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Finding the principal pointsof a random variable

Published online by Cambridge University Press:  15 August 2002

Emilio Carrizosa ,
E. Conde ,
A. Castaño  and
D. Romero–Morales
Emilio Carrizosa
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla, Spain.
E. Conde
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla, Spain.
A. Castaño
Affiliation:
Departamento de Matemáticas, E.U. Empresariales, Universidad de Cádiz, C/ Por Vera, N. 54, Jerez de la Frontera, Cádiz, Spain.
D. Romero–Morales
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla, Spain. Faculty of Economics and Business Administration, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands.
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Abstract

The p-principal points of a random variable X with finitesecond momentare those ppoints in ${\mathbb R}$ minimizing the expected squared distance from X tothe closest point.Although the determination of principal points involves in general theresolution of a multiextremal optimization problem, existing procedures inthe literature provide just a local optimum. In this paper we show thatstandard Global Optimization techniques can be applied.

Keywords

Principal pointsd.c. functionsbranch and bound.

Information

Type
Research Article
Information
RAIRO - Operations Research , Volume 35 , Issue 3 , July 2001 , pp. 315 - 328
Copyright
© EDP Sciences, 2001

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References

E. Carrizosa, E. Conde, A. Casta no, I. Espinosa, I. González and D. Romero-Morales, Puntos principales: Un problema de Optimización Global en Estadística, Presented at XXII Congreso Nacional de Estadística e Investigación Operativa. Sevilla (1995).
D.R. Cox, A use of complex probabilities in the theory of stochastic processes, in Proc. of the Cambridge Philosophical Society, Vol. 51 (1955) 313-319.
Flury, B., Principal points. Biometrika 77 (1990) 33-41. CrossRef
Flury, B. and Tarpey, T., Representing a Large Collection of Curves: A Case for Principal Points. Amer. Statist. 47 (1993) 304-306.
R. Fourer, D.M. Gay and B.W. Kernigham, AMPL, A modeling language for Mathematical Programming. The Scientific Press, San Francisco (1993).
Gelenbe, E. and Muntz, R.R., Probabilistic Models of Computer Systems-Part I. Acta Inform. 7 (1976) 35-60. CrossRef
Horst, R., Algorithm, An for Nonconvex Programming Problems. Math. Programming 10 (1976) 312-321. CrossRef
R. Horst and H. Tuy, Global Optimization. Deterministic Approaches. Springer-Verlag, Berlin (1993).
Lloyd, S.P., Least Squares Quantization in PCM. IEEE Trans. Inform. Theory 28 (1982) 129-137. CrossRef
Li, L. and Flury, B., Uniqueness of principal points for univariate distributions. Statist. Probab. Lett. 25 (1995) 323-327. CrossRef
Pötzelberger, K. and Felsenstein, K., An asymptotic result on principal points for univariate distribution. Optimization 28 (1994) 397-406. CrossRef
Rowe, S., Algorithm, An for Computing Principal Points with Respect to a Loss Function in the Unidimensional Case. Statist. Comput. 6 (1997) 187-190. CrossRef
Tarpey, T., Two principal points of symmetric, strongly unimodal distributions. Statist. Probab. Lett. 20 (1994) 253-257. CrossRef
Tarpey, T., Principal points and self-consistent points of symmetric multivariate distributions. J. Multivariate Anal. 53 (1995) 39-51. CrossRef
Tarpey, T., Li, L. and Flury, B., Principal points and self-consistent points of elliptical distributions. Ann. Statist. 23 (1995) 103-112. CrossRef
Zoppè, A., Principal points of univariate continuous distributions. Statist. Comput. 5 (1995) 127-132. CrossRef
Zoppè, A., Uniqueness, On and Symmetry of self-consistent points of univariate continuous distribution. J. Classification 14 (1997) 147-158. CrossRef