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A problem that is often encountered in non-linear regression is instability in the computation of parameter estimates. Such instability may arise from models with too many parameters, resulting in estimating equations that are ill conditioned and consequent problems of identification. This paper shows that in some cases these difficulties can be explained by the presence of an embedded model, a special case of the original model involving fewer parameters but one that is not readily identified from the parameterization of the model

SUMMARY Two distinct problems can arise in maximum likelihood estimation in certain threeparameter problems. One problem, associated with the fitting of a threshold parameter, occurs if the fitted distribution has to be very positively skewed. This problem is well known. However, there is a second and unrelated difficulty which occurs quite often in practice and which arises when the distribution is not at all skew. This problem is not so well understood. It is shown in this paper that, within the three-parameter model, there is an embedded twoparameter special case which corresponds to infinite parameter values in the original model. The problem arises when this embedded model is the best fit to the data. The problem is shown to be easily resolved by first carrying out a check to see whether the embedded model should be fitted instead of the three-parameter model. Formal tests for this are discussed. The gamma, inverse Gaussian, log-normal and Weibull distributions are examples where the problem occurs. Numerical examples are provided for illustration.

SUMMARY. The results of a survey of the macro‐invertebrates of the polluted River Ely, South Wales, are used as a basis for comparing several classification methods which have been used previously in river survey work to determine species groupings. The methods compared are product‐moment correlation (clustered by the nearest neighbour technique), Kendall's tau coefficient (clustered by the nearest neighbour and average linkage techniques), and Squared Euclidean‐Distance coefficient (clustered by nearest neighbour and Ward's techniques). The species groupings determined by these methods were influenced both by the association coefficient and the technique used to cluster it. Some species were grouped together by all or most of the methods. The ecological validity of these robust groups is examined. A clear recommendation regarding the most appropriate method is frustrated by incomplete knowledge of the ecological requirements of most of the aquatic macro‐invertebrates used in the data‐set. However, Kendall's tau coefficient clustered by the average linkage technique appeared to produce ecologically meaningful species groups. Product‐moment correlation was also reasonably successful and since it is based on absolute abundance data whereas Kendall's tau coefficient is based on relative abundance data, the use of the two together is recommended for determining robust groups.

Previously suggested methods for constructing confidence bands for cumulative distribution functions have been based on the classical Kolmogorov-Smirnov test for an empirical distribution function. This paper gives a method based on maximum likelihood estimation of the parameters. The method is described for a general continuous distribution. Detailed results are given for a location-scale parameter model, which includes the normal and extreme-value distributions as special cases. Results are also given for the related lognormal and Weibull distributions. The formulas derived for these distributions give a band with exact confidence coefftcient. A chi-squared approximation, which avoids the use of special tables, is also described. An example is used to compare the resulting bands with those obtained by previously published methods.

On suggere que l'utilisation du produit des densites pour la fonction de vraisemblance est une mauvaise application de la methode du maximum de vraisemblance dans des cas non reguliers