The Estimation of Five - Parameter Mixed Normal Distributions by 1

SUMMARY Fisher's method of maximum likelihood breaks down when applied to the problem of estimating the five parameters of a mixture of two normal densities from a continuous random sample of size n. The two alternatives considered here are moment estimates on the one hand, and multinomial maximum likelihood and minimum x 2 estimates obtained by grouping the underlying variable on the other. The methods are compared both for bias (to n-l) , and mean-squared-error (to n-2) for a variety of mixed distributions. In terms of bias, there seems to be little to choose between them. As regards mean-squared-error, the grouped estimates are shown to be more accurate than the moment estimates for most distributions, though the moment estimates do seem to be preferable for distributions which are particularly difficult to estimate. It is also found that the accuracy levels of the grouped maximum likelihood and minimum X 2 estimates do not differ greatly. For-2 any of the estimates, we show in many cases that the n term of the mean-squared-2 error cannot be neglected unless the sample size is very large indeed, and that only very large samples give an accuracy level which most experimenters would find acceptable.