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Maximum spacing methods and limit theorems for statistics based on spacings
Umeå University, Faculty of Science and Technology, Mathematical statistics.
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 [en]
Estimation, spacings, maximum spacing method, consistency, (^-divergence, goodness of fit, unimodal density, entropy estimation, uniform distribution
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-85176ISBN: 91-7191-328-9 (print)OAI: oai:DiVA.org:umu-85176DiVA: diva2:691979
Projects
digitalisering@umu
Available from: 2014-01-30 Created: 2014-01-29 Last updated: 2014-01-30Bibliographically approved
List of papers
1. Strong consistency of the maximum spacing estimate
Open this publication in new window or tab >>Strong consistency of the maximum spacing estimate
1996 (English)In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 55, 55-72 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
American Mathematical Society (AMS), 1996
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:umu:diva-81167 (URN)
Available from: 2013-10-02 Created: 2013-10-02 Last updated: 2014-01-30Bibliographically approved
2. Maximum spacing estimates based on different metrics
Open this publication in new window or tab >>Maximum spacing estimates based on different metrics
1997 (English)Report (Other academic)
Abstract [en]

The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general method for estimating param­eters in univariate continuous distributions and is known to give consistent and asymptotically efficient estimates under general conditions. This 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. In the present paper, the ideas behind the MSP metod axe extended and a class of estimation methods is derived from approximations of certain information mea­sures, i.e. the ^-divergences introduced by Csiszâr (1963). We call these methods generalized maximum spacing (GMSP) methods, and it will be shown under gen­eral conditions that they give consistent estimates. GMSP methods have the advantage that they work also in situations where the ML method breaks down, e.g. due to an unbounded likelihood function. Other properties, such as asymp­totic normality and the behaviour of the estimates when the assigned model is only approximately true, will be discussed.[1]

[1] Research was supported by MISTRA, the Foundation for Strategic Environmental Research.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1997. 18 p.
Series
Research report / Department of Mathematical Statistics, Umeå University, ISSN 1401-730X ; 1997:5
Keyword
Estimation, Spacings, Maximum spacing method, Consistency, ^-divergence
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-85171 (URN)
Projects
digitalisering@umu
Available from: 2014-01-29 Created: 2014-01-29 Last updated: 2014-01-30Bibliographically approved
3. Generalized maximum spacing estimators
Open this publication in new window or tab >>Generalized maximum spacing estimators
1997 (English)Report (Other academic)
Abstract [en]

The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general method for estimating param­eters in univariate continuous distributions and is known to give consistent and asymptotically efficient estimates under general conditions. This method can be derived from an approximation based on simple spacings of the Kullback-Leibler information.

In the present paper, we introduce a class of estimation methods, derived from approximations based on mth order spacings of certain information measures, i.e. the ^-divergences introduced by Csiszâr (1963). The introduced class of methods includes the MSP method as a special case. A subclass of these methods was considered earlier in Ranneby and Ekström (1997), i.e. those based on first order spacings. Here it is found that such methods can be improved by using high order spacings. We also show that the suggested methods give consistent estimates under general conditions. [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 ; 1997:6
Keyword
Estimation, Spacings, Consistency, (^-divergence, Maximum spacing method
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-85172 (URN)
Projects
digitalisering@umu
Available from: 2014-01-29 Created: 2014-01-29 Last updated: 2014-01-30Bibliographically approved
4. Strong limit theorems for sums of logarithms of mth order spacings
Open this publication in new window or tab >>Strong limit theorems for sums of logarithms of mth order spacings
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
Spacings, Strong limit theorems, Entropy estimation, Uniform distribution, Goodness of fit
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-85173 (URN)
Projects
digitalisering@umu
Available from: 2014-01-29 Created: 2014-01-29 Last updated: 2014-01-30Bibliographically approved
5. Consistency of generalized maximum spacing estimates
Open this publication in new window or tab >>Consistency of generalized maximum spacing estimates
1997 (English)Report (Other academic)
Abstract [en]

General methods for the estimation of distributions can be derived from approx­imations of certain information measures. For example, both the maximum like­lihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kuliback-Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distri­butions is obtained from an approximation based on spacings of an information measure, were used by Ranneby and Ekström (1997) (using simple spacings) and Ekström (1997) (using high order spacings) to obtain a class of estimation methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao and Hahn (1996) for the MSP method. In particular, it will be shown that GMSP methods give Ll consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1997. 15 p.
Series
Research report / Department of Mathematical Statistics, Umeå University, ISSN 1401-730X ; 1997:8
Keyword
Estimation, Spacings, Consistency, Maximum spacing method, Unimodal density
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-85174 (URN)
Projects
digitalisering@umu
Available from: 2014-01-29 Created: 2014-01-29 Last updated: 2014-01-30Bibliographically approved

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