Log-stable concentration distributions of trace elements in biomedical samples

In the present paper, which follows our earlier observation that the asymmetric and long-tailed concentration distributions of trace elements in biomedical samples, measured by the X-ray fluorescence techniques, can be modeled by the log-stable distributions, further specific aspects of this observation are discussed. First, we demonstrate that, typically, for a quite substantial fraction (10-20%) of trace elements studied in different kinds of biomedical samples, the measured concentration distributions are described in fact by the 'symmetric' log-stable distributions, i.e. the asymmetric distributions which are described by the symmetric stable distributions. This observation is, in fact, expected for the random multiplicative process, which models the concentration distributions of trace elements in the biomedical samples. The log-stable nature of concentration distribution of trace elements results in several problems of statistical nature, which have to be addressed in XRF data analysis practice. Consequently, in the present paper, the following problems, namely (i) the estimation of parameters for stable distributions and (ii) the testing of the log-stable nature of the concentration distribution by using the Anderson-Darling (A2) test, especially for symmetric stable distributions, are discussed in detail. In particular, the maximum likelihood estimation and Monte Carlo simulation techniques were used, respectively, for estimation of stable distribution parameters and calculation of the critical values for the Anderson-Darling test. The discussed ideas are exemplified by the results of the study of trace element concentration distributions in selected biomedical samples, which were obtained by using the X-ray fluorescence (XRF, TXRF) methods