The Annals of Statistics

On the asymptotics of constrained local $M$-estimators

Alexander Shapiro

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Abstract

We discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and,more generally, $M$-estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set $S$.We give a counterexample showing that regularity of $S$ in the sense of Clarke is not sufficient for asymptotic equivalence of $\sqrt{n}$-consistent locally optimal $M$-estimators.We argue further that stronger properties, such as so-called near convexity or prox-regularity of $S$ are required in order to ensure that any two $\sqrt{n}$-consistent locally optimal $M$-estimators have the same asymptotics.

Article information

Source
Ann. Statist. Volume 28, Number 3 (2000), 948-960.

Dates
First available: 12 March 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1015952006

Mathematical Reviews number (MathSciNet)
MR1792795

Digital Object Identifier
doi:10.1214/aos/1015952006

Zentralblatt MATH identifier
01828970

Subjects
Primary: 62F12: Asymptotic properties of estimators

Keywords
Maximum likelihood contrained $M$-estimation asymptotic distribution tangent cones Clarke regularity prox-regularity metric projection

Citation

Shapiro, Alexander. On the asymptotics of constrained local $M$-estimators. The Annals of Statistics 28 (2000), no. 3, 948--960. doi:10.1214/aos/1015952006. http://projecteuclid.org/euclid.aos/1015952006.


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References

  • Aubin, J. P. and Frankowska, H. (1990). Set-Valued Analysis. Birkh¨auser, Boston.
  • Chernoff, H. (1954). On the distribution of the likelihood ratio. Ann. Math. Statist. 25 573-578.
  • Clarke, F. H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York.
  • Geyer, C. J. (1994). On the asymptotics of constrained M-estimation. Ann. Statist. 22 1993-2010.
  • Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 1 221-233. Univ. California Press, Berkeley.
  • Poliquin, R. A. and Rockafellar, R. T. (1996). Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348 1805-1838.
  • Poliquin, R. A., Rockafellar, R. T. and Thibault, L. (2000). Local differentiability of distance functions. Trans. Amer. Math. Soc. To appear. Robinson, S. M. (1976a). Stability theorems for systems of inequalities, II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13 497-513. Robinson, S. M. (1976b). Regularity and stability for convex multivalued functions. Math. Oper. Res. 1 130-143.
  • Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Springer, New York.
  • Rubinstein, R. Y. and Shapiro, A. (1993). Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method. Wiley, New York.
  • Self, S. G. and Liang, K. Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Amer. Statist. Assoc. 82 605-610.
  • Shapiro, A. (1994). Existence and differentiability of metric projections in Hilbert spaces. SIAM J. Optim. 4 130-141.
  • Shapiro, A. (1989). Asymptotic properties of statistical estimators in stochastic programming. Ann. Statist. 17 841-858.
  • Shapiro, A. and Al-Khayyal, F. (1993). First order conditions for isolated locally optimal solutions. J. Optim. Theory Appl. 77 189-196.
  • Ursescu, C. (1975). Multifunctions with convex closed graph. CzechoslovakMath. J. 25 438-441.