The Implementation of Binned Kernel Density Estimation to Determine Open
  Clusters' Proper Motions: Validation of the Method
  Stellar membership determination of an open cluster is an important process
to do before further analysis. Basically, there are two classes of membership
determination method: parametric and non-parametric. In this study, an
alternative of non-parametric method based on Binned Kernel Density Estimation
that accounts measurements errors (simply called BKDE-e) is proposed. This
method is applied upon proper motions data to determine cluster's membership
kinematically and estimate the average proper motions of the cluster. Monte
Carlo simulations show that the average proper motions determination using this
proposed method is statistically more accurate than ordinary Kernel Density
Estimator (KDE). By including measurement errors in the calculation, the mode
location from the resulting density estimate is less sensitive to non-physical
or stochastic fluctuation as compared to ordinary KDE that excludes measurement
errors. For the typical mean measurement error of 7 mas/yr, BKDE-e suppresses
the potential of miscalculation by a factor of two compared to KDE. With median
accuracy of about 93%, BKDE-e method has comparable accuracy with respect to
parametric method (modified Sanders algorithm). Application to real data from
The Fourth USNO CCD Astrograph Catalog (UCAC4), especially to NGC 2682 is also
performed. The mode of member stars distribution on Vector Point Diagram is
located at $\mu_{\alpha}\cos\delta=-9.94\pm0.85$ mas/yr and
$\mu_{\delta}=-4.92\pm0.88$ mas/yr. Although the BKDE-e performance does not
overtake parametric approach, it serves a new view of doing membership
analysis, expandable to astrometric and photometric data or even in binary
cluster search.
