Strong Law of Large Numbers for Additive Extremum Estimators

Extremum estimators are obtained by maximizing or minimizing a function of the sampleand of the parameters relatively to the parameters. When the function to maximize or min-imize is the sum of subfuctions each depending on one observation the extremum estimatorsare additive. Maximum likelihood estimators are extremum additive whenever the observa-tions are independent. Another instance of additive extremum estimators are the least squaresestimators for multiple regressions when the usual assumptions hold. A strong law of largenumbers is derived for additive extremum estimators. This law requires only the existence offirst 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.