most important circular distribution was introduced by Von Mises (1918) which takes the form:

[Formulas]

where Io(k) is a modified Bessel function, u0 is the mean direction and k is the concentration parameter of the distribution. Watson & Williams (1956) laid the foundation of analysis of variance type techniques for the two-dimensional case of circular data using the Von Mises distribution. Stephens (1962a,b; 1969, 1972). Upton (1974) and Stephens (1982) made further improvements to Watson & Williams’ work. In this paper the authors will discuss the pitfalls of the methods adopted by Stephens (1982) and present a unified analysis of variance type approach for circular data.     

Bayesian inference for stable Lévy–driven stochastic differential equations with high‐frequency data

In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure‐jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi‐likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings.     

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.     

Goodness of fit tests with interval censored data

Cramér–von Mises type goodness of fit tests for interval censored data case 2 are proposed based on a resampling method called the leveraged bootstrap, and their asymptotic consistency is shown. The proposed tests are computationally efficient, and in fact can be applied to other types of censored data, including right censored data, doubly censored data and (mixture of) case k interval censored data. Some simulation results and an example from AIDS research are presented.     

Forecasting future values of changing sequences

Sequences of independent random variables are observed and on the basis of these observations future values of the process are forecast. The Bayesian predictive density of k future observations for normal, exponential, and binomial sequences which change exactly once are analyzed for several cases. It is seen that the Bayesian predictive densities are mixtures of standard probability distributions. For example, with normal sequences the Bayesian predictive density is a mixture of either normal or t-distributions, depending on whether or not the common variance is known. The mixing probabilities are the same as those occurring in the corresponding posterior distribution of the mean(s) of the sequence. The predictive mass function of the number of future successes that will occur in a changing Bernoulli sequence is computed and point and interval predictors are illustrated.     

Exact Likelihood Inference for k Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of k competing products with regard to their reliability. In this paper, when a joint progressively Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. Their conditional moment generating functions and exact densities are obtained, using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are discussed. An empirical evaluation of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities and average widths. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

Second-order biases of the maximum likelihood estimates in von Mises regression models

This paper discusses issues related to the improvement of maximum likelihood estimates in von Mises regression models. It obtains general matrix expressions for the second-order biases of maximum likelihood estimates of the mean parameters and concentration parameters. The formulae are simple to compute, and give the biases by means of weighted linear regressions. Simulation results are presented assessing the performance of corrected maximum likelihood estimates in these models.