This chapter outlines parametric empirical Bayes confidence intervals. Empirical Bayes modeling assumes the distributions π for the parameters θ= (θ1, …, θk) exist, with π taken from a known class Π of possible parameter distributions. Π is considered independent N (u, A) distributions on Rk. It is called parametric empirical Bayes problem, because πɛ Π is determined by the parameters (u, A) and so is a parametric family of distributions. A simulation presented in the chapter was used to determine that the intervals ▪ ±si and ▪ ±1.96si contain the true values θi in at least 68 percent and 95 percent of the cases. Empirical Bayes estimators, or Stein's estimator, can lead to misestimation of components that the statistician or his clients care about when exchangeability in the prior distribution is implausible. The term empirical Bayes, which is used for non-parametric empirical Bayes problems, actually fits the parametric empirical Bayes case too. The empirical Bayes methods in general and parametric empirical Bayes methods in particular provide a way to utilize this additional information by obtaining more precise estimates and estimating their precision.

The “what if” method uses frequentist computations as a way of thinking about prior distributions. Several applications to priors on infinite dimensional parameter spaces give backing to the widely held belief that it is hard to understand priors on large spaces. Examples include Dirichlet priors, and priors on the continuous functions. In both settings, new examples of inconsistent Bayes rules are given.

This chapter reviews autocorrelation-robust design of experiments. There has been considerable recent research interest in designing experiments under error processes, which have spatial autocorrelation. The paper by Kiefer and Wynn gives a thorough foundation for optimum treatment design in one dimension in the presence of autocorrelated errors. A generalization of the Shepp result on mixing allows us to claim that the full Π-set is the convex hull of the 6 extreme π-vectors. However, these extreme patterns do not seem to have been exhibited before. Pseudo random patterns by Kiefer and Wynn show how full-length error correcting codes can be used to generate sequences with πr = ▪ (r = 1, …, p) that is the same π-values as would be obtained (with probability one) by an infinite Bernoulli sequence. A trivial method of extending the πr = ▪ property to higher dimensions is to take the Kronecker sum of the codes with respect to the group GF(qm). Two-dimensional pattern with π = (½, ½, ½, ½) is easily seen by taking the Kronecker sum of the code 001 (0).

This chapter discusses frequentist inferences. The major operational difference between Bayesian and frequentist inferences is that in the latter, one must choose a reference set for the sample to obtain inferential probabilities. In the matter of choosing a reference set, Sherlock Holmes was right, and that many frequentist inferences are inadequate because of erroneous choices made prior to the experiment. Most of the mathematical development has to do with predata analysis. However, the question remain was that, “Is such-and-such likely to be a good procedure?” It is evident that relative frequency in a real sequence of repeated experiments is simply not a rich enough interpretation of probability. The brief general discussion has raised questions about the difference between pre- and postdata probability calculations, the legitimacy of exact theory when integrated with practical application, and careful specification and understanding of what a statistical model means in practical terms. The purpose of the more detailed discussion which follows is to consider four topics where naive application of frequentist statistical theory can lead to incorrect or unhelpful inferences, whereas careful attention to the above questions can lead to sensible frequentist inferences. The four topics to be discussed are likelihood inference, inference from transformed data, randomization in design of experiments and surveys, and robust estimation. The Bayesian analysis has the general advantage of responding to specific features of the data. The unconditional distribution theory prevalent in robustness literature is fine for choosing estimates which are generally good but not for analysis of a particular data set.

This chapter presents a case study of the robustness of Bayesian methods of inference estimating the total in a finite population using transformations to normality. Bayesian intervals provide interval estimates that can legitimately be interpreted as such or at least to offer guidance as to when the intervals that are provided can be safely interpreted in this manner. The potential application of the statistical methods is often demonstrated either theoretically, from artificial data generated following some convenient analytic form, or from real data without a known correct answer. The case study presented here uses a small, real data set with a known value for the quantity to be estimated. It is surprising and instructive to see the care that may be needed to arrive at satisfactory inferences with real data. Simulation techniques are not needed to estimate totals routinely in practice. If stratification variables were available, that is, categorizing the municipalities into villages, towns, cities, and boroughs of New York City, to estimate the population total from a sample of 100, oversampling the large municipalities would be highly desirable. Robustness is not a property of data alone or questions alone, but particular combinations of data, questions and families of models. In many problems, statisticians may be able to define the questions being studied so as to have robust answers.

