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A Continuous Representation of the Family of Stable Law Distributions
Author(s) -
Cheng R. C. H.,
Liu W. B.
Publication year - 1997
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/1467-9868.00059
Subject(s) - representation (politics) , interpolation (computer graphics) , mathematics , bounded function , parametric statistics , stability (learning theory) , limit (mathematics) , cartesian coordinate system , mathematical analysis , law , computer science , statistics , geometry , motion (physics) , artificial intelligence , machine learning , politics , political science
Conventional parametric representations of stable law distributions do not allow all members of the family to be obtained as continuous limits of the parameters. Model building (or simulation) using such representations will be numerically unstable near such limits in consequence. Existing tables are not satisfactory near such limits as interpolation cannot be carried out. We show that these difficulties are overcome by using a new shifted Cartesian representation which characterizes the entire stable law family in a completely continuous way. Standardization is still possible with this representation so that tabulation, using just two bounded parameters, can be carried out. Its use is illustrated in a non‐regular threshold estimation problem involving stable distributions which are discontinuous limits in conventional representations.