Improving variance function estimation in semiparametric longitudinal data analysis

We propose an efficient and robust method for variance function estimation in semiparametric longitudinal data analysis. The method utilizes a local log‐linear approximation for the variance function and adopts a generalized estimating equation approach to account for within subject correlations. We show theoretically and empirically that our method outperforms estimators using working independence that ignores the correlations. The Canadian Journal of Statistics 39: 656–670; 2011. © 2011 Statistical Society of Canada     

Noninformative priors for linear combinations of normal means with unequal variances

For normal populations with unequal variances, we develop matching priors and reference priors for a linear combination of the means. Here, we find three second-order matching priors: a highest posterior density (HPD) matching prior, a cumulative distribution function (CDF) matching prior, and a likelihood ratio (LR) matching prior. Furthermore, we show that the reference priors are all first-order matching priors, but that they do not satisfy the second-order matching criterion that establishes the symmetry and the unimodality of the posterior under the developed priors. The results of a simulation indicate that the second-order matching prior outperforms the reference priors in terms of matching the target coverage probabilities, in a frequentist sense. Finally, we compare the Bayesian credible intervals based on the developed priors with the confidence intervals derived from real data.     

Empirical study of robust estimation methods for PAR models with application to the air quality area

Abstract

This paper compares three estimators for periodic autoregressive (PAR) models. The first is the classical periodic Yule-Walker estimator (YWE). The second is a robust version of YWE (RYWE) which uses the robust autocovariance function in the periodic Yule-Walker equations, and the third is the robust least squares estimator (RLSE) based on iterative least squares with robust versions of the original time series. The daily mean particulate matter concentration (PM10) data is used to illustrate the methodologies in a real application, that is, in the Air Quality area.     

Deletion residuals in the detection of heterogeneity of variances in linear regression

The heterogeneity of error variance often causes a huge interpretive problem in linear regression analysis. Before taking any remedial measures we first need to detect this problem. A large number of diagnostic plots are now available in the literature for detecting heteroscedasticity of error variances. Among them the ‘residuals’ and ‘fits’ (R–F) plot is very popular and commonly used. In the R–F plot residuals are plotted against the fitted responses, where both these components are obtained using the ordinary least squares (OLS) method. It is now evident that the OLS fits and residuals suffer a huge setback in the presence of unusual observations and hence the R–F plot may not exhibit the real scenario. The deletion residuals based on a data set free from all unusual cases should estimate the true errors in a better way than the OLS residuals. In this paper we propose ‘deletion residuals’ and the ‘deletion fits’ (DR–DF) plot for the detection of the heterogeneity of error variances in a linear regression model to get a more convincing and reliable graphical display. Examples show that this plot locates unusual observations more clearly than the R–F plot. The advantage of using deletion residuals in the detection of heteroscedasticity of error variance is investigated through Monte Carlo simulations under a variety of situations.     

Do people think about absolute or relative price differences when choosing between substitute goods?

The article presents experimental evidence that shows that people often consider relative price differences in addition to absolute differences when choosing between substitute goods. Because the choice between substitute goods is a very common one, this is an important finding. The experiment uses scenarios in various consumption categories: hotel rooms, flights, and books. Subjects were either students or participants in an economics conference. The data allow to reject the hypothesis that people think only about relative price differences in favor of the hypothesis that people think about both relative and absolute price differences. Whether the price given to the subjects is that of the high-quality good or of the low-quality good makes a large difference, a result that is related to the endowment effect and the status quo bias. Implications of the results for business strategy and other areas are also discussed.     

An F Approximation to the Distribution of a Linear Combination of Chi-squared Variables.

A three-parameter F approximation to the distribution of a positive linear combination of central chi-squared variables is described. It is about as easy to implement as the Satterthwaite-Welsh and Hall-Buckley-Eagleson approximations. Some reassuring properties of the F approximation are derived, and numerical results are presented. The numerical results indicate that the new approximation is superior to the Satterthwaite approximation and, for some purposes, better than the Hall-Buckley-Eagleson approximation. It is not quite as good as the Gamma-Weibull approximation due to Solomon and Stephens, but is easier to implement because iterative methods are not required.     

First‐order bias correction for fractionally integrated time series

Most of the long memory estimators for stationary fractionally integrated time series models are known to experience non‐negligible bias in small and finite samples. Simple moment estimators are also vulnerable to such bias, but can easily be corrected. In this article, the authors propose bias reduction methods for a lag‐one sample autocorrelation‐based moment estimator. In order to reduce the bias of the moment estimator, the authors explicitly obtain the exact bias of lag‐one sample autocorrelation up to the order n⁻¹. An example where the exact first‐order bias can be noticeably more accurate than its asymptotic counterpart, even for large samples, is presented. The authors show via a simulation study that the proposed methods are promising and effective in reducing the bias of the moment estimator with minimal variance inflation. The proposed methods are applied to the northern hemisphere data. The Canadian Journal of Statistics 37: 476–493; 2009 © 2009 Statistical Society of Canada