A new approach to statistical efficiency of weighted least squares fitting algorithms for reparameterization of nonlinear regression models

We study nonlinear least-squares problem that can be transformed to linear problem by change of variables. We derive a general formula for the statistically optimal weights and prove that the resulting linear regression gives an optimal estimate (which satisfies an analogue of the Rao-Cramer lower bound) in the limit of small noise.     

Parameterization and inference for nonparametric regression problems

We consider local likelihood or local estimating equations, in which a multivariate function () is estimated but a derived function () of () is of interest. In many applications, when most naturally formulated the derived function is a non-linear function of (). In trying to understand whether the derived non-linear function is constant or linear, a problem arises with this approach: when the function is actually constant or linear, the expectation of the function estimate need not be constant or linear, at least to second order. In such circumstances, the simplest standard methods in nonparametric regression for testing whether a function is constant or linear cannot be applied. We develop a simple general solution which is applicable to nonparametric regression, varying-coefficient models, nonparametric generalized linear models, etc. We show that, in local linear kernel regression, inference about the derived function () is facilitated without a loss of power by reparameterization so that () is itself a component of (). Our approach is in contrast with the standard practice of choosing () for convenience and allowing ()> to be a non-linear function of (). The methods are applied to an important data set in nutritional epidemiology.     

Bayesian Estimation and Prediction for the Beta-Binomial Model

The beta-binomial distribution, which is generated by a simple mixture model, has been widely applied in the social, physical, and health sciences. Problems of estimation, inference, and prediction have been addressed in the past, but not in a Bayesian framework. This article develops Bayesian procedures for the beta-binomial model and, using a suitable reparameterization, establishes a conjugate-type property for a beta family of priors. The transformed parameters have interesting interpretations, especially in marketing applications, and are likely to be more stable. More specifically, one of these parameters is the market share and the other is a measure of the heterogeneity of the customer population. Analytical results are developed for the posterior and prediction quantities, although the numerical evaluation is not trivial. Since the posterior moments are more easily calculated, we also propose the use of posterior approximation using the Pearson system. A particular case (when there are two trials), which occurs in taste testing, brand choice, media exposure, and some epidemiological applications, is analyzed in detail. Simulated and real data are used to demonstrate the feasibility of the calculations. The simulation results effectively demonstrate the superiority of Bayesian estimators, particularly in small samples, even with uniform (“non-informed”) priors. Naturally, “informed” priors can give even better results. The real data on television viewing behavior are used to illustrate the prediction results. In our analysis, several problems with the maximum likelihood estimators are encountered. The superior properties and performance of the Bayesian estimators and the excellent approximation results are strong indications that our results will be potentially of high value in small sample applications of the beta-binomial and in cases in which significant prior information exists.     

Computation of Two- and Three-Dimensional Confidence Regions With the Likelihood Ratio

The asymptotic results pertaining to the distribution of the log-likelihood ratio allow for the creation of a confidence region, which is a general extension of the confidence interval. Two- and three-dimensional regions can be displayed visually to describe the plausible region of the parameters of interest simultaneously. While most advanced statistical textbooks on inference discuss these asymptotic confidence regions, there is no exploration of how to numerically compute these regions for graphical purposes. This article demonstrates the application of a simple trigonometric transformation to compute two- and three-dimensional confidence regions; we transform the Cartesian coordinates of the parameters to create what we call the radial profile log-likelihood. The method is applicable to any distribution with a defined likelihood function, so it is not limited to specific data distributions or model paradigms. We describe the method along with the algorithm, follow with an example of our method, and end with an examination of computation time. Supplementary materials for this article are available online.     

Compatible prior distributions for directed acyclic graph models

Summary.  The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given directed acyclic graph model, using a conditioning approach. We define a class of parameterizations that is consistent with the modular structure of the directed acyclic graph and derive a procedure, that is invariant within this class, which we name reference conditioning.     

Nuisance Parameters and the Use of Exploratory Graphical Methods in a Bayesian Analysis

Consider the problem of inference about a parameter θ in the presence of a nuisance parameter v. In a Bayesian framework, a number of posterior distributions may be of interest, including the joint posterior of (θ, ν), the marginal posterior of θ, and the posterior of θ conditional on different values of ν. The interpretation of these various posteriors is greatly simplified if a transformation (θ, h(θ, ν)) can be found so that θ and h(θ, v) are approximately independent. In this article, we consider a graphical method for finding this independence transformation, motivated by techniques from exploratory data analysis. Some simple examples of the use of this method are given and some of the implications of this approximate independence in a Bayesian analysis are discussed.     

The Mean Squared Error Optimum Design Criterion for Parameter Estimation in Nonlinear Regression Models

In the context of nonlinear regression models, we propose an optimal experimental design criterion for estimating the parameters that account for the intrinsic and parameter-effects nonlinearity. The optimal design criterion proposed in this article minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated using the curvature array. The design criterion reduces to the D-optimal design criterion if there are no intrinsic and parameter-effects nonlinearity in the model, and depends on the scale parameter estimator and on the reparameterization used. Some examples, using a well known nonlinear kinetics model, demonstrate the application of the proposed criterion to nonsequential design of experiments as compared with the D-optimal criterion.     

A robust approach to joint modeling of mean and scale covariance for longitudinal data

In this paper, we propose a multivariate t$t$ regression model with its mean and scale covariance modeled jointly for the analysis of longitudinal data. A modified Cholesky decomposition is adopted to factorize the dependence structure in terms of unconstrained autoregressive and scale innovation parameters. We present three distinct representations of the log-likelihood function of the model and study the associated properties. A computationally efficient Fisher scoring algorithm is developed for carrying out maximum likelihood estimation. The technique for the prediction of future responses in this context is also investigated. The implementation of the proposed methodology is illustrated through two real-life examples and extensive simulation studies.     

New large-sample confidence intervals for a linear combination of binomial proportions

In this paper, we consider the problem wherein one desires to estimate a linear combination of binomial probabilities from k>2$k>2$ independent populations. In particular, we create a new family of asymptotic confidence intervals, extending the approach taken by Beal [1987. Asymptotic confidence intervals for the difference between two binomial parameters for use with small samples. Biometrics 73, 941–950] in the two-sample case. One of our new intervals is shown to perform very well when compared to the best available intervals documented in Price and Bonett [2004. An improved confidence interval for a linear function of binomial proportions. Comput. Statist. Data Anal. 45, 449–456]. Furthermore, our interval estimation approach is quite general and could be extended to handle more complicated parametric functions and even to other discrete probability models in stratified settings. We illustrate our new intervals using two real data examples, one from an ecology study and one from a multicenter clinical trial.