Discussiones
Mathematicae Probability and Statistics 21(2) (2001) 81-88

## 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