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Astron. Astrophys. 363, 1155-1165 (2000)

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Appendix A: The Gaussian kernel method

Consider a sample [FORMULA] of size n: ([FORMULA]) of identically distributed, independent random variables. We intend to provide an estimate of the probability density function of the generic variable X, namely [FORMULA], defined by the property:

[EQUATION]

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 [FORMULA] 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 [FORMULA] is denoted [FORMULA], and is given, after convolution, by:

[EQUATION]

where [FORMULA] 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 [FORMULA] has to tend to zero when n goes to infinity, but not too fast:

[EQUATION]

The `good' properties of [FORMULA] are:

[EQUATION]

which ensure the convergence of [FORMULA] in probability to the true density function [FORMULA] of the random variable X. Therefore:

[EQUATION]

A useful kernel is the Gaussian kernel which, moreover, has the property to be [FORMULA]:

[EQUATION]

where [FORMULA], and [FORMULA] is the standard deviation.

In our case, [FORMULA], and [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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