Eliciting expert judgements about a set of proportions

Eliciting expert knowledge about several uncertain quantities is a complex task when those quantities exhibit associations. A well-known example of such a problem is eliciting knowledge about a set of uncertain proportions which must sum to 1. The usual approach is to assume that the expert's knowledge can be adequately represented by a Dirichlet distribution, since this is by far the simplest multivariate distribution that is appropriate for such a set of proportions. It is also the most convenient, particularly when the expert's prior knowledge is to be combined with a multinomial sample since then the Dirichlet is the conjugate prior family. Several methods have been described in the literature for eliciting beliefs in the form of a Dirichlet distribution, which typically involve eliciting from the expert enough judgements to identify uniquely the Dirichlet hyperparameters. We describe here a new method which employs the device of over-fitting, i.e. eliciting more than the minimal number of judgements, in order to (a) produce a more carefully considered Dirichlet distribution and (b) ensure that the Dirichlet distribution is indeed a reasonable fit to the expert's knowledge. The method has been implemented in a software extension of the Sheffield elicitation framework (SHELF) to facilitate the multivariate elicitation process.     

The effects of model mis-specifications in linear measurement error models

In the literature, there are many results on the consequences of mis-specified models for linear models with error in the response only, see, e.g., Seber(1977). There are also discussions of estimation for the model writh errors both in the response and in the predictor variables (called measurement error models; see, e.g., Fuller(1987)). In this paper, we consider the problem of model mis-specification for measurement error models. Only a few special cases have been tackled in the past (Edland, 1996; Carroll and Ruppert, 1996 and Lakshminarayanan Amp; Gunst, 1984); we deal with the situation here in some generality. Results have been obtained as follows: (a) When a model is under-fitted, the estimate of the variance of the measurement error will be asymptotically biased, as will the regression coefficients, and the asymptotic biases in the estimates of the regression coefficients will always exist for under-fitted models. Even orthogonality of the variables in the model will not make the biases vanish. (b)For over-fitting, the estimates of the variances of measurement errors and of the regression coefficients are asymptotically unbiased. However, the variance of the estimated regression coefficients will increase. Over-fitting will cause larger changes in the variances of the estimated parameters in measurement error models than in no measurement error models.