Nonparametric function estimation from inversely sampled record‐breaking data

Abstract

Abstract Often, in industrial stress testing, meteorological data analysis, and other similar situations, measurements may be made sequentially and only values smaller than all previous ones are recorded. When the number of records is fixed in advance, the data are referred to as inversely sampled record‐breaking data. This paper is concerned with nonparametric estimation of the distribution and density functions from such data (successive minima). For a single record‐breaking sample, consistent estimation is not possible except in the extreme left tail of the distribution. Hence, replication is required, and for m such independent record‐breaking samples, the estimators are shown to be strongly consistent and asymptotically normal as m ∞ →. Computer simulations are used to investigate the effect of the bandwidth on the mean squared errors and biases of the smooth estimators, and are also used to provide a comparison of their performance with the analogous estimators obtained under random sampling for record values.