Estimation of Parameters for a Mixture of Normal Distributions

Abstract

n observations are taken from a mixture of K normal subpopulations, where the value of K is known. It is assumed that these n observations are given as N frequencies from equally spaced intervals. Initial guesses of the K means, K variances, and K -1 proportions are made using the maximum likelihood estimates for a single truncated normal population as derived by Hald. Then an approximation to the likelihood function of the entire sample is used, and attempts to maximize this yield two iteration formulas. In practice, the method of steepest descent always converged, although the rate was not always fast. Special cases of equal variances and variances proportional to the square of the mean are also considered.

Mobile fishing gear reduces benthic megafaunal production on Georges Bank

“…The NORMSEP program (Hasselblad 1966) was used to identify cohorts in the length-frequency distributions, which were aggregated at 5 mm intervals of shell height. For this species and site only, we used the increment-summation (Crisp 1984) method to estimate production.…”

Growth and reproduction of the dolphinfish (<i>Coryphaena equiselis</i> and <i>Coryphaena hippurus</i>) in the Canary Islands, Central-East Atlantic (preliminary results)

“…These were fitted at the 95% confidence level by the computer programme NORMSEP (Hasselblad, 1966) included in the FISAT software (Gayanilo et al 1994).…”

A mixture likelihood approach for generalized linear models

“…A specific form of the density of the observations in each of the underlying populations is specified, and the purpose of the finite mixture approach is to decompose the sample into its mixture components. Specific forms of densities which have been extensively used include the normal (e.g., Hasselblad 1966;Day 1969;Wolfe 1970), exponential (Thomas 1966;Teicher 1961) and bernouili densities (the latter models are typically known as latent structure models, e.g., Lazarsfeld and Henry 1968;Goodman 1974).…”

“…Pearson 1894; Charlier and Wicksell 1924;Quandt and Ramsey 1978), but attention later focused on graphical techniques for the detection of (univariate) mixtures (e.g., Harding 1948;Cassie 1954;Bhattacharya 1967;Fowlkes 1979). Hasselblad (1966Hasselblad ( , 1969 was among the first to use maximum likelihood estimation for mixtures of two or more distributions from the exponential family. As maximum likelihood has been shown to be superior to the method of moments for the estimation of finite mixtures (cf.…”

“…While most of the mixture likelihood approaches (beginning with Newcomb 1886, and later Hasselblad 1966, and Wolfe 1970 have used iterative schemes corresponding to particular instances of the EM algorithm, the formal applicability of this EM algorithm with its attractive convergence properties of the likelihood solution to finite mixture problems was recognized only after the developments of Dempster, Laird and Rubin (1977), which were later supplemented by Boyles (1983) and Wu (1983). (See also Redner and Walker 1984;Titterington 1990.)…”