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Strong limit theorems for sums of logarithms of mth order spacings
Umeå University, Faculty of Science and Technology, Mathematical statistics.
1997 (English)Report (Other academic)
Abstract [en]

Several strong limit theorems axe proved for sums of logarithms of mth order spacings from general distributions. In all given results, the order of the spac­ings is allowed to increase to infinity with the sample size. These results provide a nonparametric strongly consistent estimator of entropy as well as a charac­terization of the uniform distribution on [0,1]. Furthermore, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives. [1]

[1] Research was supported by The Bank of Sweden Tercentenary Foundation.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 1997. , 14 p.
Series
Research report / Department of Mathematical Statistics, Umeå University, ISSN 1401-730X ; 1997:7
Keyword [en]
Spacings, Strong limit theorems, Entropy estimation, Uniform distribution, Goodness of fit
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-85173OAI: oai:DiVA.org:umu-85173DiVA: diva2:691966
Projects
digitalisering@umu
Available from: 2014-01-29 Created: 2014-01-29 Last updated: 2014-01-30Bibliographically approved
In thesis
1. Maximum spacing methods and limit theorems for statistics based on spacings
Open this publication in new window or tab >>Maximum spacing methods and limit theorems for statistics based on spacings
1997 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general estimation method for continuous univariate distributions. The MSP method, which is closely related to the maximum likelihood (ML) method, can be derived from an approximation based on simple spacings of the Kullback-Leibler information. It is known to give consistent and asymptotically efficient estimates under general conditions and works also in situations where the ML method fails, e.g. for the three parameter Weibull model.

In this thesis it is proved under general conditions that MSP estimates of parameters in the Euclidian metric are strongly consistent. The ideas behind the MSP method are extended and a class of estimation methods is introduced. These methods, called generalized MSP methods, are derived from approxima­tions based on sum-functions of rath order spacings of certain information mea­sures, i.e. the ^-divergences introduced by Csiszår (1963). It is shown under general conditions that generalized MSP methods give consistent estimates. In particular, it is proved that generalized MSP methods give L1 consistent esti­mates in any family of distributions with unimodal densities, without any further conditions on the distributions. Other properties such as distributional robust­ness are also discussed. Several limit theorems for sum-functions of rath order spacings are given, for ra fixed as well as for the case when ra is allowed to in­crease to infinity with the sample size. These results provide a strongly consistent nonparametric estimator of entropy, as well as a characterization of the uniform distribution. Further, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1997. 30 p.
Keyword
Estimation, spacings, maximum spacing method, consistency, (^-divergence, goodness of fit, unimodal density, entropy estimation, uniform distribution
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-85176 (URN)91-7191-328-9 (ISBN)
Projects
digitalisering@umu
Available from: 2014-01-30 Created: 2014-01-29 Last updated: 2014-01-30Bibliographically approved

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