## Appendix A: The Gaussian kernel methodConsider a sample of size
If we estimate the probability density function through the derivative of the empirical repartition function, we obtain a sum of Dirac functions. However, if we convolute these Dirac functions by a kernel which has `good' properties, the function obtained is then a continuous function and an estimator of the probability density function (Devroye 1986). The estimator of the density function of the sample is denoted , and is given, after convolution, by: where is a parameter which is
fixed with the choice of the kernel applied. This parameter is used to
smooth the shape of the curve, so that
has to tend to zero when The `good' properties of are: which ensure the convergence of in probability to the true density function of the random variable X. Therefore: A useful kernel is the Gaussian kernel which, moreover, has the property to be : where , and is the standard deviation. In our case, , and . © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |