Inference on parameters of Watson distributions and application to classification of observations. (English) Zbl 1479.62033
Summary: In this paper, we derive a class of equivariant estimators of the directional parameter of the Watson distribution with a known concentration parameter. When all parameters are unknown, we derive restricted maximum likelihood estimators (MLEs) of the concentration parameters and Bayes estimators of the parameters under a noninformative prior. An improved likelihood ratio test is proposed to test equality of directional parameters of several Watson distributions with a common concentration parameter. We derive rules to classify axial data into one of the Watson populations on the hypersphere when all parameters are unknown. We propose classification rules based on the MLEs and the Bayes estimators of the parameters. The likelihood ratio-based rule, predictive Bayes rule, and kernel density classifier have been derived for two Watson populations. Moreover, the rules are compared using simulations.
MSC:
| 62H11 | Directional data; spatial statistics |
| 62F03 | Parametric hypothesis testing |
| 62F10 | Point estimation |
| 62F15 | Bayesian inference |
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
Keywords:
Bayes estimator; classification rule; equivariant estimator; improved likelihood ratio test; probability of misclassification; restricted MLEReferences:
| [1] | Fisher, N. I.; Lewis, T.; Embleton, B., Statistical Analysis of Spherical Data (1993), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0782.62059 |
| [2] | Bagchi, P.; Guttman, I., Theoretical considerations of the multivariate von Mises-Fisher distribution, J. Appl. Stat., 15, 2, 149-169 (1988) |
| [3] | Bagchi, P., Empirical Bayes estimation in directional data, J. Appl. Stat., 21, 4, 317-326 (1994) |
| [4] | Nunez-Antonio, G.; Gutiérrez-Pena, E., A Bayesian analysis of directional data using the von Mises-Fisher distribution, Comm. Statist. Simulation Comput., 34, 4, 989-999 (2005) · Zbl 1078.62025 |
| [5] | Singh, H.; Misra, N.; Li, S., Estimation of order restricted concentration parameters of von Mises distributions, Comm. Statist. Simulation Comput., 34, 1, 21-40 (2005) · Zbl 1061.62034 |
| [6] | Kumar, Kanika S.; Sengupta, A., A unified approach to decision-theoretic properties of the MLEs for the mean directions of several Langevin distributions, J. Multivariate Anal., 133, 160-172 (2015) · Zbl 1302.62017 |
| [7] | Rumcheva, P.; Presnell, B., An improved test of equality of mean directions for the Langevin-von Mises-Fisher distribution, Aust. N. Z. J. Stat., 59, 1, 119-135 (2017) · Zbl 1373.62071 |
| [8] | Best, D. J.; Fisher, N. I., Goodness-of-fit and discordancy tests for samples from the Watson distribution on the sphere, Aust. J. Stat., 28, 1, 13-31 (1986) |
| [9] | Figueiredo, A., Multi-sample likelihood ratio tests based on bipolar Watson distributions defined on the hypersphere, Comm. Statist. Theory Methods, 36, 4, 815-820 (2007) · Zbl 1109.62046 |
| [10] | Figueiredo, A., Two-way ANOVA for the Watson distribution defined on the hypersphere, Statist. Papers, 49, 2, 363-376 (2008) · Zbl 1168.62360 |
| [11] | Figueiredo, A., Multi-sample tests for axial data from Watson distributions, AStA Adv. Stat. Anal., 93, 4, 371-386 (2009) · Zbl 1331.62092 |
| [12] | Sra, S.; Karp, D., The multivariate Watson distribution: Maximum-likelihood estimation and other aspects, J. Multivariate Anal., 114, 256-269 (2013) · Zbl 1258.62062 |
| [13] | Nascimento, A.; da Silva, R. C.; Amaral, J. A., Distance-based hypothesis tests on the Watson distribution, Comm. Statist. Simulation Comput., 49, 9, 2225-2238 (2020) · Zbl 07552794 |
| [14] | Fallaize, C. J.; Kypraios, T., Exact Bayesian inference for the Bingham distribution, Stat. Comput., 26, 1-2, 349-360 (2016) · Zbl 1342.62031 |
