## STRONG LAW OF LARGE NUMBERS FOR ADDITIVE EXTREMUM ESTIMATORS

João Tiago Mexia and Pedro Corte Real

Universidade Nova de Lisboa, Departamento de Matemática
da Faculdade de Ciências e Tecnologia
Quinta da Torre, 2825-114 Monte de Caparica
e-mail:
parcr@mail.fct.unl.pt

## Abstract

Extremum estimators are obtained by maximizing or minimizing a function of the sample and of the parameters relatively to the parameters. When the function to maximize or minimize is the sum of subfunctions each depending on one observation, the extremum estimators are additive. Maximum likelihood estimators are extremum additive whenever the observations are independent. Another instance of additive extremum estimators are the least squares estimators for multiple regressions when the usual assumptions hold. A strong law of large numbers is derived for additive extremum estimators. This law requires only the existence of first order moments and may be of interest in connection with maximum likelihood estimators, since the usual assumption that the observations are identically distributed is discarded.

Keywords: Kolmogorov's strong law of large numbers, multiple regression, almost sure convergence, additive extremum estimators.

2000 Mathematics Subject Classification: 60F15, 62F10.

## References

 [1] N. Bac Van, Strong convergence of least squares estimates in polynomial regression with random explanatory variables, Acta Mathematica Vietnamica 23 (2) (1998), 195-205. [2] N. Bac Van, Strong convergence of least squares estimates in polynomial regression with random explanatory variables, Acta Mathematica Vietnamica 19 (1) (1994), 111-137. [3] J. Galambos, Advanced Probability Theory, Marcel Dekker 1988. [4] J.T. Mexia and P.C. Real, Extension of Kolmogorov's strong law to multiple regression, 23rd European Meeting of Statisticians, Funchal (Madeira Island), August 13-18 2001, Revista de Estatística, 2 Quadrimestre de 2001, page 277-278. [5] D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks 1991. [6] S. Zacks, The Theory of Statistical Inference, John Wiley 1971.

Received 10 June 2001
Revised 26 August 2001