Robust sequential designs for nonlinear regression

The authors introduce the formal notion of an approximately specified nonlinear regression model and investigate sequential design methodologies when the fitted model is possibly of an incorrect parametric form. They present small‐sample simulation studies which indicate that their new designs can be very successful, relative to some common competitors, in reducing mean squared error due to model misspecifi‐cation and to heteroscedastic variation. Their simulations also suggest that standard normal‐theory inference procedures remain approximately valid under the sequential sampling schemes. The methods are illustrated both by simulation and in an example using data from an experiment described in the chemical engineering literature.     

Methods for clarifying criteria for study continuation at interim analysis

In monitoring clinical trials, the question of futility, or whether the data thus far suggest that the results at the final analysis are unlikely to be statistically successful, is regularly of interest over the course of a study. However, the opposite viewpoint of whether the study is sufficiently demonstrating proof of concept (POC) and should continue is a valuable consideration and ultimately should be addressed with high POC power so that a promising study is not prematurely terminated. Conditional power is often used to assess futility, and this article interconnects the ideas of assessing POC for the purpose of study continuation with conditional power, while highlighting the importance of the POC type I error and the POC type II error for study continuation or not at the interim analysis. Methods for analyzing subgroups motivate the interim analyses to maintain high POC power via an adjusted interim POC significance level criterion for study continuation or testing against an inferiority margin. Furthermore, two versions of conditional power based on the assumed effect size or the observed interim effect size are considered. Graphical displays illustrate the relationship of the POC type II error for premature study termination to the POC type I error for study continuation and the associated conditional power criteria.     

On generalized moment identity and its applications: a unified approach

In this paper, we obtain a generalized moment identity for the case when the distributions of the random variables are not necessarily purely discrete or absolutely continuous. The proposed identity is useful to find the generator which has been used for the approximation of distributions by Stein's method. Apparently, a new approach is discussed for the approximation of distributions by Stein's method. We bring the characterization based on the relationship between conditional expectations and hazard measure in our unified framework. As an application, a new lower bound to the mean-squared error is obtained and it is compared with Bayesian Cramer–Rao bound.     

A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models

This article develops a method for testing the goodness-of-fit of a given parametric autoregressive conditional duration model against unspecified nonparametric alternatives. The test statistics are functions of the residuals corresponding to the quasi maximum likelihood estimate of the given parametric model, and are easy to compute. The limiting distributions of the test statistics are not free from nuisance parameters. Hence, critical values cannot be tabulated for general use. A bootstrap procedure is proposed to implement the tests, and its asymptotic validity is established. The finite sample performances of the proposed tests and several other competing ones in the literature, were compared using a simulation study. The tests proposed in this article performed well consistently throughout, and they were either the best or close to the best. None of the tests performed uniformly the best. The tests are illustrated using an empirical example.     

Uniform Ergodicity of the Particle Gibbs Sampler

The particle Gibbs sampler is a systematic way of using a particle filter within Markov chain Monte Carlo. This results in an off‐the‐shelf Markov kernel on the space of state trajectories, which can be used to simulate from the full joint smoothing distribution for a state space model in a Markov chain Monte Carlo scheme. We show that the particle Gibbs Markov kernel is uniformly ergodic under rather general assumptions, which we will carefully review and discuss. In particular, we provide an explicit rate of convergence, which reveals that (i) for fixed number of data points, the convergence rate can be made arbitrarily good by increasing the number of particles and (ii) under general mixing assumptions, the convergence rate can be kept constant by increasing the number of particles superlinearly with the number of observations. We illustrate the applicability of our result by studying in detail a common stochastic volatility model with a non‐compact state space.     

The Extensively Corrected Score for Measurement Error Models

In measurement error problems, two major and consistent estimation methods are the conditional score and the corrected score. They are functional methods that require no parametric assumptions on mismeasured covariates. The conditional score requires that a suitable sufficient statistic for the mismeasured covariate can be found, while the corrected score requires that the object score function can be estimated without bias. These assumptions limit their ranges of applications. The extensively corrected score proposed here is an extension of the corrected score. It yields consistent estimations in many cases when neither the conditional score nor the corrected score is feasible. We demonstrate its constructions in generalized linear models and the Cox proportional hazards model, assess its performances by simulation studies and illustrate its implementations by two real examples.     

A Model Specification Test For GARCH(1,1) Processes

We provide a consistent specification test for generalized autoregressive conditional heteroscedastic (GARCH (1,1)) models based on a test statistic of Cramér‐von Mises type. Because the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner, we propose a model‐based (semiparametric) bootstrap method to approximate critical values of the test and to verify its asymptotic validity. Finally, we illuminate the finite sample behaviour of the test by some simulations.     

Instrument Assisted Regression for Errors in Variables Models with Binary Response

We study errors‐in‐variables problems when the response is binary and instrumental variables are available. We construct consistent estimators through taking advantage of the prediction relation between the unobservable variables and the instruments. The asymptotic properties of the new estimator are established and illustrated through simulation studies. We also demonstrate that the method can be readily generalized to generalized linear models and beyond. The usefulness of the method is illustrated through a real data example.