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Advances in flexible parametric distribution theory
Published in Christophe Ley, Thomas Verdebout, Modern Directional Statistics, 2017
Christophe Ley, Thomas Verdebout
For instance, if fZ is the normal density, one obtains the well-known wrapped normal distribution. The latter, like the majority of known wrapped distributions on the circle, suffers however from a major drawback: (2.1) does not simplify to a closed form. As a consequence, wrapped circular densities are often not easy to handle, the best known exception being the wrapped Cauchy distribution; see Section 2.2.2.Conditioning approach: express a distribution on R2 $ \mathbb{R}^{2} $ as the joint distribution of the polar coordinates length and angle (r,Θ) $ (r, \iTheta ) $ , and consider then the distribution of the angle conditionally on the restriction r = 1. A well-known example is the von Mises distribution (see Section 2.2.2), obtained by conditioning a bivariate normal distribution with mean (cos(μ),sin(μ))′ $ ( {\text{cos }}(\mu ), {\text{sin }}(\mu ))^{'} $ and covariance matrix
Gauge R & R studies for angular measurements
Published in Quality Engineering, 2021
Michael S. Hamada, Christine M. Anderson-Cook, Brian P. Weaver
In this article, we have introduced a wrapped normal variance component model for analyzing angular measurements from a gauge R & R study. The model is analogous to the variance components (or random effects) model for continuous normal measurements. The wrapped normal distribution is a proper distribution for angular measurements because its support is and is location-invariant to the choice of the range of the measurements and the location of the measurements. Other angular distributions such as the von Mises distribution (Fisher 1996) can be used; e.g., the R package circular (Agostinelli and Lund 2017) has functions for the von Mises random number generator and probability density function (rvonmises, dvonmises). We also saw in the location angle gauge R & R study example that a mixed-effects could be used when the “part” effects are fixed, i.e., the parts are intentionally selected and do not represent a random sample.