Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T04:23:01.714Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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