Bayes factors for goodness of fit testing

We propose the use of the generalized fractional Bayes factor for testing fit in multinomial models. This is a non-asymptotic method that can be used to quantify the evidence for or against a sub-model. We give expressions for the generalized fractional Bayes factor and we study its properties. In particular, we show that the generalized fractional Bayes factor has better properties than the fractional Bayes factor.     

Estimating and testing generalized linear models under inequality restrictions

We consider maximum likelihood estimation and likelihood ratio tests under inequality restrictions on the parameters. A special case are order restrictions, which may appear for example in connection with effects of an ordinal qualitative covariate. Our estimation approach is based on the principle of sequential quadratic programming, where the restricted estimate is computed iteratively and a quadratic optimization problem under inequality restrictions is solved in each iteration. Testing for inequality restrictions is based on the likelihood ratio principle. Under certain regularity assumptions the likelihood ratio test statistic is asymptotically distributed like a mixture of χ², where the weights are a function of the restrictions and the information matrix. A major problem in theory is that in general there is no unique least favourable point. We present some empirical findings on finite-sample behaviour of tests and apply the methods to examples from credit scoring and dentistry.     

Residual components in generalized linear models

The adequacy of a postulated generalized linear model can often be improved by transforming predictors and/or including additional explanatory variables. To assess the fit relative to a given predictor, we define its corresponding residual component. Asymptotic bias and variance of the residual component are considered, paying particular attention to the case that the presumed model is valid.     

Building adaptive estimating equations when inverse of covariance estimation is difficult

Summary. To construct an optimal estimating function by weighting a set of score functions, we must either know or estimate consistently the covariance matrix for the individual scores. In problems with high dimensional correlated data the estimated covariance matrix could be unreliable. The smallest eigenvalues of the covariance matrix will be the most important for weighting the estimating equations, but in high dimensions these will be poorly determined. Generalized estimating equations introduced the idea of a working correlation to minimize such problems. However, it can be difficult to specify the working correlation model correctly. We develop an adaptive estimating equation method which requires no working correlation assumptions. This methodology relies on finding a reliable approximation to the inverse of the variance matrix in the quasi-likelihood equations. We apply a multivariate generalization of the conjugate gradient method to find estimating equations that preserve the information well at fixed low dimensions. This approach is particularly useful when the estimator of the covariance matrix is singular or close to singular, or impossible to invert owing to its large size.     

A generalized Pearson system useful in reliability analysis

In the present note, we study an extended class of Pearson system of distributions in the context of reliability. It is shown that the proposed class of models can be characterized by a relatioaship between the failure rate and the conditional moments. Further, we develop a procedure to identify an increasing (decreasing) failure rate model in the generalized Pearson system.     

Oil and gas depletion: Diffusion models and forecasting under strategic intervention

Crude oil and natural gas depletion may be modelled by a diffusion process based upon a constrained life-cycle. Here we consider the Generalized Bass Model. The choice is motivated by the realistic assumption that there is a self-evident link between oil and gas extraction and the spreading of the modern technologies in wide areas such as transport, heating, cooling, chemistry and hydrocarbon fuels consumption. Such a model may include deterministic or semi-deterministic regulatory interventions. Statistical analysis is based upon nonlinear methodologies and more flexible autoregressive structure of residuals. The technical aim of this paper is to outline the meaningful hierarchy existing among the components of such diffusion models. Statistical effort in residual component analysis may be read as a significant confirmation of a well-founded diffusion process under rare but strong deterministic shocks. Applications of such ideas are proposed with reference to world oil and gas production data and to particular regions such as mainland U.S.A., U.K., Norway and Alaska. The main results give new evidence in time-peaks location and in residual times to depletion.     

Count data and treatment heterogeneity in 2×2 crossover trials

Count data are routinely assumed to have a Poisson distribution, especially when there are no straightforward diagnostic procedures for checking this assumption. We reanalyse two data sets from crossover trials of treatments for angina pectoris , in which the outcomes are counts of anginal attacks. Standard analyses focus on treatment effects, averaged over subjects; we are also interested in the dispersion of these effects (treatment heterogeneity). We set up a log-Poisson model with random coefficients to estimate the distribution of the treatment effects and show that the analysis is very sensitive to the distributional assumption; the population variance of the treatment effects is confounded with the (variance) function that relates the conditional variance of the outcomes, given the subject's rate of attacks, to the conditional mean. Diagnostic model checks based on resampling from the fitted distribution indicate that the default choice of the Poisson distribution for the analysed data sets is poorly supported. We propose to augment the data sets with observations of the counts, made possibly outside the clinical setting, so that the conditional distribution of the counts could be established.     

Exact Inference for a Simple Step-stress Model with Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.     

Simultaneous Small Sample Inference for Linear Combinations of Generalized Linear Model Parameters

A method is proposed to construct simultaneous confidence intervals for multiple linear combinations of generalized linear model parameters, that uses a multivariate normal- or t-distribution together with the signed likelihood root statistic. In an application to a case study simultaneous confidence bands for logistic regression are calculated. A simulation study based on the example evaluation suggests superior performance compared to the common Wald-type approaches. The proposed methods are readily implemented in the R extension package mcprofile.