Topics in circular statistics / S. Rao Jammalamadaka, A. SenGupta

Incluye bibliografía: páginas 295-309 e índice: páginas 311-319

1 Introduction.. 1.1 Introduction.. 1.2 Applications and Background.. 1.2.1 Some Examples.. 1.2.2 The Need for Appropriate Analysis.. 1.3 Descriptive Statistics.. 1.3.1 Measure of Center.. 1.3.2 Circular Distance and Measure of Dispersion.. 1.3.3 Higher Moments.. 2 Circular Probability Distributions.. 2.1 Introduction.. 2.1.1 Some Methods of Obtaining Circular Distributions.. 2.2 Circular Distributions.. 2.2.1 Uniform Distribution.. 2.2.2 Cardioid Distribution.. 2.2.3 A Triangular Distribution.. 2.2.4 Circular Normal (CN Distribution.. 2.2.5 Offset Normal Distribution.. 2.2.6 Wrapped Normal (WN Distribution.. 2.2.7 Wrapped Cauchy (WC Distribution.. 2.2.8 General Wrapped Stable (WS Distributions.. 2.2.9 Variations of the CN Distribution.. 2.2.10 A Circular Beta Model.. 2.2.11 Asymmetric Circular Distributions.. 2.3 Bivariate Circular Distributions.. 2.3.1 A Bivariate von Mises Distribution.. 2.3.2 Wrapped Bivariate Normal Distribution.. 2.3.3 Circular-Linear Distribution.. 2.4 Generation of Circular Random Variables.. 2.5 Appendix: Curved Exponential Families and CND.. 3 Some Sampling Distributions.. 3.1 Introduction.. 3.2 Generalized Pearson's Random Walk Problem.. 3.2.1 The General Case.. 3.3 Sampling Distributions for CN Distribution.. 3.3.1 Distribution of (C, S.. 3.3.2 Distribution of R.. 3.3.3 Distribution of V.. 3.4 Some Large Sample Results.. 3.4.1 Series Approximations for R and V.. 3.4.2 Central Limit Type Results.. 3.4.3 Large Sample Results for Statistics Based on Moments.. 3.4.4 First Significant Digit Phenomenon.. 3.5 Two or More Samples.. 3.6 Approximate Distributions for Large k.. 4 Estimation of Parameters.. 4.1 Introduction.. 4.2 CN Distribution.. 4.2.1 Estimating the Parameters of a CN Distribution.. 4.2.2 Optimal Properties of the MLEs.. 4.3 CN Mixtures.. 4.4 ML Estimation for the WC Distribution.. 4.5 Circular Beta Distribution.. 4.6 WS Family.. 4.7 Confidence Intervals.. 4.8 Appendix: Proofs

5 Tests for Mean Direction and Concentration.. 5.1 Introduction.. 5.2 Single Population.. 5.2.1 Tests for Mean Direction.. 5.2.2 Higher-Order Power Comparison.. 5.2.3 Tests for the Concentration Parameter.. 5.3 Two or More Populations.. 5.3.1 Comparing Mean Directions and Approximate ANOVA.. 5.3.2 Tests for Concentration Parameters.. 6 Tests for Uniformity.. 6.1 Introduction.. 6.2 Uniformity against WS Alternatives.. 6.2.1 LMP Test Against the WS Family.. 6.2.2 Monotonicity of Power Function.. 6.2.3 Consistency and Other Optimal Properties.. 6.3 Uniformity versus WSM Alternatives.. 6.3.1 Monotonicity of the Power Function.. 6.3.2 Consistency and Other Optimal Properties.. 6.3.3 Optimal Test with Unknown p.. 6.4 LMP Invariant Test for Unknown µ.. 6.4.1 Monotonicity of the Power Function.. 6.4.2 Consistency and Other Optimal Properties.. 6.5 Appendix: Proofs.. 7 Nonparametric Testing Procedures.. 7.1 Introduction.. 7.2 One-Sample Problem and Goodness-of-fit.. 7.2.1 Tests Based on Empirical Distribution Functions.. 7.2.2 x2 and Other Tests.. 7.2.3 Tests Based on Sample Arc-Lengths.. 7.3 Two-Sample Problems.. 7.3.1 Two-Sample Tests based on Edf's.. 7.3.2 Wheeler and Watson Test.. 7.3.3 Tests based on Spacing-Frequencies.. 7.4 Multi-Sample Tests.. 7.4.1 Homogeneity Tests in Large Samples.. 8 Circular Correlation and Regression.. 8.1 Introduction.. 8.2 A Circular Correlation Measure, pc.. 8.2.1 pc for Some Parametric Models.. 8.3 Rank Correlation.. 8.4 Other Measures of Circular Correlation.. 8.5 Circular-Linear Correlation.. 8.6 Circular-Circular Regression.. 8.6.1 Estimation of Regression Coefficients.. 8.6.2 Determination of m.. 8.7 Circular-Linear Regression.. 8.8 Linear-Circular Regression.. 8.9 Testing Stochastic Independence of Circular Variables.. 9 Predictive Inference for Directional Data.. 9.1 Introduction.. 9.2 Prediction Under a von Mises Model.. 9.3 Generalized von Mises-type Distributions

9.3.1 -Modal von Mises Distribution.. 9.4 Bayesian Predictive Density.. 9.4.1 Case of Large k.. 9.5 Robustness Against Symmetric Wrapped Stable Family.. 9.6 HPD Computation and An Example.. 10 Outliers and Related Problems.. 10.1 Introduction.. 10.2 Diagnostic Tools.. 10.3 Tests for Mixtures.. 10.3.1 Introduction.. 10.3.2 The LMPI Test.. 10.3.3 Asymptotic Distribution of T.. 10.3.4 Monotonicity and Consistency Properties.. 10.3.5 Non-regular Cases.. 10.4 Slippage Problem and Outliers.. 10.4.1 Introduction.. 10.4.2 A Decision Theoretic Approach.. 10.4.3 The LRT.. 10.4.4 Simulations and An Example.. 10.5 Concluding Remarks.. 11 Change-point Problems.. 11.1 Introduction.. 11.2 Tests for Change in Mean Direction.. 11.2.1 K Known Case.. 11.2.2 K Unknown Case.. 11.2.3 Simulated Critical Values.. 11.2.4 Power Comparisons.. 11.2.5 Robustness.. 11.3 Tests for Change in µ and/or k.. 11.4 An Example.. 11.5 Nonparametric Approaches.. 11.6 Other Approaches.. 12 Miscellaneous Topics.. 12.1 Introduction.. 12.2 An Entropy-based Test for the CND.. 12.3 Classification and Discriminant Analysis.. 12.4 Factorial Designs.. 12.4.1 Likelihood Ratio Tests.. 12.5 Bayesian Analysis.. 12.6 Sequential Methods.. 12.7 Shape Analysis.. 12.8 Stochastic Processes and Time Series.. 12.9 Density Estimation.. 12.10 Periodic Smoothing Splines.. A Some Facts on Bessel Functions.. B How to Use the CircStats Package.. B.l Installation Instructions for the CircStats Library.. B.l.l Windows Version.. B.1.2 UNIX Version.. B.2 CircStats Functions.. Bibliography.. Author Index.. Subject Index.. Notations and Abbreviations Index

This research monograph on circular data analysis covers some recent advances in the field, besides providing a brief introduction to, and a review of, existing methods and models. The primary focus is on recent research into topics such as change-point problems, predictive distributions, circular correlation and regression, etc. An important feature of this work is the S-plus subroutines provided for analyzing actual data sets. Coupled with the discussion of new theoretical research, the book should benefit both the researcher and the practitioner. eng