Implicit models of Gaussian mixture densities and locally optimum detectors

In the classical kernel approximation of a probability density function, a fixed zero-mean smooth approximation of the Dirac delta function is translated to each sample value of the random variable to form a family of density functions. The sample values are the means of the density functions in the corresponding family. The kernel approximation is the average over all members of this family. In certain applications, there may be too few sample values to give a good approximation. In this paper, we propose an approach that utilizes the sample values as the variances of a family of density functions. This approach works particularly well with a Gaussian mixture density.