Abstract
Consider a sequence of estimators \(\hat \theta _n \) 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 \(\hat \theta _n \) is further than ɛ away from θ 0 when ɛ → 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 \(\hat \theta _n \) 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) 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 ɛ 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.
Similar content being viewed by others
References
R. Adler and L. Brown, “Tail Behavior for Suprema of Empirical Processes”, Ann. Probab. 14, 1–30 (1986).
P. Andersen, Ø. Borgan, R. Gill, and N. Keiding, Statistical Models Based on Counting Processes, in Springer Series in Statistics (Springer, 1993).
R. Bahadur, “Rates of Convergence of Estimates and Test Statistics”, Ann. Math. Statist 38, 303–324 (1967).
U. Bandyopadhyay, R. Das, and A. Biswas, “Fixed Width Confidence Interval of P(X < Y) in Partial Sequential Sampling Scheme”, Sequential Anal. 22, 75–94 (2003).
P. Bickel, A. Klaassen, Y. Ritov, and J. Wellner, Efficient and Adaptive Inference in Semi-Parametric Models (Johns Hopkins University Press, Baltimore, 1993).
V. de la Peña and E. Giné, Decoupling: From Dependence to Independence (Springer, 1999).
R. Dudley, “Frechét Differentiability, p-Variation and Uniform Donsker Classes”, Ann. Probab. 20, 1968–1982 (1992).
R. Dudley, Uniform Central Limit Theorems (Cambridge Univ. Press, Cambridge, 1999).
N. L. Hjort and G. Fenstad, “On the Last Time and the Number of Times an Estimator is More than ɛ from Its Target Value”, Ann. Statist. 20, 469–489 (1992).
N. L. Hjort and G. Fenstad, “Second-Order Asymptotics for the Number of Times an Estimator is More than ɛ from Its Target Value”, J. Statist. Plann. and Inference 48, 261–275 (1995).
N. L. Hjort and R. Z. Khasminskii, “On the Time a Diffusion Process Spends along a Line”, Stochastic Process. Appl. 47, 229–247 (1993).
C. Kao, “On the Time and the Excess of Linear Boundary Crossings of Sample Sums”, Ann. Statist. 6, 191–199 (1978).
A. Koning and V. Protasov, “Tail Behavior of Gaussian Processes with Applications to the Brownian Pillow”, J. Multivar. Anal. 87, 370–397 (2003).
P. Massart, “About the Constants in Talagrand’s Concentration Inequalities for Empirical Processes”, Ann. Probab. 863–884 (2000).
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, in Comprehensive Studies in Mathematics (Springer, 1999), 3rd ed.
H. Robbins, D. Siegmund, and J. Wendel, “The Limiting Distribution of the Last Time S n ≥ nɛ”, Proc. Nat. Acad. Sci. USA 61, 1228–1230 (1968).
W. Römisch, “Delta Method, Infinite Dimensional”, in Encyclopedia of Statistical Sciences (Wiley, New York, 2005), 2nd ed.
P. Sen, Sequential Nonparametrics (Wiley, New York, 1981).
R. Serfling, Approximation Theorems of Mathematical Statistics (Wiley, New York, 1980).
J. Shao, Mathematical Statistics. in Springer Texts in Statistics (Springer, 2003).
A. Shapiro, “Asymptotics of Minimax Stochastic Programs”, Statist. and Probab. Letters 78, 150–157 (2008).
A. Shapiro and A. Ruszczynski, Lectures on Stochastic Programming, Preprint (Georgia Tech., 2008).
G. Shorack and J. Wellner, Empirical Processes with Applications to Statistics (Wiley, New York, 1986).
W. Stute, “Last Passage Times of M-Estimators”, Scand. J. of Statist. 10, 301–305 (1983).
M. Talagrand, “The Glivenko-Cantelli Problem”, Ann. Probab. 15, 837–870 (1987).
M. Talagrand, “New Concentration Inequalities in Product Spaces”, Inventiones Mathematicae 126, 505–563 (1996).
A.W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, in Springer Series in Statistics (Springer, 1996).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Grønneberg, S., Hjort, N.L. On the errors committed by sequences of estimator functionals. Math. Meth. Stat. 20, 327–346 (2011). https://doi.org/10.3103/S106653071104003X
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S106653071104003X
Keywords
- the last n
- Hadamard-differentiable statistical functionals
- sequential confidence regions
- Gaussian processes
- the Nelson-Aalen estimator