Bayesian assessment of goodness of fit against nonparametric alternatives

The classical chi‐square test of goodness of fit compares the hypothesis that data arise from some parametric family of distributions, against the nonparametric alternative that they arise from some other distribution. However, the chi‐square test requires continuous data to be grouped into arbitrary categories. Furthermore, as the test is based upon an approximation, it can only be used if there are sufficient data. In practice, these requirements are often wasteful of information and overly restrictive. The authors explore the use of the fractional Bayes factor to obtain a Bayesian alternative to the chi‐square test when no specific prior information is available. They consider the extent to which their methodology can handle small data sets and continuous data without arbitrary grouping.     

The Mean,Median, and Confidence Intervals of the Kaplan-Meier Survival Estimate—Computations and Applications

This short note points out estimators of the mean, median, and the associated confidence intervals of the Kaplan-Meier product limit estimate. Some uses of the estimator of the mean are described. In addition, differences among popular software packages in the calculation of both the mean and median and associated confidence intervals are demonstrated and are due to default settings in the software. Simple examples of the calculations are provided using S-Plus, R, SAS, Stata, and SPSS.     

Construction of improved estimators of multinomial proportions

The improved large sample estimation theory for the probabilities of multi¬nomial distribution is developed under uncertain prior information (UPI) that the true proportion is a known quantity. Several estimators based on pretest and the Stein-type shrinkage rules are constructed. The expressions for the bias and risk of the proposed estimators are derived and compared with the maximum likelihood (ml) estimators. It is demonstrated that the shrinkage estimators are superior to the ml estimators. It is also shown that none of the preliminary test and shrinkage estimators dominate each other, though they perform y/ell relative to the ml estimators. The relative dominance picture of the estimators is presented. A simulation study is carried out to assess the performance of the estimators numerically in small samples.     

Smooth tests of fit for censored gmma samples

In this paper properties of two estimators of Cpm are investigated in terms of changes in the process mean and variance. The bias and mean squared error of these estimators are derived. It can be shown that the estimate of Cpm proposed by Chan, Cheng and Spiring (1988) has smaller bias than the one proposed by Boyles (1991) and also has a smaller mean squared error under certain conditions. Various approximate confidence intervals for Cpm are obtained and are compared in terms of coverage probabilities, missed rate and average interval width.     

Restricted Ridge Estimators of the Parameters in Semiparametric Regression Model

In this article, we introduce a ridge estimator for the vector of parameters β in a semiparametric model when additional linear restrictions on the parameter vector are assumed to hold. We also obtain the semiparametric restricted ridge estimator for the parametric component in the semiparametric regression model. The ideas in this article are illustrated with a data set consisting of housing prices and through a comparison of the performances of the proposed and related estimators via a Monte Carlo simulation.     

Two Kinds of Restricted Estimators in Singular Linear Regression

In this article, the parameter estimators in singular linear model with linear equality restrictions are considered. The restricted root estimator and the generalized restricted root estimator are proposed and some properties of the estimators are also studied. Furthermore, we compare them with the restricted unified least squares estimator and show their sufficient conditions under which their superior over the restricted unified least squares estimator in terms of mean squares error, and discuss the choice of the unknown parameters of the generalized restricted root estimator.     

Approximations to the powers of the likelihood ratio tests: the loop ordering and slippage alternatives

Approximations to the power functions of the likelihood ratio tests of homogeneity of normal means against the simple loop ordering at slippage alternatives are considered. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a simple loop ordering is appropriate. The accuracy of the several moment approximations are studied for the case of known variances and it is found that for powers in the range typically of interest, the two-moment approximation seems quite adequate. Approximations based on mixtures of noncentral F variables are developed for the case of unknown variances. The critical values of the test statistics are also tabulated for selected levels of significance.     

Questions & Answers: The Role of Mathematical Models in Clinical Research

Given any generalized inverse (X'X)^? appropriate to normal equations X'Xb ⁰ = X'y for the linear model y = Xb + e, a procedure is given for obtaining from it a generalized inverse appropriate to a restricted model having restrictions P'b = 0 for P'b nonestimable.     

Stepwise Regression in Mixed Quantitative Linear Models with Autocorrelated Errors

ABSTRACT

In the stepwise procedure of selection of a fixed or a random explanatory variable in a mixed quantitative linear model with errors following a Gaussian stationary autocorrelated process, we have studied the efficiency of five estimators relative to Generalized Least Squares (GLS): Ordinary Least Squares (OLS), Maximum Likelihood (ML), Restricted Maximum Likelihood (REML), First Differences (FD), and First-Difference Ratios (FDR). We have also studied the validity and power of seven derived testing procedures, to assess the significance of the slope of the candidate explanatory variable x ₂ to enter the model in which there is already one regressor x ₁. In addition to five testing procedures of the literature, we considered the FDR t-test with n ? 3 df and the modified t-test with n? ? 3 df for partial correlations, where n? is Dutilleul's effective sample size. Efficiency, validity, and power were analyzed by Monte Carlo simulations, as functions of the nature, fixed vs. random (purely random or autocorrelated), of x ₁ and x ₂, the sample size and the autocorrelation of random terms in the regression model. We report extensive results for the autocorrelation structure of first-order autoregressive [AR(1)] type, and discuss results we obtained for other autocorrelation structures, such as spherical semivariogram, first-order moving average [MA(1)] and ARMA(1,1), but we could not present because of space constraints. Overall, we found that:

1. the efficiency of slope estimators and the validity of testing procedures depend primarily on the nature of x ₂, but not on that of x ₁;

2. FDR is the most inefficient slope estimator, regardless of the nature of x ₁ and x ₂;

3. REML is the most efficient of the slope estimators compared relative to GLS, provided the specified autocorrelation structure is correct and the sample size is large enough to ensure the convergence of its optimization algorithm;

4. the FDR t-test, the modified t-test and the REML t-test are the most valid of the testing procedures compared, despite the inefficiency of the FDR and OLS slope estimators for the former two;

5. the FDR t-test, however, suffers from a lack of power that varies with the nature of x ₁ and x ₂; and

6. the modified t-test for partial correlations, which does not require the specification of an autocorrelation structure, can be recommended when x ₁ is fixed or random and x ₂ is random, whether purely random or autocorrelated. Our results are illustrated by the environmental data that motivated our work.

     

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.