Estimation of circular-circular probability distribution (original) (raw)

Density estimation for circular data observed with errors

Biometrics, 2021

Until now the problem of estimating circular densities when data are observed with errors has been mainly treated by Fourier series methods. We propose kernel‐based estimators exhibiting simple construction and easy implementation. Specifically, we consider three different approaches: the first one is based on the equivalence between kernel estimators using data corrupted with different levels of error. This proposal appears to be totally unexplored, despite its potential for application also in the Euclidean setting. The second approach relies on estimators whose weight functions are circular deconvolution kernels. Due to the periodicity of the involved densities, it requires ad hoc mathematical tools. Finally, the third one is based on the idea of correcting extra bias of kernel estimators which use contaminated data and is essentially an adaptation of the standard theory to the circular case. For all the proposed estimators, we derive asymptotic properties, provide some simulatio...

A class of goodness-of-fit tests for circular distributions based on trigonometric moments

2019

We propose a class of goodness–of–fit test procedures for arbitrary parametric families of circular distributions with unknown parameters. The tests make use of the specific form of the characteristic function of the family being tested, and are shown to be consistent. We derive the asymptotic null distribution and suggest that the new method be implemented using a bootstrap resampling technique that approximates this distribution consistently. As an illustration, we then specialize this method to testing whether a given data set is from the von Mises distribution, a model that is commonly used and for which considerable theory has been developed. An extensive Monte Carlo study is carried out to compare the new tests with other existing omnibus tests for this model. An application involving five real data sets is provided in order to illustrate the new procedure.

Some Statistical Methods for Bivariate Circular Data

Journal of the Royal Statistical Society: Series B (Methodological), 1982

A function of the smallest singular value of the cross product matrix between two circular unit vectors is suggested as a measure of correlation. It possesses most of the desirable properties for a correlation coefficient given by Jupp and Mardia (1980). The concept of cluster dependence is introduced. A good predictor of one unit vector given the other is shown to depend on the type of dependence observed. The wrapped bivariate normal is characterized.

Automatic bandwidth selection for circular density estimation

Computational Statistics & Data Analysis, 2008

Given angular data θ 1 ,. .. , θ n ∈ [0, 2π) a common objective is to estimate the density. In the case that a kernel estimator is used, bandwidth selection is crucial to the performance. This paper obtains a "plug-in rule" for the bandwidth, which is based on the concentration of a reference density, namely, the von Mises distribution. It is seen that this is equivalent to the usual Euclidean plug-in rule in the case that the concentration becomes large. In the case that the concentration parameter is unknown, alternative methods are explored which are intended to be robust to departures from the reference density. Simulations indicate that "wrapped estimators" can perform well in this context.

On Smooth Density Estimation for Circular Data 1

2017

Fisher (1989: J. Structural Geology 11, 775-778) outlined an adaptation of the linear kernel estimator for density estimation that is commonly used in applications. However, better alternatives are now available based on circular kernels; see e.g. Di Marzio, Panzera, and Taylor, 2009: Statistics & Probability Letters 79, 2066-2075. This paper provides a short review on modern smoothing methods for density and distribution functions dealing with the circular data. We highlight the usefulness of circular kernels for smooth density estimation in this context and contrast it with smooth density estimation based on orthogonal series. It is seen that the wrapped Cauchy kernel as a choice of circular kernel appears as a natural candidate as it has a close connection to orthogonal series density estimation on a unit circle. In the literature the use of von Mises circular kernel is investigated (see Taylor, 2008: Computational Statistics & Data Analysis 52, 3493-3500), that requires numerica...

Nonparametric estimating equations for circular probability density functions and their derivatives

Electronic Journal of Statistics

We propose estimating equations whose unknown parameters are the values taken by a circular density and its derivatives at a point. Specifically, we solve equations which relate local versions of population trigonometric moments with their sample counterparts. Major advantages of our approach are: higher order bias without asymptotic variance inflation, closed form for the estimators, and absence of numerical tasks. We also investigate situations where the observed data are dependent. Theoretical results along with simulation experiments are provided.

Estimation of circular statistics in the presence of measurement bias

arXiv (Cornell University), 2022

Background and objective. Circular statistics and Rayleigh tests are important tools for analyzing the occurrence of cyclic events. However, current methods fail in the presence of measurement bias, such as incomplete or otherwise nonuniform sampling. Consider, for example, studying 24-cyclicity but having data not recorded uniformly over the full 24-hour cycle. The objective of this paper is to present a method to estimate circular statistics and their statistical significance even in this circumstance. Methods. We present our objective as a special case of a more general problem: estimating probability distributions in the context of imperfect measurements, a highly studied problem in high energy physics. Our solution combines 1) existing approaches that estimate the measurement process via numeric simulation and 2) innovative use of linear parametrizations of the underlying distributions. We compute the estimation error for several toy examples as well as a realworld example: analyzing the 24-hour cyclicity of an electrographic biomarker of epileptic tissue controlled for state of vigilance. Results. Our method shows low estimation error. In a real-world example, we observed the corrected moments had a root mean square residual less than 0.007. We additionally found that, even with unfolding, Rayleigh test statistics still often underestimate the p-values (and thus overestimate statistical significance) in the presence of non-uniform sampling. Numerical estimation of statistical significance, as described herein, is thus preferable. Conclusions. The presented methods provide a robust solution to addressing incomplete or otherwise non-uniform sampling. The general method presented is also applicable to a wider set of analyses involving estimation of the true probability distribution adjusted for imperfect measurement processes.

On Some Circular Distributions Induced by Inverse Stereographic Projection

Sankhya B, 2019

In earlier studies of circular data, the corresponding probability distributions considered were mostly assumed to be symmetric. However, the assumption of symmetry may not be meaningful for some data. Thus there has been increased interest, more recently, in developing skewed circular distributions. In this article we introduce three skewed circular models based on inverse stereographic projection, originally introduced by Minh and Farnum (2003), by considering three different versions of skewed-t considered in the literature, namely skewed-t by Azzalini (1985), two-piece skewed-t (Fernández and Steel, 1998) and skewedt by Jones and Faddy (2003). Shape properties of the resulting circular distributions along with estimation of parameters using maximum likelihood are also discussed in this article. Further, real data sets are used to illustrate the application of the new models. It is found that Azzalini and Jones-Faddy skewed-t versions are good competitors, however, the Jones-Faddy version is computationally more tractable.

A General Approach for Obtaining Wrapped Circular Distributions via Mixtures

Sankhya A, 2017

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