MSP-Partitions and Unbiased Quantizations: A Review of Results
DOI:
https://doi.org/10.17713/ajs.v31i2&3.482Abstract
We resume recent developments in the theory of unbiased quantizations of probability distributions. Starting with variance-minimizing partitions, we review concept such as f-information, maximum support plane partition and quantizations, and motivate the definition of unbiased quantizations. The obtained results have applications in statistical inference and in the theory of comparison of experiments.References
D. Blackwell. Comparison of experiments. In Proc. 2nd Berkeley Symp. Math. Statistics Prob., pages 93–102, 1951.
D. Blackwell. Equivalent comparisons of experiments. Ann. Math. Statistics, 24:265–272, 1953.
H.-H. Bock. Automatische Klassifikation. Vandenhoeck und Ruprecht, 1974.
H.-H. Bock. A clustering technique for maximizing ϕ-divergence, noncentrality and discriminating power. In Analyzing and Modeling Data and Knowledge, pages 19–36, Heidelberg, 1992. Springer-Verlag.
B.A. Flury. Principal points. Biometrika, 77:33–41, 1990.
S. Graf and H. Luschgy. Foundations of quantization for probability distributions. volume 1730 of Lecture Notes in Math. Springer-Verlag, Heidelberg, 2000.
T. Kohonen. Self-organization and associative memory. Springer-Verlag, New York, 1984.
J.A. Mazanec and H. Strasser. A nonparametric approach to perceptions-based market segmentation: Foundations. In Interdisciplinary Studies in Economics and Management
Springer-Verlag, Heidelberg, 2000.
K. Pötzelberger. Admissible unbiased quantizations: Distributions with linear components, 2000a. Submitted.
K. Pötzelberger. The general quantization problem for distributions with regular support. Math. Methods of Statistics, 9:176–198, 2000b.
K. Pötzelberger. Admissible unbiased quantizations: Distributions with linear components. Math. Methods of Statistics, 2002. To appear.
K. Pötzelberger and H. Strasser. Clustering and quantization by msp-partitions. Statistics and Decisions, 19:331–371, 2001.
J. Rahnenführer. Multivariate permutation tests for the k-sample problem with clustered data, 2000. Submitted.
G. Steiner. Quantization and clustering with maximal information: Algorithms and numerical experiments. PhD thesis, Vienna University of Economics and Business Administration, Austria, 1999.
V. Strassen. The existence of probablity measures with given marginals. Ann. Math. Statist., 36:423–439, 1965.
H. Strasser. Mathematical Theory of Statistics. De Gruyter, Berlin, 1985.
H. Strasser. Towards a statistical theory of optimal quantization. In W. Gaul, O. Opitz, and M. Schader, editors, Data Analysis: Scientific Modeling and Practical Application, pages 369–383, Heidelberg, 2000. Springer-Verlag.
E. Torgersen. Comparison of statistical experiments. Cambridge University Press, Cambridge, 1991.
P.L. Zador. Development and evaluation of procedures for quantizing multivariate distributions. PhD thesis, Stanford University, 1964.
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