Finding the principal points of a random variable

Carrizosa, Emilio; Conde, E.; Castaño, A.; Romero-Morales, D.

Carrizosa, Emilio ; Conde, E. ; Castaño, A.¹ ; Romero-Morales, D.²

¹ Departamento de Matemáticas, E.U. Empresariales, Universidad de Cádiz, C/ Por Vera, N. 54, Jerez de la Frontera, Cádiz, Spain.
² Faculty of Economics and Business Administration, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 3, pp. 315-328.

Résumé

The $p$ -principal points of a random variable $X$ with finite second moment are those $p$ points in $ℝ$ minimizing the expected squared distance from $X$ to the closest point. Although the determination of principal points involves in general the resolution of a multiextremal optimization problem, existing procedures in the literature provide just a local optimum. In this paper we show that standard Global Optimization techniques can be applied.

Mots-clés : principal points, d.c. functions, branch and bound

Affiliations des auteurs :

Carrizosa, Emilio ; Conde, E. ; Castaño, A. ¹ ; Romero-Morales, D. ²

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     author = {Carrizosa, Emilio and Conde, E. and Casta\~no, A. and Romero-Morales, D.},
     title = {Finding the principal points of a random variable},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {315--328},
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     number = {3},
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     url = {https://www.numdam.org/item/RO_2001__35_3_315_0/}
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Carrizosa, Emilio; Conde, E.; Castaño, A.; Romero-Morales, D. Finding the principal points of a random variable. RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 3, pp. 315-328. https://www.numdam.org/item/RO_2001__35_3_315_0/

Bibliographie
Cité par

[1] E. Carrizosa, E. Conde, A. Castaño, 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).

[2] 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. | MR | Zbl

[3] B. Flury, Principal points. Biometrika 77 (1990) 33-41. | MR | Zbl

[4] B. Flury and T. Tarpey, Representing a Large Collection of Curves: A Case for Principal Points. Amer. Statist. 47 (1993) 304-306.

[5] R. Fourer, D.M. Gay and B.W. Kernigham, AMPL, A modeling language for Mathematical Programming. The Scientific Press, San Francisco (1993).

[6] E. Gelenbe and R.R. Muntz, Probabilistic Models of Computer Systems-Part I. Acta Inform. 7 (1976) 35-60. | MR | Zbl

[7] R. Horst, An Algorithm for Nonconvex Programming Problems. Math. Programming 10 (1976) 312-321. | MR | Zbl

[8] R. Horst and H. Tuy, Global Optimization. Deterministic Approaches. Springer-Verlag, Berlin (1993). | MR | Zbl

[9] S.P. Lloyd, Least Squares Quantization in PCM. IEEE Trans. Inform. Theory 28 (1982) 129-137. | MR | Zbl

[10] L. Li and B. Flury, Uniqueness of principal points for univariate distributions. Statist. Probab. Lett. 25 (1995) 323-327. | MR | Zbl

[11] K. Pötzelberger and K. Felsenstein, An asymptotic result on principal points for univariate distribution. Optimization 28 (1994) 397-406. | MR | Zbl

[12] S. Rowe, An Algorithm for Computing Principal Points with Respect to a Loss Function in the Unidimensional Case. Statist. Comput. 6 (1997) 187-190.

[13] T. Tarpey, Two principal points of symmetric, strongly unimodal distributions. Statist. Probab. Lett. 20 (1994) 253-257. | MR | Zbl

[14] T. Tarpey, Principal points and self-consistent points of symmetric multivariate distributions. J. Multivariate Anal. 53 (1995) 39-51. | MR | Zbl

[15] T. Tarpey, L. Li and B. Flury, Principal points and self-consistent points of elliptical distributions. Ann. Statist. 23 (1995) 103-112. | MR | Zbl

[16] A. Zoppè, Principal points of univariate continuous distributions. Statist. Comput. 5 (1995) 127-132.

[17] A. Zoppè, On Uniqueness and Symmetry of self-consistent points of univariate continuous distribution. J. Classification 14 (1997) 147-158. | MR | Zbl