Fixed-width interval estimation of the minimum point of a regression function based on the Kiefer-Wolfowitz procedure

A version of the central limit theorem for the Kiefer-Wolfowitz procedure is stated. One constructs an asymptotically consistent fixed-width confidence interval for the minimum of a regression function. It is shown that the first moment of the corresponding stopping rule is finite. Both the construction and properties of the estimates of unknown parameters and an adaptive version of the procedure are presented.     

Bayesian experimental design for a toxicokinetic-toxicodynamic model

The aim of this study is to apply the Bayesian method of identifying optimal experimental designs to a toxicokinetic-toxicodynamic model that describes the response of aquatic organisms to time dependent concentrations of toxicants. As for experimental designs, we restrict ourselves to pulses and constant concentrations. A design of an experiment is called optimal within this set of designs if it maximizes the expected gain of knowledge about the parameters. Focus is on parameters that are associated with the auxiliary damage variable of the model that can only be inferred indirectly from survival time series data. Gain of knowledge through an experiment is quantified both with the ratio of posterior to prior variances of individual parameters and with the entropy of the posterior distribution relative to the prior on the whole parameter space. The numerical methods developed to calculate expected gain of knowledge are expected to be useful beyond this case study, in particular for multinomially distributed data such as survival time series data.     

Optimal design for nonlinear regression models

Two results for D _θ-optimal designs for nonlinear regression models are shown to follow directly from approximate design theory. The first result considered is one concerning the replication of exact designs with minimum support, first established by Atkinson and Hunter and by M.J. Box in 1968, while the second pertains to a heteroscedastic model introduced by Velilla and Llosa in 1992. An illustrative example is provided.     

Bayesian likelihood methods for estimating the end point of a distribution

Summary.  We consider maximum likelihood methods for estimating the end point of a distribution. The likelihood function is modified by a prior distribution that is imposed on the location parameter. The prior is explicit and meaningful, and has a general form that adapts itself to different settings. Results on convergence rates and limiting distributions are given. In particular, it is shown that the limiting distribution is non-normal in non-regular cases. Parametric bootstrap techniques are suggested for quantifying the accuracy of the estimator. We illustrate performance by applying the method to multiparameter Weibull and gamma distributions.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

On generalised estimating equations for vector regression

Generalised estimating equations (GEE) for regression problems with vector‐valued responses are examined. When the response vectors are of mixed type (e.g. continuous–binary response pairs), the GEE approach is a semiparametric alternative to full‐likelihood copula methods, and is closely related to Prentice & Zhao's mean‐covariance estimation equations approach. When the response vectors are of the same type (e.g. measurements on left and right eyes), the GEE approach can be viewed as a ‘plug‐in’ to existing methods, such as the vglm function from the state‐of‐the‐art VGAM package in R. In either scenario, the GEE approach offers asymptotically correct inferences on model parameters regardless of whether the working variance–covariance model is correctly or incorrectly specified. The finite‐sample performance of the method is assessed using simulation studies based on a burn injury dataset and a sorbinil eye trial dataset. The method is applied to data analysis examples using the same two datasets, as well as to a trivariate binary dataset on three plant species in the Hunua ranges of Auckland.     

Asymptotic results for estimation and testing variances in regression models

Usually the variance of independent observations resulting from a linear or a nonlinear relationship is estimated by the Least-Squares residual estimator. In this paper its asymptotic properties are investigated. Further the asymptotic behaviour of tests for one-sided hypotheses on the variance is studied. The paper splits into two parts, the first one concerned with linear and the second one with nonlinear models.     

Comparison of design for quadratic regression on cubes

Designs for quadratic regression are considered when the possible choices of the controllable variable are points x=(x₁,x₂,…,x_q) in the q-dimensional cube of side 2. The designs that are optimum with respect to such criteria as those of D-, A-, and E-optimality are compared in their performance relative to these and other criteria. Some of the results are developed algebraically; others, numerically. The possible supports of E-optimum designs are much more numerous than the D-optimum supports characterized earlier. The A-optimum design appears to be fairly robust in its efficiency, under variation of criterion.     

Bayesian optimal sequential design for nonparametric regression via inhomogeneous evolutionary MCMC

We develop a novel computational methodology for Bayesian optimal sequential design for nonparametric regression. This computational methodology, that we call inhomogeneous evolutionary Markov chain Monte Carlo, combines ideas of simulated annealing, genetic or evolutionary algorithms, and Markov chain Monte Carlo. Our framework allows optimality criteria with general utility functions and general classes of priors for the underlying regression function. We illustrate the usefulness of our novel methodology with applications to experimental design for nonparametric function estimation using Gaussian process priors and free-knot cubic splines priors.     

A Bayesian sequential design with binary outcome

Several researchers have proposed solutions to control type I error rate in sequential designs. The use of Bayesian sequential design becomes more common; however, these designs are subject to inflation of the type I error rate. We propose a Bayesian sequential design for binary outcome using an alpha‐spending function to control the overall type I error rate. Algorithms are presented for calculating critical values and power for the proposed designs. We also propose a new stopping rule for futility. Sensitivity analysis is implemented for assessing the effects of varying the parameters of the prior distribution and maximum total sample size on critical values. Alpha‐spending functions are compared using power and actual sample size through simulations. Further simulations show that, when total sample size is fixed, the proposed design has greater power than the traditional Bayesian sequential design, which sets equal stopping bounds at all interim analyses. We also find that the proposed design with the new stopping for futility rule results in greater power and can stop earlier with a smaller actual sample size, compared with the traditional stopping rule for futility when all other conditions are held constant. Finally, we apply the proposed method to a real data set and compare the results with traditional designs.     

