Maximum spacing methods and limit theorems for statistics based on spacings
1997 (English)Doctoral thesis, comprehensive summary (Other academic)
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.
Estimation, spacings, maximum spacing method, consistency, (^-divergence, goodness of fit, unimodal density, entropy estimation, uniform distribution
Probability Theory and Statistics
IdentifiersURN: urn:nbn:se:umu:diva-85176ISBN: 91-7191-328-9OAI: oai:DiVA.org:umu-85176DiVA: diva2:691979
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