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Appendix A: Testing for goodness fit of three-parameter distributions

When fitting a probability density function (PDF) to a set of N data , two parameters often have to be estimated that typically are linked to its position and shape (e.g. in the case of the Gaussian PDF these parameters are the mean and the standard deviation). There are situations, however, where the PDF, φ(x) is defined in a domain x_min − ∞, with x_min an unknown quantity; i.e., the parameters to estimate are three. In this case, the method of ML often fails to provide useful parameter estimates. For example, this happens with PDFs, such as the Schechter one, for which φ(x_min) = ∞. A possible way out is an approach based on the use of the empirical cumulative distribution function (ECDF) given by (A.1)with { x_i } sorted in increasing order. The idea is that, if an ECDF is plotted versus the values of the CDF Φ(x_i) corresponding to the true PDF φ(x_i), then the resulting points distribute along a straight line with unit slope. An estimate of the parameters can therefore be obtained from the PDF whose CDF provides the ECDF-CDF point distribution closest, in the least-squares sense, to such line. In practice, the three parameters are iteratively changed, the corresponding Φ(x_i) evaluated and finally the sum of the squared distances of the ECDF-CDF point distribution from the straight line computed. In this operation, it is useful to weight the data to give more importance to the tails of the PDF that are more able to distinguish the various types. The weights { w_i } can be defined in terms of (A.2)\newpage \noindentFigure A.1 shows the results for the data in the K-band for the sample of late-type galaxies when this method is applied with two PDFs given by a log-normal and a Schechter distribution. The superiority of the fit provided by the former is evident. In particular, the root-mean square of the distances for the log-normal PDF is 1.1 × 10^-3 vs. 2.1 × 10^-3 for the Schechter one.

	Fig. A.1 Goodness test for the three parameter log-normal and Schechter probability density function for the late type galaxies in the K-band.
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© ESO, 2014