A Bayesian sequential design with adaptive randomization for 2‐sided hypothesis test

Bayesian sequential and adaptive randomization designs are gaining popularity in clinical trials thanks to their potentials to reduce the number of required participants and save resources. We propose a Bayesian sequential design with adaptive randomization rates so as to more efficiently attribute newly recruited patients to different treatment arms. In this paper, we consider 2‐arm clinical trials. Patients are allocated to the 2 arms with a randomization rate to achieve minimum variance for the test statistic. Algorithms are presented to calculate the optimal randomization rate, critical values, and power for the proposed design. Sensitivity analysis is implemented to check the influence on design by changing the prior distributions. Simulation studies are applied to compare the proposed method and traditional methods in terms of power and actual sample sizes. Simulations show that, when total sample size is fixed, the proposed design can obtain greater power and/or cost smaller actual sample size than the traditional Bayesian sequential design. Finally, we apply the proposed method to a real data set and compare the results with the Bayesian sequential design without adaptive randomization in terms of sample sizes. The proposed method can further reduce required sample size.     

A note on the efficiency of composite quantile regression

Composite quantile regression (CQR) is motivated by the desire to have an estimator for linear regression models that avoids the breakdown of the least-squares estimator when the error variance is infinite, while having high relative efficiency even when the least-squares estimator is fully efficient. Here, we study two weighting schemes to further improve the efficiency of CQR, motivated by Jiang et al. [Oracle model selection for nonlinear models based on weighted composite quantile regression. Statist Sin. 2012;22:1479–1506]. In theory the two weighting schemes are asymptotically equivalent to each other and always result in more efficient estimators compared with CQR. Although the first weighting scheme is hard to implement, it sheds light on in what situations the improvement is expected to be large. A main contribution is to theoretically and empirically identify that standard CQR has good performance compared with weighted CQR only when the error density is logistic or close to logistic in shape, which was not noted in the literature.     

An approximate Bayesian up-down method for estimating a percentage point on a dose-response curve

While the up-down method for estimating a percentage point on a dose-response curve has received considerable attention, a general Bayesian solution to the up-down design and estimation has never been presented, probably due to its computational complexity both in design and use. This paper presents a theoretical approach for up-down experimental designs with unknown location and slope parameters, and a practical approach for their use. The simplex method is used to find the optimal starting dose level and step sizes that minimize the expected root mean square error for a fixed number of observations and a reduced number of step sizes. The Bayesian estimate is then approximated by a polynomial formula. The coefficients of the formula are also chosen using simplex minimization. Two example solutions are given with uniform-uniform and normal-gamma joint prior distributions, showing that the simplifying assumptions make the method far easier to use with only a marginal increase in expected root mean square error. We show how to adapt these prior distributions to a wide range of frequently encountered applications.     

Adaptive estimating function inference for nonstationary determinantal point processes

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this article, we establish asymptotic normality of estimating function estimators in a very general setting of nonstationary point processes. We then adapt this result to the case of nonstationary determinantal point processes, which are an important class of models for repulsive point patterns. In practice, often first‐ and second‐order estimating functions are used. For the latter, it is a common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance, which is usually specified in an ad hoc manner. We suggest instead a data‐driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non‐stationary determinantal point processes and by an application to a real dataset.     

Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D‐optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian _q‐optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.     

Bayesian two-stage optimal design for mixture models

In this paper, a Bayesian two-stage D–D optimal design for mixture experimental models under model uncertainty is developed. A Bayesian D-optimality criterion is used in the first stage to minimize the determinant of the posterior variances of the parameters. The second stage design is then generated according to an optimalityprocedure that collaborates with the improved model from the first stage data. The results show that a Bayesian two-stage D–D-optimal design for mixture experiments under model uncertainty is more efficient than both the Bayesian one-stage D-optimal design and the non-Bayesian one-stage D-optimal design in most situations. Furthermore, simulations are used to obtain a reasonable ratio of the sample sizes between the two stages.     

A cubic smoothing spline based lack of fit test for nonlinear regression models

The use of a statistic based on cubic spline smoothing is considered for testing nonlinear regression models for lack of fit. The statistic is defined to be the Euclidean squared norm of the smoothed residual vector obtained from fitting the nonlinear model, The asymptotic distribution of the statistic is derived under suitable smooth local alternatives and a numerical example is presented.     

Variance function in regression analysis of longitudinal data using the generalized estimating equation approach

Longitudinal or clustered response data arise in many applications such as biostatistics, epidemiology and environmental studies. The repeated responses cannot in general be assumed to be independent. One method of analysing such data is by using the generalized estimating equations (GEE) approach. The current GEE method for estimating regression effects in longitudinal data focuses on the modelling of the working correlation matrix assuming a known variance function. However, correct choice of the correlation structure may not necessarily improve estimation efficiency for the regression parameters if the variance function is misspecified [Wang YG, Lin X. Effects of variance-function misspecification in analysis of longitudinal data. Biometrics. 2005;61:413–421]. In this connection two problems arise: finding a correct variance function and estimating the parameters of the chosen variance function. In this paper, we study the problem of estimating the parameters of the variance function assuming that the form of the variance function is known and then the effect of a misspecified variance function on the estimates of the regression parameters. We propose a GEE approach to estimate the parameters of the variance function. This estimation approach borrows the idea of Davidian and Carroll [Variance function estimation. J Amer Statist Assoc. 1987;82:1079–1091] by solving a nonlinear regression problem where residuals are regarded as the responses and the variance function is regarded as the regression function. A limited simulation study shows that the proposed method performs at least as well as the modified pseudo-likelihood approach developed by Wang and Zhao [A modified pseudolikelihood approach for analysis of longitudinal data. Biometrics. 2007;63:681–689]. Both these methods perform better than the GEE approach.