Compact hypothesis and extremal set estimators

João Tiago Mexia ; Pedro Corte Real

Discussiones Mathematicae Probability and Statistics, Tome 23 (2003), p. 103-121 / Harvested from The Polish Digital Mathematics Library

Résumé

In extremal estimation theory the estimators are local or absolute extremes of functions defined on the cartesian product of the parameter by the sample space. Assuming that these functions converge uniformly, in a convenient stochastic way, to a limit function g, set estimators for the set ∇ of absolute maxima (minima) of g are obtained under the compactness assumption that ∇ is contained in a known compact U. A strongly consistent test is presented for this assumption. Moreover, when the true parameter value $\vec{β₀}^{k}$ is the sole point in ∇, strongly consistent pointwise estimators, ${ ^{k}: n ∈ ℕ }$ for $\vec{β₀}^{k}$ are derived and confidence ellipsoids for $\vec{β₀}^{k}$ centered at $^{k}$ are obtained, as well as, strongly consistent tests. Lastly an application to binary data is presented.

Détail

Publié le : 2003-01-01

Zbl 1049.62026

EUDML-ID : urn:eudml:doc:287599

Bibliographie

[000] [1] T. Amemiya, Advanced Econometrics, Harvard University Press 1985.

[001] [2] J. Burguete, R. Gallant and G. Souza, On Unification of the Asymptotic Theory of Nonlinear Econometric Models, Econometric Reviews, 1, (1982), 151-190. | Zbl 0496.62093

[002] [3] D.R. Cox and E.J. Snell, The Analysis of Binary Data, Chapman & Hall, London, 2nd Ed., 1989. | Zbl 0729.62004

[003] [4] D. J. Finney, Probit Analysis, Cambridge University Press 1971. | Zbl 0227.62065

[004] [5] C. Gourieroux, A. Monfort and A. Trognon, Pseudo Maximum Likelihood Methods: Theory, Econometrica 52 (3), (1984).

[005] [6] C. Gourieroux, A. Monfort and A. Trognon, Pseudo Maximum Likelihood Methods: Applications To Poisson Models, Econometrica 52 (3) (1984).

[006] [7] R. Gallant and A. Holly, Statistical Inference in an Implicit, Nonliner, Simultaneous Equation Model in the Context of Maximum Likelihood Estimation, Econometrica 48 (1980), 697-720. | Zbl 0456.62022

[007] [8] J. Hoffman-Jorgensen, The Theory Of Analytic Sets, Aarhus Universitet, Matematisk Institut, Various Publications Series, No.10, March 1970.

[008] [9] J. Hoffman-Jorgensen, Asymptotic Likelihood Theory, Aarhus Universitet, Matematisk Institut, Various Publications Series, No. 40, March 1992.

[009] [10] D.E. Jennings, Judging Inference Adequacy in Logist Regression, Journal of American Statistical Association 81 (1986), 471-476. | Zbl 0594.62072

[010] [11] R. Jennrich, Asymptotic Properties of Nonlinear Least Squares Estimators, The Annals of Mathematical Statistics 40 (1969), 633-643. | Zbl 0193.47201

[011] [12] J.D. Jobson and W.A. Fuller, Least Squares Estimation When the Covariance Matrix and Parameter Vector Are Functionally Related, Journal of the American Statistical Association 75 (1980), 176-181. | Zbl 0437.62064

[012] [13] J.T. Mexia and P. Corte Real, Strong Law of Large Numbers for Additive Extremum Estimators, Discussiones Mathematicae - Probability and Statistics 21 (2002), 81-88. | Zbl 1013.62075

[013] [14] J.T. Mexia, Asymptotic Chi-squared Tests, Designs, and Log-Linear Models, Trabalhos de Investigaçäo No. 1, 1992. | Zbl 0801.62045

[014] [15] J.T. Mexia, Controlled Heteroscedasticity, Quocient Vector Spaces and F Tets for Hypothesis On Mean Vectors, Trabalhos de Investigaçäo No. 1, 1991.

[015] [16] E. Malinvaud, The Consistency of Nonlinear Regressions, The Annals of Mathematical Statistics 41 (1970), 956-969. | Zbl 0206.20504

[016] [17] U. Mosco, Convergence Of Convex Sets Of Solutions Of Variational Inequalies, Advances in Mathematics 3 (1969), 510-585. | Zbl 0192.49101

[017] [18] J.T. Oliveira, Statistical Choice of Univariate Extreme Models, Statistical Distributions in Scientific Work, Taillie et al. (eds) - Reichel, Dordrecht, 6 (1980), 367-382.

[018] [19] H. Scheffé, The Analysis of Variance, John Willey & Sons, New York 1959. | Zbl 0086.34603

[019] [20] A. Wald, Note On The Consistency of The Maximum Likelihood Estiamte, Annals of Mathematics and Statistics 20 (1949), 595-601. | Zbl 0034.22902

[020] [21] R.A. Wijsman, Convergence of Sequences of Convex Sets II, Transactions of American Mathematical Society 123 (1966), 32-45. | Zbl 0146.18204

[021] [22] D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks 1991. | Zbl 0722.60001