  Exoplanet searches using radial velocity (RV) and microlensing (ML) produce
samples of "projected" mass and orbital radius, respectively. We present a new
method for estimating the probability density distribution (density) of the
unprojected quantity from such samples. For a sample of n data values, the
method involves solving n simultaneous linear equations to determine the
weights of delta functions for the raw, unsmoothed density of the unprojected
quantity that cause the associated cumulative distribution function (CDF) of
the projected quantity to exactly reproduce the empirical CDF of the sample at
the locations of the n data values. We smooth the raw density using
nonparametric kernel density estimation with a normal kernel of bandwidth
\sigma. We calibrate the dependence of \sigma on n by Monte Carlo experiments
performed on samples drawn from a theoretical density, in which the integrated
square error is minimized. We scale this calibration to the ranges of real RV
samples using the Normal Reference Rule. The resolution and amplitude accuracy
of the estimated density improve with n. For typical RV and ML samples, we
expect the fractional noise at the PDF peak to be approximately 80 n^{-\log 2}.
For illustrations, we apply the new method to 67 RV values given a similar
treatment by Jorissen et al. in 2001, and to the 308 RV values listed at
exoplanets.org on 20 October 2010. In addition to analyzing observational
results, our methods can be used to develop measurement
requirements--particularly on the minimum sample size n--for future programs,
such as the microlensing survey of Earth-like exoplanets recommended by the
Astro 2010 committee.
