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.     

On the accuracy in high‐dimensional linear models and its application to genomic selection

Genomic selection is today a hot topic in genetics. It consists in predicting breeding values of selection candidates, using the large number of genetic markers now available owing to the recent progress in molecular biology. One of the most popular methods chosen by geneticists is ridge regression. We focus on some predictive aspects of ridge regression and present theoretical results regarding the accuracy criteria, that is, the correlation between predicted value and true value. We show the influence of singular values, the regularization parameter, and the projection of the signal on the space spanned by the rows of the design matrix. Asymptotic results in a high‐dimensional framework are given; in particular, we prove that the convergence to optimal accuracy highly depends on a weighted projection of the signal on each subspace. We discuss on how to improve the prediction. Last, illustrations on simulated and real data are proposed.     

Restricted ridge estimator in generalized linear models: Monte Carlo simulation studies on Poisson and binomial distributed responses

It is known that collinearity among the explanatory variables in generalized linear models (GLMs) inflates the variance of maximum likelihood estimators. To overcome multicollinearity in GLMs, ordinary ridge estimator and restricted estimator were proposed. In this study, a restricted ridge estimator is introduced by unifying the ordinary ridge estimator and the restricted estimator in GLMs and its mean squared error (MSE) properties are discussed. The MSE comparisons are done in the context of first-order approximated estimators. The results are illustrated by a numerical example and two simulation studies are conducted with Poisson and binomial responses.     

Adaptive ridge estimator in a linear regression model with spherically symmetric error under constraint

Abstract

We consider adaptive ridge regression estimators in the general linear model with homogeneous spherically symmetric errors. A restriction on the parameter of regression is considered. We assume that all components are non negative (i.e. on the positive orthant). For this setting, we produce under general quadratic loss such estimators whose risk function dominates that of the least squares provided the number of regressors in the least fore.     

On the inadmssibelity of ridge estimator in a linear model

The necessary and sufficient conditions for the inadmissibility of the ridge regression is discussed under two different criteria, namely, average loss and Pitman nearness. Although the two criteria are very different, same conclusions are obtained. The loss functions considered in this article are th likelihood loss function and the Mahalanobis loss function. The two loss functions are motivated from the point of view of classification of two normal populations. Under the Mahalanobis loss it is demonstrated that the ridge regression is always inadmissible as long as the errors are assumed to be symmetrically distributed about the origin.     

PDCA循环原理在大学毕业生就业管理工作中的应用

如果把“大学毕业生就业工作”看作一个项目,那么应用项目质量管理理论中的PDCA循环原理,科学制定就业工作计划和目标,认真组织实施就业辅导和就业管理工作,及时检查就业结果,积极处理就业工作存在的问题,既可以强化当前大学毕业生就业工作的管理、也能够提高大学毕业生的就业质量。     

Sparse alternatives to ridge regression: a random effects approach

In a calibration of near-infrared (NIR) instrument, we regress some chemical compositions of interest as a function of their NIR spectra. In this process, we have two immediate challenges: first, the number of variables exceeds the number of observations and, second, the multicollinearity between variables are extremely high. To deal with the challenges, prediction models that produce sparse solutions have recently been proposed. The term ‘sparse’ means that some model parameters are zero estimated and the other parameters are estimated naturally away from zero. In effect, a variable selection is embedded in the model to potentially achieve a better prediction. Many studies have investigated sparse solutions for latent variable models, such as partial least squares and principal component regression, and for direct regression models such as ridge regression (RR). However, in the latter, it mainly involves an L₁ norm penalty to the objective function such as lasso regression. In this study, we investigate new sparse alternative models for RR within a random effects model framework, where we consider Cauchy and mixture-of-normals distributions on the random effects. The results indicate that the mixture-of-normals model produces a sparse solution with good prediction and better interpretation. We illustrate the methods using NIR spectra datasets from milk and corn specimens.     

A note on the moments of stochastic shrinkage parameters in ridge regression

The purpose of this note is to gain insight on the performance of two well known operational Ridge Regression estimators by deriving the moments of their stochastic shrinkage parameters. We also show that, under certain conditions, one of them has bounded moments.     

Residents,recreationists, ridge regression,and revenues:the five r's of florida's motor- tax

Ridge regression is applied to the question of how to revise Florida's motor-fuel tax to remedy the shortfall of its transportation revenues. Along the way a model of gasoline consumption by residents and tourists is presented, and the differences in consumption patterns between the two types of purchasers are reported.