Advanced SearchSearch Tips
A Projected Exponential Family for Modeling Semicircular Data
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
A Projected Exponential Family for Modeling Semicircular Data
Kim, Hyoung-Moon;
  PDF(new window)
For modeling(skewed) semicircular data, we derive a new exponential family of distributions. We extend it to the l-axial exponential family of distributions by a projection for modeling any arc of arbitrary length. It is straightforward to generate samples from the l-axial exponential family of distributions. Asymptotic result reveals that the linear exponential family of distributions can be used to approximate the l-axial exponential family of distributions. Some trigonometric moments are also derived in closed forms. The maximum likelihood estimation is adopted to estimate model parameters. Some hypotheses tests and confidence intervals are also developed. The Kolmogorov-Smirnov test is adopted for a goodness of t test of the l-axial exponential family of distributions. Samples of orientations are used to demonstrate the proposed model.
Uniformly minimum variance unbiased estimator;maximum likelihood estimator;skewed l-axial data;Kolmogorov-Smirnov test;delta method;
 Cited by
Ahn, B. J. and Kim, H. M. (2008). A new family of semicircular models: The semicircular Laplace distributions, Communications of the Korean Statistical Society, 15, 775-781. crossref(new window)

Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization, SIAM Journal Scientific Computing, 16, 1190-1208. crossref(new window)

Casella, G. and Berger, R. L. (2002). Statistical Inference, 2nd Edition, Duxbury Press.

Fisher, N. I. (1993). Statistical Analysis of Circular Data, Cambridge University Press.

Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products(7th edition), Academic Press.

Guardiola, J. H. (2004). The Semicircular Normal Distribution, Ph. D. Dissertation, Baylor University, Institute of Statistics.

Healy, M. J. R. (1968). Multivariate normal plotting, Applied Statistics, 17, 157-161. crossref(new window)

Jammalamadaka, S. R. and Kozubowski, T. J. (2003). A New family of circular models: The wrapped Laplace distributions, Advances and Applications in Statistics, 3, 77-103.

Jammalamadaka, S. R. and Kozubowski, T. J. (2004). New families of wrapped distributions for modeling skew circular data, Communications in Statistics-Theory and Method, 33, 2059-2074. crossref(new window)

Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics, World Scientific Publishing, Singapore.

Jones, M. C. and Pewsey, A. (2005). A Family of symmetric distributions on the circle, Journal of the American Statistical Association, 100, 1422-1428. crossref(new window)

Jones, T. A. (1968). Statistical analysis of orientation data, Journal of Sedimentary Petrology, 38, 61-67.

Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation, 2nd edition, Springer Science, New York.

Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd edition, Springer Science, New York.

Mardia, K. V. and Jupp, P. (2000). Directional Statistics, John Wiley and Sons, Chichester.

Pewsey, A. (2000). The wrapped skew-normal distribution on the circle, Communications in Statistics-Theory and Method, 29, 2459-2472. crossref(new window)

Pewsey, A. (2002). Testing circular symmetry, The Canadian Journal of Statistics, 30, 591-600. crossref(new window)

Pewsey, A. (2004). Testing for circular reflective symmetry about a known median axis, Journal of Applied Statistics, 31, 575-585 crossref(new window)

Pewsey, A. (2006). Modelling asymmetrically distributed circular data using the wrapped skew-normal distribution, Environmental and Ecological Statistics, 13, 257-269. crossref(new window)

Pewsey, A. (2008). The wrapped stable family of distributions as a flexible model for circular data, Computational Statistics & Data Analysis, 52, 1516-1523. crossref(new window)

Pewsey, A., Lewis, T. and Jones, M. C. (2007). The wrapped t family of circular distributions, Australian & New Zealand Journal of Statistics, 49, 79-91. crossref(new window)

Smirnov, N. V. (1948). Tables for estimating the goodness of fit of empirical distributions, Annals of Mathematical Statistics, 19, 279-281. crossref(new window)