Comparison of naïve,Kenward–Roger,and parametric bootstrap interval approaches to small-sample inference in linear mixed models

In mixed models the mean square error (MSE) of empirical best linear unbiased estimators generally cannot be written in closed form. Unlike traditional methods of inference, parametric bootstrapping does not require approximation of this MSE or the test statistic distribution. Data were simulated to compare coverage rates for intervals based on the naïve MSE approximation and the method of Kenward and Roger, and parametric bootstrap intervals (Efron's percentile, Hall's percentile, bootstrap-t). The Kenward–Roger method performed best and the bootstrap-t almost as well. Intervals were also compared for a small set of real data. Implications for minimum sample size are discussed.     

The parametric hypothesis test of multiple uniform populations

X₁, X₂, …, X_k are k(k ? 2) uniform populations which each X_i follows U(0, θ_i). This note shows the test statistic for the null hypothesis H₀: θ₁ = θ₂ = ??? = θ_k by using the order statistics.     

Why are mediation effects so small?

Identifying mediators in variable chains as part of a causal mediation analysis can shed light on issues of causation, assessment, and intervention. However, coefficients and effect sizes in a causal mediation analysis are nearly always small. This can lead those less familiar with the approach to reject the results of causal mediation analysis. The current paper highlights five factors that contribute to small path coefficients in mediation research: loss of information when measuring relationships across time, controlling for prior levels of a predicted variable, adding control variables to the analysis, ignoring measurement error in one’s variables, and using multiple mediators. It is argued that these issues are best handled by increasing the statistical power of the analysis, identifying the optimal temporal interval between variables, using bootstrapped confidence intervals to analyze the results, and finding alternate ways of assessing the meaningfulness of the indirect effect.     

中国教育收益率统计估计与分析 ——基于参数和半参数估计方法的比较

 本文分别运用参数和半参数估计方法,就如何更准确有效测度中国教育收益率作深入探讨。利用CHNS数据对我国1989年至2006年的教育收益率进行估计,并采用Hausman检验法对两种估计方法结果进行检验。结果表明,从估计效率看,前者比后者的效率更高;但从估计效果看,后者才是一致性估计。而且我们发现虽然教育收益率整体呈逐渐增加趋势,但与实物资本收益率相比仍偏低。中国的教育和劳动力市场亟需加大投资和改革力度。     

A fiducial p-value approach for comparing heteroscedastic regression models

To study the equality of regression coefficients in several heteroscedastic regression models, we propose a fiducial-based test, and theoretically examine the frequency property of the proposed test. We numerically compare the performance of the proposed approach with the parametric bootstrap (PB) approach. Simulation results indicate that the fiducial approach controls the Type I error rates satisfactorily regardless of the number of regression models and sample sizes, whereas the PB approach tends to be a little of liberal in some scenarios. Finally, the proposed approach is applied to an analysis of a real dataset for illustration.     

Simulation-Based Confidence Intervals for Functions With Complicated Derivatives

In many scientific problems, the quantity of interest is a function of parameters that index the model, and confidence intervals are constructed by applying the delta method. However, when the function of interest has complicated derivatives, this standard approach is unattractive and alternative algorithms are required. This article discusses a simple simulation-based algorithm for estimating the variance of a transformation, and demonstrates its simplicity and accuracy by applying it to several statistical problems.     

Power of One-Sample Location Tests Under Distributions with Equal Lévy Distance

In this article, we study the power of one-sample location tests under classical distributions and two supermodels which include the normal distribution as a special case. The distributions of the supermodels are chosen in such a way that they have equal distance to the normal as the logistic, uniform, double exponential, and the Cauchy, respectively. As a measure of distance we use the Lévy metric. The tests considered are two parametric tests, the t-test and a trimmed t-test, and two nonparametric tests, the sign test and the Wilcoxon signed-rank tests. It turns out that the power of the tests, first of all, does not depend on the Lévy distance but on the special chosen supermodel.     

Comparison of two methods in estimating standard error of the method of simulated moments estimators for generalized linear mixed models

ABSTRACT

The likelihood of a generalized linear mixed model (GLMM) often involves high-dimensional integrals, which in general cannot be computed explicitly. When direct computation is not available, method of simulated moments (MSM) is a fairly simple way to estimate the parameters of interest. In this research, we compared parametric bootstrap (PB) and nonparametric bootstrap methods (NPB) in estimating the standard errors of MSM estimators for GLMM. Simulation results show that when the group size is large, the PB and NPB perform similarly; when group size is medium, NPB performs better than PB in estimating standard errors of the mean.     

Nonparametric Estimation of the Multivariate Distribution Function in a Censored Regression Model with Applications

In a regression model with univariate censored responses, a new estimator of the joint distribution function of the covariates and response is proposed, under the assumption that the response and the censoring variable are independent conditionally to the covariates. This estimator is based on the conditional Kaplan–Meier estimator of Beran (1981 Beran , R. ( 1981 ). Nonparametric regression with randomly censored survival data. Technical Report, University of California, Berkeley, California . [Google Scholar]), and happens to be an extension of the multivariate empirical distribution function used in the uncensored case. We derive asymptotic i.i.d. representations for the integrals with respect to the measure defined by this estimated distribution function. These representations hold even in the case where the covariates are multidimensional under some additional assumption on the censoring. Applications to censored regression and to density estimation are considered.