Reconciling some apparently incompatible approaches to statistical inference
Creators: name: Brewer, Kenneth Ronald Walter
Publisher: name: The Australian National University
Year: 1996

description: This thesis consists of an Introduction and two Topics. The Introduction deals with the underlying theme of the thesis and why these two particular topics were chosen to illustrate it. It is argued that the contradictions we meet in statistics (as in the physical sciences) may only appear to be such as a result of our imperfect understanding, and that in consequence it is often healthy to hold them in a creative tension. Topic A deals with the design-based and model-based approaches to sampling inference in the context of business surveys. Each has its strong advocates and those who hold to either view typically regard the other as misguided and worthless. It is shown that both have their advantages and disadvantages; also that it is possible to devise schemes which enable both to be used at the same time, reaping nearly all the advantages and avoiding nearly all the disadvantages of both, and obtaining new insights and options into the bargain. Part I is almost purely expository, but Parts II and III indicate how the synthesis can be achieved, in the contexts of stratified and unequal probability designs respectively. Part m includes an empirical study. Part IV provides an assessment of what can be achieved by using the two approaches simultaneously. Topic B deals principally with the disparate approaches used by Frequentist and Bayesian statisticians to the testing of a precise null hypothesis against a composite alternative. It is shown that starting from identical evidence they can easily come to opposite conclusions as to the choice that should be made between the two, particularly when the problem is ill-specified, but also even when it is not. The author was trained as a Frequentist, and although now leaning to the view that Bayesian inference, when correctly applied, can usually supply a deeper insight, he is acutely aware of certain areas where current Bayesian practice is wholly inadequate. The current Bayesian procedure for testing a precise null hypothesis against a composite alternative is the most remarkable instance of this failure, and it turns out that when a more robust Bayesian test is specified, the relevant test statistic is closely related to that already used by Frequentists. It is argued that, properly understood, the two approaches are both valid, and each can provide useful insights for the other. The hypothesis testing problem is here approached via Stone's (1976) notorious Treasure Hunt in Flatland, which caught his Bayesian contemporaries sadly off-balance. It transpires that there is actually a better solution to Stone's problem than the one used by his Classical Statistician, and that it can be found straightforwardly using "empirical Bayes." A fortunate ambiguity in Stone's own diagram enables both the strengths and the weaknesses of Bayesian inference to be exhibited. If (as in Part I) the Treasure Layers followed a clockwise path through a certain loop, the relevant hypothesis test is between two precise hypotheses, a task that current Bayesian practice can handle quite readily. If, however, the path they followed included an anticlockwise loop (as in Part II), the test is between a precise null and a potentially diffuse alternative. Current Bayesian practice in this situation requires a subjective specification of the alternative prior, and it is shown that this can lead to very different answers, some of which are highly counterintuitive. The properties of the newly specified Bayesian test statistic are considered in Part III, and this is where the relationship with Frequentist inference is discovered. Once again, the conclusion is that the apparent contradictions are best held in creative tension.