Sister chromatid exchange data and Gram-Charlier series

Bowman et al. [K.O. Bowman, M.A. Kastenbaum, L.R. Shenton, Fitting multi-parameter distributions to SCE data, Mutat. Res., 358 (1996) 15-24.] showed how discrete Pearson and discrete Johnson translation-system distributions may be fitted to sister chromatid exchange (SCE) data presented by Bender et al. [M.A. Bender, R.J. Pearston, R.C. Leonard, B.E. Pyatt, P.C. Gooch, On the distribution of spontaneous SCE in human peripheral blood lymphocytes, Mutat. Res., 281 (1992) 227-232.]. When their performances were measured by the chi-squared test of goodness of fit, these distributions proved to be only moderately better alternatives to the poorly fitting Poisson, binomial, and negative binomial distributions. In this paper, we extend our search for better characterizations of the SCE data by calling upon the Gram-Charlier type B approximation of the negative binomial distribution. We introduce an innovative extension of methods described in a little-known paper by Aitken and Gonin [A.C. Aitken, H.T. Gonin, On fourfold sampling with and without replacement, Proc. R. Soc. Edinburgh, 55 (1934) 114-125.], and show how this leads to fits of the SCE data that, in general, are within acceptable levels of probability. Moreover, we show how a theorem by Cramér [H. Cramér, Mathematical Methods of Statistics, Princeton Univ. Press, 1946.], relating to the scale factor m2/m'1 and its asymptotic distribution, may be used to discriminate between smokers and nonsmokers of the same gender.