This chapter reviews the statistical aspects of some philosophies of inference and modeling. Frequentist model-checking needs few assumptions about alternative models, while Bayesian assumptions always reduce to a grand model involving models across models. An attempt was made to extend the likelihood principle ideas from the inferential to the modeling situation. However, the most important topic at issue has been as to whether and how a model should be checked out against the data, with respect to a completely general alternative. At some point in the analysis, it is necessary to make model assumptions, which cannot be checked out against the data but which relate either to the scientific background or the need to provide some sensible conclusions. It is not the model itself which should be checked but the scientific conclusions given the model. The model is itself an artifact, which enables to think more closely about the data in relation to its scientific background. The validity of the scientific conclusions given the model can only be discussed by reference to the scientific background and to the plausibility of the intuitive thought underlying the statistical analysis; an expert judgment is called for. However, a good future for pragmatic Bayesians is seen, who are more concerned with the extraction of meaningful practical conclusions and the interpretation of statistical data in relation to their scientific background and who are prepared to think more broadly than permitted by the constraints of mathematical axioms.

This chapter focuses on purposes and limitations of data analysis. Data analysis consists of the middle sequence of steps between design and decision making, wherein knowledge obtained from data is described and quantified. One of the tensions in the field of data analysis is captured by the alternative terms exploratory data analysis (EDA) and statistical modeling (SM). The essence of EDA or SM is data reduction and manipulation so as to extract and exhibit comprehensible structure. Both EDA and SM cycle back and forth between model fitting procedures and diagnostic model-checking procedures. EDA generally starts the process with summaries and displays, which means starting on the diagnostic side of the cycle rather than the fitting side. While the nonprobabilistic tools and skills of EDA are necessary for statistical practice, they are only sometimes sufficient. Computing solutions require hardware and software, of course, but theoreticians are greatly needed for mathematical, numerical, and statistical analyses associated with algorithm development. Computing problems appear in nonprobabilistic EDA, in retrospective CDA, in prospective CDA, and in IRDA. Instruction in statistics badly needs to convey not only technical content but also a real sense of the functional contributions of data analysis technology to resolving borderline issues in many sciences and professions.

This chapter explores the estimation of variance of the ratio estimator under simple random sampling. The majority of estimators are design-based, that is, their justification and choice are based on the performance according to the probability mechanism that generates the sample. The inference should be made conditional on the observed sample and a hypothetical super population model. The sampling design becomes irrelevant. Other estimators, for example, the jackknife, may not be justified exclusively by either approach. The theoretical comparison of various variance estimators is made by assuming that the x and y populations satisfy some linear regression models or super populations. Although the results are sometimes exact, dependence on the super population parameters can be delicate. The study of estimation of variance of the ratio estimator is restricted to the bias behavior of the variance estimators when the true parameters deviate from the assumed ones. In estimating the population mean, the purpose of variance estimation is for assessing the variability of the ratio estimator rather than for estimating the variance itself. Variance estimators that estimate the conditional MSE of the ratio estimator better tend to give more reliable interval estimates of the population mean.

This chapter presents an apology for Ecumenism in statics. Bays model had the difficulty that by supposing all possible sets of assumptions known a priori, it discredits the possibility of new discovery. It had been argued that if significance tests are to be employed to check the model, then it is necessary to state in advance the level of significance α that is to be used and that no rational basis exists for making such a choice. The present proposals excluded the possibilities of attempts to base estimation on sampling theory, using point estimates and confidence intervals and attempts to base criticism and hypothesis testing entirely on Bayesian theory. The Bayes' approach provides some assurance against incredibility as it requires that all assumptions of the model be clearly visible and available for criticism. The scientific method employs and requires not one but two kinds of inference, namely, criticism and estimation; once this is understood, the statistical advances made in recent years in Bayesian methods, data analysis, robust and shrinkage estimators can be seen as a cohesive whole.

This chapter focuses on the robustness of a hierarchical model for multinomial and contingency tables. Hierarchical Bayesian models can be used in a completely Bayesian manner or in a manner that has been called semi-, or pseudo-, or quasi Bayesian. In the latter case, the hyperparameters, or possibly the hyperhyperparameters, etc., are estimated by non-Bayesian methods, or at least by some not purely Bayesian method such as by maximum likelihood. In the so-called non-Bayesian statistics, the use of the Ockham-Duns razor is sometimes called the principle of parsimony, and it encourages one to avoid having more parameters than are necessary. In hierarchical Bayesian methods, one similarly uses a principle of parsimony or hyper-razor. There are at least two different ways to test a model. One is by means of significance tests after observations are made. Another is by examining the robustness of a model, that is, by seeing if small changes in the model lead to small changes in the implications. Tests for robustness can sometimes be carried out by the device of imaginary results before making observations. When calculating a Bayes factor F1, reasonable robustness with respect to the choice of hyperhyperparameters has also been found.