No abstract

Model checking with discrete data regressions can be dif®cult because the usual methods such as residual plots have complicated reference distributions that depend on the parameters in the model. Posterior predictive checks have been proposed as a Bayesian way to average the results of goodness-of-®t tests in the presence of uncertainty in estimation of the parameters. We try this approach using a variety of discrepancy variables for generalized linear models ®tted to a historical data set on behavioural learning. We then discuss the general applicability of our ®ndings in the context of a recent applied example on which we have worked. We ®nd that the following discrepancy variables work well, in the sense of being easy to interpret and sensitive to important model failures: structured displays of the entire data set, general discrepancy variables based on plots of binned or smoothed residuals versus predictors and speci®c discrepancy variables created on the basis of the particular concerns arising in an application. Plots of binned residuals are especially easy to use because their predictive distributions under the model are suf®ciently simple that model checks can often be made implicitly. The following discrepancy variables did not work well: scatterplots of latent residuals de®ned from an underlying continuous model and quantile±quantile plots of these residuals.

item response model, local item dependence, test speededness,

We present families of nonparametric estimators for the conditional tail index of a Paretotype distribution in presence of random covariates. These families are constructed from locally weighted sums of power transformations of excesses over a high threshold. The asymptotic properties of the proposed estimators are derived under some assumptions on the conditional response distribution, the weight function and the density function of the covariates. We also introduce bias-corrected versions of the estimators for the conditional tail index, and propose in this context a consistent estimator for the second order tail parameter. The finite sample performance of some specific examples from our classes of estimators is illustrated with a small simulation experiment.