Bayesian sample-size determination for one and two Poisson rate parameters with applications to quality control

We formulate Bayesian approaches to the problems of determining the required sample size for Bayesian interval estimators of a predetermined length for a single Poisson rate, for the difference between two Poisson rates, and for the ratio of two Poisson rates. We demonstrate the efficacy of our Bayesian-based sample-size determination method with two real-data quality-control examples and compare the results to frequentist sample-size determination methods.     

Maximum Likelihood Inference for Log-linear Models Subject to Constraints of Double Monotone Dependence

To model an hypothesis of double monotone dependence between two ordinal categorical variables A and B usually a set of symmetric odds ratios defined on the joint probability function is subject to linear inequality constraints. Conversely in this paper two sets of asymmetric odds ratios defined, respectively, on the conditional distributions of A given B and on the conditional distributions of B given A are subject to linear inequality constraints. If the joint probabilities are parameterized by a saturated log-linear model, these constraints are nonlinear inequality constraints on the log-linear parameters. The problem here considered is a non-standard one both for the presence of nonlinear inequality constraints and for the fact that the number of these constraints is greater than the number of the parameters of the saturated log-linear model.This work has been supported by the COFIN 2002 project, references 2002133957_002, 2002133957_004. Preliminary findings have been presented at SIS (Società Italiana di Statistica) Annual Meeting, Bari, 2004.     

Importance of interpolation when constructing double-bootstrap confidence intervals

We show that, in the context of double-bootstrap confidence intervals, linear interpolation at the second level of the double bootstrap can reduce the simulation error component of coverage error by an order of magnitude. Intervals that are indistinguishable in terms of coverage error with theoretical, infinite simulation, double-bootstrap confidence intervals may be obtained at substantially less computational expense than by using the standard Monte Carlo approximation method. The intervals retain the simplicity of uniform bootstrap sampling and require no special analysis or computational techniques. Interpolation at the first level of the double bootstrap is shown to have a relatively minor effect on the simulation error.     

A unified approach to regression analysis under double-sampling designs

We propose a unified approach to the estimation of regression parameters under double-sampling designs, in which a primary sample consisting of data on the rough or proxy measures for the response and/or explanatory variables as well as a validation subsample consisting of data on the exact measurements are available. We assume that the validation sample is a simple random subsample from the primary sample. Our proposal utilizes a specific parametric model to extract the partial information contained in the primary sample. The resulting estimator is consistent even if such a model is misspecified, and it achieves higher asymptotic efficiency than the estimator based only on the validation data. Specific cases are discussed to illustrate the application of the estimator proposed.     

Confidence intervals for secondary parameters following a sequential test

In sequential studies, formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest. Valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study. However, the completed data set may warrant a more detailed analysis, involving the estimation of parameters corresponding to effects that were not used to determine when to stop, and yet correlated with those that were. This paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used. The method assumes that information accumulates on the primary and secondary parameters at proportional rates. This requirement will be valid in many potential applications, but only in limited situations in survival analysis.     

Diagnostic checks for discrete data regression models using posterior predictive simulations

Model checking with discrete data regressions can be difficult 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-fit 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 fitted to a historical data set on behavioural learning. We then discuss the general applicability of our findings in the context of a recent applied example on which we have worked. We find 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 specific 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 sufficiently simple that model checks can often be made implicitly. The following discrepancy variables did not work well: scatterplots of latent residuals defined from an underlying continuous model and quantile–quantile plots of these residuals.     

Profile upper Confidence Limits from Discrete Data

When the data are discrete, standard approximate confidence limits often have coverage well below nominal for some parameter values. While ad hoc adjustments may largely solve this problem for particular cases, Kabaila & Lloyd (1997) gave a more systematic method of adjustment which leads to tight upper limits, which have coverage which is never below nominal and are as small as possible within a particular class. However, their computation for all but the simplest models is infeasible. This paper suggests modifying tight upper limits by an initial replacement of the unknown nuisance parameter vector by its profile maximum likelihood estimator. While the resulting limits no longer possess the optimal properties of tight limits exactly, the paper presents both numerical and theoretical evidence that the resulting coverage function is close to optimal. Moreover these profile upper limits are much (possibly many orders of magnitude) easier to compute than tight upper limits.     

Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

Summary.  Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models , where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

Some results on generalized ageing classes

In this paper we provide new results about generalized ageing classes on the excess lifetime of a renewal process. We also obtain some characterizations of generalized ageing classes by means of the residual life at random time.