Smoothed Nonparametic Density Estimation for Censored or Truncated Samples

Abstract

Nonparametric distribution estimators such as the Kaplan-Meier [5] estimator for censored data and the Lynden-Bell [66] estimator for truncated data have been applied to astronomical data with good results. Such es-timators, however, have limitations including failure to gracefully account for uncertainties in the detected or censored data [3]and the inability to directly estimate the differential distribution or frequency function. Both of these limitations are related to the treatment of the empirical source distribution, ρ(x),as discrete,ie,asa sum of delta functions:

$$p\left( x \right) = \sum\limits_{i = 1}^N {\delta \left( {x - {x_i}} \right)}$$

(38.1)

where {x_i} are the reported observations. While such treatment is ap-propriate for, e.g., survival analysis, where loss and failure times are well known, it may be problematic in astronomical applications where luminosi-ties, source distances, and flux limits have large uncertainies. Data smooth-ing has been applied in similar situations to uncensored and untruncated data [4,7]. This poster and the accompanying reprints describe the appli-cation of data smoothing to truncated samples using the QSO luminosity function as a specific example.