RECENT DEVELOPMENTS IN DIRECTIONAL STATISTICS: DISTRIBUTIONAL ASPECTS(<Special Section>Spatial Statistics)

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  • Shimizu Kunio
    Department of Mathematics, Faculty of Science and Technology, Keio University

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  • 方向統計学の最近の発展(<特集>時空間統計)
  • 方向統計学の最近の発展
  • ホウコウ トウケイガク ノ サイキン ノ ハッテン

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Abstract

The paper reviews recent developments in (a) distributions on the circle and on manifolds such as sphere, torus, cylinder and disc, (b) regressions, and (c) inferences including test of symmetry, test of independence, and estimation and/or test of mean direction and concentration in von Mises-Fisher distributions. About distributions on the circle, the paper deals with the following. A symmetric Pearson type VII ditribution (t as a special case) on the circle includes the von Mises distribution as a limit. The Jones-Pewsey distribution extends the t-distribution on the circle and it contains cardioid, wrapped Cauchy, and Cartright's power-of-cosine or Minh-Farnum distributions as special cases. Relation between t- and Jones-Pewsey distributions on the sphere is discussed and an asymmetric t is proposed. The paper also reviews the development of wrapped skew Laplace, wrapped skew normal, and wrapped symmetric α-stable distributions. About distributions on manifolds such as torus, cylinder and disc, the following are discussed. A submodel of Mardia's bivariate von Mises distribution has a concise expression of the normalizing constant. Maximizing the entropy gives some models on the cylinder. One of which is a distribution whose angular marginal distribution is a wrapped Cauchy and conditional distribution is a von Mises. Another has a generalized von Mises and an exponential as conditional distributions. Similar to bivariate distributions on the torus whose marginals are specified, four dimensional distributions with specified bivariate marginals on the cylinder are proposed and are applicable to distributions of wind direction and speed observed at two sites. The Mobius transformation of a bivariate beta (Pearson type II) provides a skew distribution on the disc. Circular-circular regression model based on von Mises errors as well as wrapped Cauchy are discussed. As for inferences of models, the paper gives a review of test of symmetry and test of mean direction and/or concentration in von Mises-Fisher distributions.

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