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

Abstract

Suppose $\hat{\theta}_n$ is a strongly consistent estimator for $\theta_0$ in some i.i.d. situation. Let $N_\varepsilon$ and $Q_\varepsilon$ be, respectively, the last $n$ and the total number of $n$ for which $\hat{\theta}_n$ is at least $\varepsilon$ away from $\theta_0$. The limit distributions for $\varepsilon^2N_\varepsilon$ and $\varepsilon^2Q_\varepsilon$ as $\varepsilon$ 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 $\|F_n - F\| \geq \varepsilon$, where $F_n$ is the empirical distribution function. Other results are reached for $\varepsilon^{5/2}N_\varepsilon$ and $\varepsilon^{5/2}Q_\varepsilon$ in the context of nonparametric density estimation, referring to the last time and the number of times where $|f_n(x) - f(x)| \geq \varepsilon$. Finally, it is shown that our results extend to several non-i.i.d. situations.