Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T08:04:18.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Last passage times of minimum contrast estimators

Part of: Stochastic analysis Distribution theory Parametric inference Limit theorems

Published online by Cambridge University Press:  09 April 2009

Arup Bose  and
Snigdhansu Chatterjee
Arup Bose
Affiliation:
Theoretical Statistics and Mathematics Unit Indian Statistical Institute203 B. T. Road Calcutta 700 035India e-mail: [email protected]
Snigdhansu Chatterjee
Affiliation:
Department of Mathematics The University of ManchesterOxford Road Manchester M139PLUK e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the last passage time and its asymptotic distribution for minimum contrast estimators defined through the minimization of a convex criterion function based on U-functionals. This includes cases of non-smooth estimators for vector valued parameters. We also derive a Bahadur-type representation and the law of iterated logarithms for such estimators.

Keywords

minimum contrast estimatorslast passage timelaw of iterated logarithmsBahadur representationOja medianL1median

MSC classification

Secondary: 62F12: Asymptotic properties of estimators 60F99: None of the above, but in this section 60H99: None of the above, but in this section 62E20: Asymptotic distribution theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 1 , August 2001 , pp. 1 - 10
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bickel, P. J. and Lehmann, E. L., ‘Descriptive statistics for nonparametric models. IV. Spread’, in: Contributions to statistics (ed. Jurečková, J.) (Academia, Prague, 1979) pp. 33–40.CrossRefGoogle Scholar
[2]Bickel, P. J. and Yahav, J. A., ‘Asymptotically optimal Bayes and minimax procedures in sequential estimation’, Ann. Math. Statist. 39 (1968), 442–456.CrossRefGoogle Scholar
[3]Bose, A., ‘Bahadur representation and other asymptotic properties of M estimates based on U functionals’, Tech. Rep. no. 19/94, (Stat. Math. Unit. ISI, Calcutta, 1994).Google Scholar
[4]Bose, A., ‘Bahadur representation of Mm estimates’, Ann. Statist. 26 (1998), 771–777.CrossRefGoogle Scholar
[5]Chaudhuri, P., ‘On a geometric notion of quantiles for multivariate data’, J. Amer. Statist. Assoc. 91 (1996), 862–872.CrossRefGoogle Scholar
[6]Choudhury, J. and Serfling, R. J., ‘Generalized order statistics, Bahadur representations, and sequential nonparametric fixed-width confidence intervals’, J. Statist. Plann. Inference 19 (1988), 269–282.CrossRefGoogle Scholar
[7]Habermann, S. J., ‘Concavity and estimation’, Ann. Statist. 17 (1989), 1631–1661.CrossRefGoogle Scholar
[8]Hollander, M. and Wolfe, D. A., Nonparametrical statistical methods (John Wiley and Sons, New York, 1973).Google Scholar
[9]Huber, P. J., ‘Robust estimation of a location parameter’, Ann. Math. Statist. 35 (1964), 73–101.CrossRefGoogle Scholar
[10]Lee, A. J., U-Statistics (Marcel Dekker, New York, 1990).Google Scholar
[11]Maritz, J. S., Wu, M. and Staudte, R. G., ‘A location estimator based on U statistic’, Ann. Statist. 5 (1977), 779–786.CrossRefGoogle Scholar
[12]Niemiro, W., ‘Asymptotics for M-estimators defined by convex minimization’, Ann. Statist. 20 (1992), 1514–1533.CrossRefGoogle Scholar
[13]Oja, H., ‘Descriptive statistics for multivariate distribution’, Statist. Probab. Letters 1 (1983), 327–333.CrossRefGoogle Scholar
[14]Robbins, H. and Siegmund, D., ‘Boundary crossing probabilities for the Wiener process and sample sums’, Ann. Math. Statist. 41 (1970), 1410–1429.CrossRefGoogle Scholar
[15]Stute, W., ‘Last passage times of M-estimators’, Scandinavian J. Statist. 10 (1983), 301–305.Google Scholar
[16]Stute, W., ‘On a class of stopping times for M-estimators’, J. Multivariate Anal. 14 (1984), 83–93.CrossRefGoogle Scholar