Statistica Sinica 27 (2017), 1661-1673

ESTIMATION OF QUANTILES FROM DATA WITH

ADDITIONAL MEASUREMENT ERRORS

Matthias Hansmann and Michael Kohler

Technische Universität Darmstadt

Abstract: In this paper we study the problem of estimating quantiles from data that contain additional measurement errors. The only assumption on these errors is that the average absolute measurement error converges to zero for sample size tending to infinity with probability one. In particular we do not assume that the measurement errors are independent with expectation zero. We show that the empirical measure based on the data with measurement errors leads to an estimator which approaches the quantile set asymptotically. Provided the quantile is uniquely determined, this implies that this quantile estimate is strongly consistent for the true quantile. If this assumption does not hold, we also show that we can construct estimators for the limits of the quantile set if the average absolute measurement error is bounded by a given sequence, that tends to zero for sample size tending to infinity with probability one. But if such a sequence, which upper bounds the measurement errors, is not given, we show that there exists no estimator that is consistent for every distribution of the underlying random variable and all data containing the measurement errors. We derive the rate of convergence of our estimator and show that the derived rate of convergence is optimal. The results are applied in simulations and in the context of experimental fatigue tests.

Key words and phrases: Consistency, experimental fatigue tests, quantile estimation, rate of convergence.