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Fitting Kent models to compositional data with small concentration. (English) Zbl 1325.62049
Summary: Compositional data can be transformed to directional data by the square root transformation and then modelled by using the Kent distribution. The current approach for estimating the parameters in the Kent model for compositional data relies on a large concentration assumption which assumes that the majority of the transformed data is not distributed too close to the boundaries of the positive orthant. When the data is distributed close to the boundaries with large variance significant folding may result. To treat this case we propose new estimators of the parameters derived based on the actual folded Kent distribution which are obtained via the EM algorithm. We show that these new estimators significantly reduce the bias in the current estimators when both the sample size and amount of folding is moderately large. We also propose using a saddlepoint density approximation for the Kent distribution normalising constant in order to more accurately estimate the shape parameters when the concentration is small or only moderately large.

MSC:
62F10 Point estimation
62-07 Data analysis (statistics) (MSC2010)
92B10 Taxonomy, cladistics, statistics in mathematical biology
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[1] Aitchison, J.: The statistical analysis of compositional data (with discussion). J. R. Stat. Soc., Ser. B, Stat. Methodol. 44, 139–177 (1982) · Zbl 0491.62017
[2] Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London (1986) · Zbl 0688.62004
[3] Chen, M., Kianifard, F.: Estimation of treatment difference and standard deviation with blinded data in clinical trials. Biom. J. 45, 135–142 (2003) · doi:10.1002/bimj.200390000
[4] Cuesta-Albertos, J.A., Cuevas, A., Fraiman, R.: On projection-based tests for directional and compositional data. Stat. Comput. 19, 367–380 (2009) · doi:10.1007/s11222-008-9098-3
[5] Jung, S., Foskey, M., Marron, J.S.: Principal arc analysis on direct product manifolds. Ann. Appl. Stat. 5, 578–603 (2011) · Zbl 1220.62077 · doi:10.1214/10-AOAS370
[6] Kent, J.T.: The Fisher-Bingham distribution on the sphere. J. R. Stat. Soc., Ser. B, Stat. Methodol. 44, 71–80 (1982) · Zbl 0485.62015
[7] Kent, J.T., Mardia, K.V., McDonnell, P.: The complex Bingham quartic distribution and shape analysis. J. R. Stat. Soc., Ser. B, Stat. Methodol. 68, 747–765 (2006) · Zbl 1110.62070 · doi:10.1111/j.1467-9868.2006.00565.x
[8] Kume, A., Walker, S.G.: Sampling from compositional and directional distributions. Stat. Comput. 16, 261–265 (2006) · doi:10.1007/s11222-006-8077-9
[9] Kume, A., Wood, A.T.A.: Saddlepoint approximations for the Bingham and Fisher-Bingham normalising constants. Biometrika 92, 465–476 (2005) · Zbl 1094.62063 · doi:10.1093/biomet/92.2.465
[10] Rivest, L.: On the information matrix for symmetric distributions on the hypersphere. Ann. Stat. 12, 1085–1089 (1984) · Zbl 0545.62035 · doi:10.1214/aos/1176346724
[11] Matz, A.W.: Maximum likelihood parameter estimation for the quartic exponential distribution. Technometrics 20, 475–484 (1978) · Zbl 0414.62027 · doi:10.1080/00401706.1978.10489702
[12] McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions, 2nd edn. Wiley, New Jersey (2008) · Zbl 1165.62019
[13] Scealy, J.L.: Modelling techniques for compositional data using distributions defined on the hypersphere. PhD thesis, Australian National University (2010)
[14] Scealy, J.L., Welsh, A.H.: Regression for compositional data by using distributions defined on the hypersphere. J. R. Stat. Soc., Ser. B, Stat. Methodol. 73, 351–375 (2011) · doi:10.1111/j.1467-9868.2010.00766.x
[15] Stephens, M.A.: Use of the von mises distribution to analyse continuous proportions. Biometrika 69, 197–203 (1982) · doi:10.1093/biomet/69.1.197
[16] Sundberg, R.: On estimation and testing for the folded normal distribution. Commun. Stat., Theory Methods 3, 55–72 (1974) · Zbl 0274.62021 · doi:10.1080/03610927408827103
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