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Usage of nonparametric model to study over rejection observed in Granger non-causality is presented.

A nonparametric Bayesian modeling is described and is compared to the one used by Shively, Kohn and Wood in their study. The former is concerned with broad features of the data, such as the general form in which a covariate enters the model or if interaction terms are required. A parametric element is needed when there is a substantive differential effect across a range of values for a covariate. For instance, a piecewise linear term, can account for most of the overall trend of the data., This is a landmark article in applied statistics. It takes a subject of interest to many, who is the best - in hockey, golf and baseball - and applies a [...]

The use of function estimation in the study done by Shively, Kohn and Wood, on selecting the correct model in a multiple predictor semiparametric regression is discussed. Function estimation is made up of a model expressed as a line plus an integrated Wiener process, wherein fitting is attained by using a state-space formulation and Kalman filtering. However, the Wiener process is a difficult procedure and the researchers should have used simpler method instead., My comments concern the questions of scientific interest in longitudinal studies and the usefulness (appropriateness) of causal inference about the effects of time-varying exposures on the prevalence of an outcome [...]

Comments made by Brumback et al., E.I. George et al., Steven N. MacEachern and Peter Muller on Shively, Kohn and Wood's study on additive nonparametric regression models are discussed. For instance, Brumback et al. recommends the use of regression splines as a prior for one-dimensional curve instead of a Wiener process before simplifying the computations and making nonparametric smoothing easier. On the other hand, George et al. argue that a conditional mean instead of the marginal mean should be used as the prior., We thank the discussants for their kind words and insightful criticisms. We first address some general concepts touched on by each of the discussants, then address each discussant's specific comments [...]

A hierarchical Bayesian approach is proposed for variable selection and function estimation in additive nonparametric Gaussian regression models and additive nonparametric binary regression models. The prior for each component function is an integrated Wiener process resulting in a posterior mean estimate that is a cubic smoothing spline. Each of the explanatory variables is allowed to be in or out of the model, and the regression functions are estimated by model averaging. To allow variable selection and model averaging, data-based priors are used for the smoothing parameter and the slope at 0 of each component function. A two-step Markov chain Monte Carlo method is used to efficiently obtain the data-based prior and to carry out variable selection and function estimation. It is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously. The methodology is illustrated for a binary regression using heart attack data. KEY WORDS: Bayesian information criterion; Binary regression; Gaussian errors; Model averaging; Posterior probabilities; Time series models., 1. INTRODUCTION This article addresses the problem of variable selection and function estimation in additive nonparametric Gaussian regression models and additive nonparametric binary regression models. For the binary model, the [...]

A 'data-based prior' is used in estimating the hyperparameter values in additive nonparametric Gaussian regression models and additive nonparametric binary regression models. An advantage of using the data-based prior is that its influence dies out asymptotically such that the problem becomes insignificant with a large sample size. Its use results in a posterior that accumulates near submodel regions. Substantial gains are also realized when one or more of the variables are not related to the dependent variable., Berry, Reese and Larkey (BRL) have developed interesting models that shed light on the structure and evolution of player performance in three distinct sports: baseball, golf and hockey. Their article [...]

The selection of the probability model, the data-based prior and the marginal prior in Shively, Kohn and Wood's study in selecting variables and estimating functions in nonparametric Gaussian models and binary regression models is discussed. They should be commended for synthesizing state-of-the-art theories and numerical methods for resolving problems related with additive nonparametric regression., 1. INTRODUCTION This article addresses a fascinating question: How can one compare the accomplishments of players in a given sport from different eras? In the sport of baseball, is it [...]