I am currently trying to estimate the density of a joint distribution of K single dimensional RVs. I have at my disposal a set of N sample points, each of which represents an outcome of the K RVs.
Some specifics about my problem:
the RVs are independent
the RVs need not belong to same family of distributions
the RVs can either all be discrete or all be continuous, but not both
I know the upper and lower bounds for each RV; in the discrete case, I know the values that the RV can take on.
Right now, I am estimating the density of each RV separately using the ksdensity function in MATLAB. The independence assumption then allows me to produce a joint density using the product of the individual densities. I am hoping to improve the precision of my estimate this by either using another method (that I can code up in MATLAB) or by playing around with the options in ksdensity (such as the kernel type, support, width of the density window).
I am specifically hoping that people can shed light on:
What method to use for the discrete case vs. the continuous case. In the discrete case, is it worth specifying the bounds and the values? In the continuous case,
Whether it ever makes sense to forget the independence assumption and estimate the joint distribution as a joint distribution
Whether anyone knows about some simple reading material on the issue.