A procedure for estimating parameters of a mixture of normal distributions fitted to data available in grouped truncated form is developed in Chapter 2. The procedure is based on an application of the EM algorithm. The technique is illustrated by modelling the distribution of red blood cell volumes of anaemic cattle as a mixture of normal distributions.
To provide standard errors for the estimated parameters of the normal mixture model fitted to grouped truncated data, an estimate of the information matrix is obtained in Chapter 3 in terms of quantities computed during the implementation of the EM algorithm. This estimated information matrix is also used to enhance the convergence rate of the EM algorithm using a Newton-type step procedure. A comparison is made of this enhanced procedure with the original procedure based on data used in the illustration of Chapter 2.
In Chapter 4, an algorithm is presented for carrying out the procedures developed in Chapters 2 and 3. An assessment of the GLIM computing approach to this problem is made. A comparison is made of the algorithm we have developed with the alternative approach of associating the class frequencies with the midpoint of the grouping intervals. This latter approach performs the parameter estimation using mixture techniques appropriate for data in the form of individual observations.
Modifications to the method of Chapter 2 are made in Chapter 5 to allow for constraints, firstly on the mixing proportions and secondly on the means, in the context of grouped data. In the first instance the mixing parameters for a three component mixture fitted to P.T.C sensitivity are constrained to follow the Hardy-Weinberg equilibrium law requiring the proportions to be in the form: (q2, 2pq and p2 In the second instance the means of components of a mixture fitted to collagen fibril diameter distributions are constrained to be equally spaced.
Missing information on one of the components of a mixture model fitted to grouped data is investigated in Chapter 6. The frequency of particles that do not emit electrons when fired at a metal surface is inferred by fitting a mixture model to the electron energy distribution of particles that emit a finite number of electrons. The electron energy distribution, observed over grouping intervals, is modelled as a mixture with normal components with constraints on the mixing proportions, the means and the variances.
Finally, in Chapter 7, techniques are presented for fitting a Laplace-normal mixture and a mixture of two skew Laplace distributions, both in the situation of grouped data. The first situation is required to model wind shear distributions measured on aircraft landings, and the second situation is required for modelling soil particle size distributions. The discussion of this last mixture model also reports on techniques for coping with data for each class interval being only available in relative frequency form.