Comparison of Parzen density and frequency histogram as estimators of probability density functions

In neurobiology, and in other fields, the frequency histogram is a traditional tool for determining the probability density function (pdf) of random processes, although other methods have been shown to be more efficient as their estimators. In this study, the frequency histogram is compared with the Parzen density estimator, a method that consists of convolving each measurement with a weighting function of choice (Gaussian, rectangular, etc) and using their sum as an estimate of the pdf of the random process. The difference in their performance in evaluating two types of pdfs that occur commonly in quantal analysis (monomodal and multimodal with equidistant peaks) is demonstrated numerically by using the integrated square error criterion and assuming a knowledge of the ''true'' pdf. The error of the Parzen density estimates decreases faster as a function of the number of observations than that of the frequency histogram, indicating that they are asymptotically more efficient. A variety of ''reasonable'' weighting functions can provide similarly efficient Parzen density estimates, but their efficiency greatly depends on their width, The optimal widths determined using the integrated square error criterion, the harmonic analysis (applicable only to multimodal pdfs with equidistant peaks), and the ''test graphs'' (the graphs of the second derivatives of the Parzen density estimates that do not assume a knowledge of the ''true'' pdf, but depend on the distinction between the ''essential features'' of the pdf and the ''random fluctuations'') were compared and found to be similar.