Sample design for observing classes of objects

We consider optimal sample designs for observing classes of objects. Suppose we will take a simple random sample of equal-sized sectors from a study population and observe the classes existing on these sectors. The classes might be many different things, for example, herbaceous plant species (in sampling quadrats), microinvertebrate species (in sampling cores), and side effects from a drug (in conducting medical trials). Under a nonparametric mixture model and data from a previous related study, we first estimate the optimal number of sample sectors of a given size. Then for negative binomial dispersions of individuals with a common aggregation parameter k, we consider the optimal size as well as number of sample sectors. A simple test exists to check our common k assumption and our optimal size method requires far less data than would be required by a grid method or other method which utilizes data from sample sectors of several different sizes.     

A note on second order slope rotatable designs over all directions

Hader and Park (1978) introduced second order slope rotatability in axial directions. Park (1987) introduced second order slope rotatabilty over all directions. It is shown that these designs have the additional properly that the sum of the variances of estimates of slopes in all directions at any point is a function of the distance of the point from the design origin.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.     

The effects of covariate adjustment in generalized linear models

Results from classical linear regression regarding the effects of covariate adjustment, with respect to the issues of confounding, the precision with which an exposure effect can be estimated, and the efficiency of hypothesis tests for no treatment effect in randomized experiments, are often assumed to apply more generally to other types of regression models. In this paper results pertaining to several generalized linear models involving a dichotomous response variable are given, demonstrating that with respect to the issues of confounding and precision, for models having a linear or log link function the results of classical linear regression do generally apply, whereas for other models, including those having a logit, probit, log-log, complementary log-log, or generalized logistic link function, the results of classical linear regression do not always apply. It is also shown, however, that for any link function, covariate adjustment results in improved efficiency of hypothesis tests for no treatment effect in randomized experiments, and hence that the classical linear regression results regarding efficiency do apply for all models having a dichotomous response variable.     

The Relative Size of Windfall Income and the Permanent Income Hypothesis

A reconciliation is offered for the diverse test results on Friedman's permanent income hypothesis. A large data sample of those receiving windfall income in the Bureau of Labor Statistics' 1972–1973 Consumer Expenditure Survey is divided according to the size of the windfall relative to estimated permanent income. A pattern of a declining marginal propensity to consume windfall income as the relative size of the windfall increases is apparent. These results support the permanent income hypothesis for relatively large windfalls.     

On the robustness of the extreme deviate test for a single multivariate outlier against heavy-tailed distributions

In this paper we assess the sensitivity of the multivariate extreme deviate test for a single multivariate outlier to non-normality in the form of heavy tails. We find that the empirical significance levels can be markedly affected by even modest departures from multivariate normality. The effects are particularly severe when the sample size is large relative to the dimension. Finally, by way of example we demonstrate that certain graphical techniques may prove useful in identifying the source of rejection for the multivariate extreme deviate test.     

Testing a main effect in a three factor mixed model

In the present paper, the author has considered and compared the power of three test procedures, based on two preliminary tests of significance (PTS), for testing a main effect in a three factor factorial experiment. Davenport and Webster (1973) type test statistics are used in the final test. Recommendation for a suitable size of PTS has been made so that the power of a test procedure is adequately high.     

Shape properties of the g-and-h and johnson families

A substantial part of examining the properties of a distributional family consists of considering shape properties. It is important that this examination is sufficiently thorough to enable understanding of the behaviour of the family, its comparison with others, and to assist in developing future families. The g-and-h distributions and the Johnson system are examined here in these terms     

Saddlepoint-Based Bootstrap Inference for Quadratic Estimating Equations

Abstract.  We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.