| [15] | McLachlan, G. J., Discriminant Analysis and Statistical Pattern Recognition (2004), John Wiley & Sons: John Wiley & Sons New Jersey · Zbl 1108.62317 |
| [16] | Conde, D.; Salvador, B.; Rueda, C.; Fernández, M. A., Performance and estimation of the true error rate of classification rules built with additional information. An application to a cancer trial, Stat. Appl. Genet. Mol., 12, 5, 583-602 (2013) |
| [17] | Batsidis, A., Errors of misclassification in discrimination with data from truncated \(t\) populations, Statist. Papers, 53, 2, 281-298 (2012) · Zbl 1440.62236 |
| [18] | Jana, N.; Kumar, S., Ordered classification rules for inverse Gaussian populations with unknown parameters, J. Stat. Comput. Simul., 89, 14, 2597-2620 (2019) · Zbl 07193857 |
| [19] | El Khattabi, S.; Streit, F., Identification analysis in directional statistics, Comput. Statist. Data Anal., 23, 1, 45-63 (1996) · Zbl 0900.62323 |
| [20] | Figueiredo, A.; Gomes, P., Discriminant analysis based on the Watson distribution defined on the hypersphere, Statistics, 40, 5, 435-445 (2006) · Zbl 1098.62080 |
| [21] | López-Cruz, P. L.; Bielza, C.; Larrañaga, P., Directional naive Bayes classifiers, Pattern Anal. Appl., 18, 2, 225-246 (2015) · Zbl 1428.62283 |
| [22] | SenGupta, A.; Roy, S., A simple classification rule for directional data, (Advances in Ranking and Selection, Multiple Comparisons, and Reliability (2005), Springer), 81-90 |
| [23] | Sengupta, A.; Ugwuowo, F. I., A classification method for directional data with application to the human skull, Comm. Statist. Theory Methods, 40, 3, 457-466 (2011) · Zbl 1208.62104 |
| [24] | Fernandes, K.; Cardoso, J. S., Discriminative directional classifiers, Neurocomputing, 207, 141-149 (2016) |
| [25] | Tsagris, M.; Alenazi, A., Comparison of discriminant analysis methods on the sphere, Commun. Stat. Case Stud. Data Anal. Appl., 5, 4, 467-491 (2019) |
| [26] | Leguey, I.; Bielza, C.; Larrañaga, P., Circular Bayesian classifiers using wrapped Cauchy distributions, Data Knowl. Eng., 122, 101-115 (2019) |
| [27] | Mardia, K. V.; Jupp, P. E., Directional Statistics (2000), John Wiley & Sons: John Wiley & Sons New York · Zbl 0935.62065 |
| [28] | Watson, G. S., Some estimation theory on the sphere, Ann. Inst. Statist. Math., 38, 2, 263-275 (1986) · Zbl 0609.62077 |
| [29] | Bingham, C., An antipodally symmetric distribution on the sphere, Ann. Statist., 2, 6, 1201-1225 (1974) · Zbl 0297.62010 |
| [30] | James, A. T., Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Stat., 35, 2, 475-501 (1964) · Zbl 0121.36605 |
| [31] | Robertson, T.; Wright, F. T.; Dykstra, R. L., Order Restricted Statistical Inference (1988), Wiley: Wiley New York · Zbl 0645.62028 |
| [32] | Kume, A.; Walker, S. G., On the Bingham distribution with large dimension, J. Multivariate Anal., 124, 345-352 (2014) · Zbl 1283.60020 |
| [33] | Ferguson, T. S., Mathematical Statistics: A Decision Theoretic Approach (1967), Academic Press: Academic Press New York · Zbl 0153.47602 |
| [34] | Lehmann, E. L.; Casella, G., Theory of Point Estimation (1998), Springer Verlag: Springer Verlag New York · Zbl 0916.62017 |
| [35] | Gray, H. L.; Baek, J.; Woodward, W. A.; Miller, J.; Fisk, M., A bootstrap generalized likelihood ratio test in discriminant analysis, Comput. Statist. Data Anal., 22, 2, 137-158 (1996) · Zbl 0900.62316 |
| [36] | Chen, Y.-H.; Wei, D.; Newstadt, G.; DeGraef, M.; Simmons, J.; Hero, A., Statistical estimation and clustering of group-invariant orientation parameters, (2015 18th International Conference on Information Fusion (2015), IEEE), 719-726 |
| [37] | Lachenbruch, P. A.; Mickey, M. R., Estimation of error rates in discriminant analysis, Technometrics, 10, 1, 1-11 (1968) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
