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On the errors committed by sequences of estimator functionals

Steffen Grønneberg, Nils Lid Hjort

Abstract

Consider a sequence of estimators $\hat θ_n$ which converges almost surely to $θ_0$ as the sample size $n$ tends to infinity. Under weak smoothness conditions, we identify the asymptotic limit of the last time $\hat θ_n$ is further than $\eps$ away from $θ_0$ when $\eps \rightarrow 0^+$. These limits lead to the construction of sequentially fixed width confidence regions for which we find analytic approximations. The smoothness conditions we impose is that $\hat θ_n$ is to be close to a Hadamard-differentiable functional of the empirical distribution, an assumption valid for a large class of widely used statistical estimators. Similar results were derived in Hjort and Fenstad (1992, Annals of Statistics) for the case of Euclidean parameter spaces; part of the present contribution is to lift these results to situations involving parameter functionals. The apparatus we develop is also used to derive appropriate limit distributions of other quantities related to the far tail of an almost surely convergent sequence of estimators, like the number of times the estimator is more than $\eps$ away from its target. We illustrate our results by giving a new sequential simultaneous confidence set for the cumulative hazard function based on the Nelson--Aalen estimator and investigate a problem in stochastic programming related to computational complexity.

On the errors committed by sequences of estimator functionals

Abstract

Consider a sequence of estimators which converges almost surely to as the sample size tends to infinity. Under weak smoothness conditions, we identify the asymptotic limit of the last time is further than away from when . These limits lead to the construction of sequentially fixed width confidence regions for which we find analytic approximations. The smoothness conditions we impose is that is to be close to a Hadamard-differentiable functional of the empirical distribution, an assumption valid for a large class of widely used statistical estimators. Similar results were derived in Hjort and Fenstad (1992, Annals of Statistics) for the case of Euclidean parameter spaces; part of the present contribution is to lift these results to situations involving parameter functionals. The apparatus we develop is also used to derive appropriate limit distributions of other quantities related to the far tail of an almost surely convergent sequence of estimators, like the number of times the estimator is more than away from its target. We illustrate our results by giving a new sequential simultaneous confidence set for the cumulative hazard function based on the Nelson--Aalen estimator and investigate a problem in stochastic programming related to computational complexity.
Paper Structure (10 sections, 7 theorems, 125 equations, 1 figure)

This paper contains 10 sections, 7 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Limit Theorems
  3. Sequential confidence sets
  4. A reduction to the Kolmogorov--Smirnov limit
  5. Gaussian Local Martingales
  6. An application to risk averse stochastic problems
  7. Further applications
  8. The multivariate case
  9. The number of $\varepsilon$-misses and two new variables
  10. Measures of asymptotic relative efficiency

Key Result

Lemma 1

Under assumptions 1-3, we have that on $l^\infty([1,\infty) \times \mathcal{F})$ where $\mathbb{Z}$ is a Kiefer-Müller process on $[1,\infty) \times \mathcal{F}$ and $\dot \phi (s^ {-1} \mathbb{Z}_{s})$ is short-hand for $\dot \phi$ evaluated at the $l^\infty(\mathcal{F})$-map $f \mapsto s^ {-1} \mathbb{Z}_{s}(f)$. The limit $\dot \

Figures (1)

  • Figure 1: Median value and lower and upper 0.05 quantiles of the variable $R(1,b)$ (the limit of $R_\varepsilon(1,b)$) for a range of $b$ values for the simple average.

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 2
  • ...and 9 more