Discussion of “ancillaries and Conditional Inference” by D.a.s. Fraser

1. THE EVOLUTION OF STATISTICAL THEORY Reliance on probability theory in statistical writing spans the spectrum from none, to fixed effects models, to random effects models, to Bayesian reasoning. One factor is the extent to which an author regards probability as a feature of the natural world. For a Bayesian, probability measures the strength of opinions, which are modeled by a sigma algebra. At the other end of the spectrum, illustrated by J. Tukey’s (1977) Exploratory Data Analysis, data-analytic algorithms are basic reality and probability models are hypothetical constructs. A second factor is the technological environment in which an author is writing. Until the late 1950’s, the tools available to a statistician consisted of mathematics, logic, mechanical calculators, and simple computers. Because calculation was laborious, writers on statistical theory thought in terms of virtual data governed by probability models involving relatively few parameters. Indeed, the great intellectual advances made in probability theory during the twentieth century made this approach the technology of choice. Thus, the hotly debated statistical theories formulated in A. Wald’s (1950) Statistical Decision Functions, R. A. Fisher’s (1956) Statistical Methods and Scientific Inference, and L. J. Savage’s (1954) The Foundations of Statistics shared a common reliance on relatively simple probability models. After 1960, results on weak convergence of probability measures provided the technology for major development of asymptotic theory in statistics. Notable achievements by 1970 included: (a) the clarification of what is meant by asymptotic optimality; (b) the understanding, through Le Cam’s work, that risks in simple parametric models can approximate risks in certain more general models; (c) the discovery of superefficient estimators whose asymptotic risk undercuts the information bound on sets of Lebesgue measure zero; and (d) the remarkable discovery, through the James-Stein estimator, that superefficient estimators for