On the Last Time and the Number of Times an Estimator is More than ε From its Target Value

• Published in: The Annals of statistics 1992-03, Vol.20 (1), p.469-489
• Main Authors: Hjort, Nils Lid, Fenstad, Grete
• Format: Article
• Language: English
• Subjects: Brownian motion; Consistent estimators; Covariance matrices; Density estimation; Distance functions; Estimators; Exact sciences and technology; Mathematics; Maximum likelihood estimation; Maximum likelihood estimators; Parametric inference; Perceptron convergence procedure; Probability and statistics; Sciences and techniques of general use; Statistical discrepancies; Statistics
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/aos/1176348533

Suppose θ̂nis a strongly consistent estimator for θ0in some i.i.d. situation. Let Nεand Qεbe, respectively, the last n and the total number of n for which θ̂nis at least ε away from θ0. The limit distributions for ε2N εand ε2Q εas ε goes to zero are obtained under natural and weak conditions. The theory covers both parametric and nonparametric cases, multidimensional parameters and general distance functions. Our results are of probabilistic interest, and, on the statistical side, suggest ways in which competing estimators can be compared. In particular several new optimality properties for the maximum likelihood estimator sequence in parametric families are established. Another use of our results is ways of constructing sequential fixed-volume or shrinking-volume confidence sets, as well as sequential tests with power 1. The paper also includes limit distribution results for the last n and the number of n for which the supremum distance |Fn- F| ≥ ε, where Fnis the empirical distribution function. Other results are reached for ε5/2Nεand ε5/2Qεin the context of nonparametric density estimation, referring to the last time and the number of times where |fn(x) - f(x)| ≥ ε. Finally, it is shown that our results extend to several non-i.i.d. situations.