The reported density estimates are kernel estimates using the Epanechnikov kernel function. For the bandwidth choice we use the cross-validation likelihood method, that is, the bandwidth is data dependent and is chosen in an objective manner. Specifically, the optimal bandwidth is given by
where 
is the kernel estimate at the point
, excluding the ith observation itself i.e.,
where 
denotes the sample standard deviation of x.
In the results reported below, the kernel function 
is
the
Epanechnikov kernel given by
