A general method of estimating parameters in continuous univariate distributions is proposed. It is especially suited to cases where one of the parameters is an unknown shifted origin. This occurs, for example, in the three-parameter lognormal, gamma and Weibull models. For such distributions it is known that maximum likelihood (ML) estimation can break down because the likelihood is unbounded and this can lead to inconsistent estimators. Properties of the proposed method are described. In particular it is shown to give consistent estimators with asymptotic efficiency equal to ML estimators when these exist. Moreover it gives consistent, asymptotically efficient estimators in situations where ML fails. Examples are given including numerical ones showing the advantages of the method.
This paper compares two methods of assessing variability in simulation output. The methods make specific allowance for two sources of variation: that caused by uncertainty in estimating unknown input parameters (parameter uncertainty), and that caused by the inclusion of random variation within the simulation model itself (simulation uncertainty). The first method is based on classical statistical differential analysis; we show explicitly that, under general conditions, the two sources contribute separately to the total variation.In the classical approach, certain sensitivity coefficients have to be estimated. The effort needed to do this becomes progressively more expensive, increasing linearly with the number of unknown parameters. Moreover there is an additional difficulty of detecting spurious variation when the number of parameters is large. It is shown that a parametric form of bootstrap sampling provides an alternative method which does not suffer from either problem.For illustration, simulation of the operation of a small (4-node) computer communication network is used to compare the accuracy of estimates using the two methods. A larger, more realistic, (30-node) network is presented showing how the bootstrap method becomes competitive when the number of unknown parameters is large.
SUMMARY
Four non‐regular estimation problems are reviewed and discussed. One (the unbounded likelihood problem) involves distributions with infinite spikes, for which maximum likelihood can fail to give consistent estimators. A comparison is made with modified likelihood and spacings methods which do give efficient estimators in this case. An application to the Box–Cox shifted power transform is given. The other three problems occur when the true parameter lies in some special subregion. In one (the constrained parameter problem) the subregion is a boundary. The other two (the embedded model and the indeterminate parameters problems) occur when the model takes on a special form in the subregion. These last two problems have previously been investigated separately. We show that they are equivalent in some situations. Both often arise in non‐linear models and we give a directed graph approach which allows for their occurrence in nested model building. It is argued that many non‐regular problems can be handled systematically without having to resort to elaborate technical assumptions. Relatively uncomplicated methods may be used provided that the underlying nature of the non‐regularity is understood.
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.
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