×

Minimum distance estimation in normed linear spaces with Donsker-classes. (English) Zbl 1282.60029

Summary: We consider minimum distance estimators where the discrepancy function is defined in terms of a supremum-norm based on a Donsker-class of functions. If the parameter set is contained in a normed linear space we prove a Portmanteau-type theorem. Here, the limit in general is not a probability measure, but an outer measure given by the hitting family of the set of all minimizing points of a certain stochastic process. In case there is exactly one minimizer one obtains traditional weak convergence.

MSC:

60F05 Central limit and other weak theorems
62E15 Exact distribution theory in statistics
60F17 Functional limit theorems; invariance principles
62F12 Asymptotic properties of parametric estimators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Billingsley, Convergence of ProbabilityMeasures (Wiley, New York, 1968). · Zbl 0172.21201
[2] J. Blackman, ”On the Approximation of a Distribution Function by an Empirical Distribution”, Ann. Math. Statist. 26, 256–267 (1955). · Zbl 0065.11403 · doi:10.1214/aoms/1177728542
[3] E. Bolthausen, ”Convergence in Distribution of Minimum-Distance Estimators”, Metrika 24, 215–227 (1977). · Zbl 0396.62022 · doi:10.1007/BF01893411
[4] M. Donsker, ”Justification and Extension of Doob’s Heuristic Approach to the Kolmogorov-Smirnov Theorems”, Ann. Math. Statist. 23, 277–281 (1952). · Zbl 0046.35103 · doi:10.1214/aoms/1177729445
[5] D. L. Donoho and R. C. Liu, ”The ’Automatic’ Robustness of Minimum Distance Functionals”, Ann. Statist. 16, 552–586 (1988). · Zbl 0684.62030 · doi:10.1214/aos/1176350820
[6] D. L. Donoho and R. C. Liu, ”Pathologies of Some Minimum Distance Estimators”, Ann. Statist. 16, 587–608 (1988). · Zbl 0684.62029 · doi:10.1214/aos/1176350821
[7] R.M. Dudley, Uniform Central Limit Theorems (Cambridge Univ. Press, Cambridge, 1999). · Zbl 0951.60033
[8] L. Dümbgen, ”The Asymptotic Behavior of Some Nonparametric Change-Point Estimators”, Ann. Statist. 19, 1471–1495 (1991). · Zbl 0776.62032 · doi:10.1214/aos/1176348257
[9] T. P. Hettmansperger, I. Hueter, and J. Hüsler, ”Minimum Distance Estimators”, J. Statist. Plann. Inference 41, 291–302 (1994). · Zbl 0803.62016 · doi:10.1016/0378-3758(94)90025-6
[10] P. Huber, Robust Statistics (Wiley, New York, 1981). · Zbl 0536.62025
[11] M. Kac, J. Kiefer, and J. Wolfowitz, ”On Tests of Normality and Other Tests of Goodness of Fit Based on Distance Methods”, Ann. Math. Statist. 26, 189–211 (1955). · Zbl 0066.12301 · doi:10.1214/aoms/1177728538
[12] H. L. Koul, Weighted Empirical Processes in Dynamic Nonlinear Models, in Lecture Notes in Statistics, 2nd ed. (Springer, New York, 2002), Vol. 166. · Zbl 1007.62047
[13] A. S. Kozek, ”On Minimum Distance Estimation Using Kologorov-Lévy Type Metrics”, Austral. & New Zealand J. Statist. 40, 317–333 (1998). · Zbl 0949.62024 · doi:10.1111/1467-842X.00036
[14] F. Liese and K.-J. Miescke, Statistical Decision Theory (Springer, New York, 2008). · Zbl 1154.62008
[15] F. Liese and I. Vajda, ”A General Asymptotic Theory of M-Estimators. I”, Math. Methods. Statist.. 12, 454–477 (2003).
[16] R. C. Littel and P. V. Rao, ”On Minimum Distance Estimation Based on the Kolmogorov-Statistic”, Commun. Statist. A-Theory Methods 11, 1793–1807 (1982). · Zbl 0507.62039 · doi:10.1080/03610928208828350
[17] P. W. Millar, ”Robust estimation via minimum distance methods”, Z. Wahrsch. verw. Gebiete 55, 72–89 (1981). · Zbl 0461.62036 · doi:10.1007/BF01013462
[18] W. C. Parr, ”Minimum Distance Estimation: a Bibliography”, Commun. Statist. A-Theory Methods 10, 1205–1224 (1981). · Zbl 0458.62035 · doi:10.1080/03610928108828104
[19] W. C. Parr and W. R. Schucany, ”Minimum Distance and Robust Estimation”, J. Amer. Statist. Assoc. 75, 615–624 (1980). · Zbl 0481.62031 · doi:10.1080/01621459.1980.10477522
[20] J. Pfanzagl, ”Consistent Estimation in the Presence of Incidental Parameters”, Metrika 15, 141–148 (1970). · Zbl 0206.20004 · doi:10.1007/BF02613567
[21] D. Pollard, ”The Minimum Distance Method of Testing”, Metrika 27, 43–70 (1980). · Zbl 0425.62029 · doi:10.1007/BF01893576
[22] W. Sahler, ”Estimation by Minimum Discrepancy Methods”, Metrika 16, 85–106 (1970). · Zbl 0221.62010 · doi:10.1007/BF02613939
[23] G. Salinetti and R. J.-R. Wets, ”On the Convergence in Distribution of Measurable Multifunctions (Random Sets), Normal Integrands, Stochastic Processes and Stochastic Infima”, Math. Oper. Res. 11, 385–419 (1986). · Zbl 0611.60004 · doi:10.1287/moor.11.3.385
[24] G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics (Wiley, New York, 1986). · Zbl 1170.62365
[25] W. Stute, ”Parameter Estimation in Smooth Empirical Processes”, Stochastic Process. Appl. 22, 223–244 (1986). · Zbl 0612.62040 · doi:10.1016/0304-4149(86)90003-7
[26] A.W. van der Vaart, Asymptotic Statistics (Cambridge Univ. Press, Cambridge, 1998). · Zbl 0910.62001
[27] A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes (Springer, New York, 1986). · Zbl 0862.60002
[28] A. Wald, ”Note on the Consistency of the Maximum Likelihood Estimate”, Ann. Math. Statist. 20, 595–601 (1949). · Zbl 0034.22902 · doi:10.1214/aoms/1177729952
[29] H. Witting and U. Müller-Funk, Mathematische Statistik II (Teubner, Stuttgart, 1995). · Zbl 0838.62001
[30] J. Wolfowitz, ”Estimation by the Minimum Distance Method”, Ann. Inst. Statist. Math. 5, 9–23 (1953). · Zbl 0051.37004 · doi:10.1007/BF02949797
[31] J. Wolfowitz, ”The Minimum Distance Method”, Ann. Math. Statist. 28, 75–88 (1957). · Zbl 0086.35403 · doi:10.1214/aoms/1177707038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.