Discussion of “Estimation of Parameters for a Mixture of Normal Distributions” by Victor Hasselblad

Abstract

I wish to congratulate Dr. Hasselblad for his most interesting and useful paper. He has provided us with maximum likelihood procedures for estimating the 3K 1 parameters of a mixed distribution consisting of K normally distributed components. As the aut,hor notes, the special case in which K = 2 was dealt with as early as 1894 by Karl Pearson using the method of moments. It has since received attention from various investigators including Charlier (1906), Charlier and Wicksell (1924), Rao (1952) myself (1965). The more general case dealt with by Hasselblad in which K >_ 3 seems to have received little if any previous attention. Prior to the advent of modern large capacity electronic computers, the formidable task of simultaneously estimating such a large number of parameters has no doubt served as a major deterrent to would-be investigators of this problem. Hasselblad’s consideration of the problem is limited to grouped data in which all class intervals are of equal width. In his formulation of the likelihood function, areas under the normal curve within the various class boundaries are approximated as products of class width times mid-interval ordinates. For sufficiently narrow grouping intervals, this approximation would certainly be satisfactory, but it could result in the introduction of troublesome errors for coarse groupings of the sample data. The estimating equat,ions are solved through iteration using the method of “steepest descent” and Newton’s method. The method of steepest descent apparently gave better results from less accurate initial approximations. For this reason the author reports that he usually ran this procedure for as many as forty iterations and then switched to Newton’s method for the concluding series of iterations. He reports that when Newton’s procedure converged, it always improved results with fewer iterations. He found that in general, the method of steepest descent appeared better on smaller amounts of data with fewer intervals whereas Newton’s procedure worked well on larger samples. Initial approximations are obtained by artificially truncating the resultant sample and employing maximum likelihood estimators for truncated samples, using previous results of Hald (1949) and previous result’s of mine (1950). I am inclined t,o believe that certain of my more recent results, Technometrics (1959) and (1961) and Biometrika (1957) might result in a reduction of the computational effort otherwise necessary.