ESTIMATION OF THE PARAMETERS OF THE MIXTURE OF AN ARBITRARY NUMBER OF EXPONENTIAL DISTRIBUTIONS
W. KRYSICKI · Demonstratio Mathematica · 1972
Introduction.K.Pearson in [l] first indicated the importance of estimating the parameters of a mixture of distributions.He gasfi a method of estimating five parameters of the mixture of two normal distributions p + + (1 -p) N(m2» 62).Other papers on this topic have been subsequently published by K.Pearson [2], Muench [3], [4],Gumbel [5], Mendenhall and Hader [6], Rider [7,8], Blischke [9], Krysicki [10, 11], K^cki [12], Wasilewski [13,14-],Cohen [15], K^cki and Krysicki [16], Behbodian [17], Falls [18].The mentioned papers deal with mixtures of distributions of continuous as well 'as discrete type, but always of two components, depending on one or two parameters.The first paper concernig the estimation of the mixture of more than two Bernoulli distributions was written by Blischke [19].He applied there factorial moments.Subsequently Hasselblad gave a method of the estimation of parameters of the mixture of k>3 normal distributions N(mit 6^) derived from the maximum likelihood method.The most recent papers in this field are papersof Kabir [21] and Gridgeman [22].The latter deals with the estimation of parameters of the mixture of normal distributions N(0, 6^).In the interesting paper [21] the author gives a method of the estimation of parameters of the mixture of k>3 distributions belonging to the class of so-called one-parameter exponential distributions introduced by Koopman and Pitman in 1936.To use this method it is necessary to restrict the range butions from the exponential family.J.