Nonparametric estimation for big-but-biased data.

Nonparametric estimation for a large-sized sample subject to sampling bias is studied in this paper. The general parameter considered is the mean of a transformation of the random variable of interest. When ignoring the biasing weight function, a small-sized simple random sample of the real population is assumed to be additionally observed. A new nonparametric estimator that incorporates kernel density estimation is proposed. Asymptotic properties for this estimator are obtained under suitable limit conditions on the small and the large sample sizes and standard and non-standard asymptotic conditions on the two bandwidths. Explicit formulas are shown for the particular case of mean estimation. Simulation results show that the new mean estimator outperforms two classical ones for suitable choices of the two smoothing parameters involved. The influence of two smoothing parameters on the performance of the final estimator is also studied, exhibiting a striking limit behavior of their optimal values. The new method is applied to a real data set from the Telco Company Vodafone ES, where a bootstrap algorithm is used to select the smoothing parameter. [ABSTRACT FROM AUTHOR]