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CHAP1 - CHAPTER 1 Introduction 1.1 Background The first...

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CHAPTER 1 Introduction 1.1 Background The first sentence of I. R. Savage’s book, Statistics: Uncertainty and Behavior (1968), is: “Probabilities are quantitative expressions of uncertainty.” Thus, for Savage, the term uncertainty denotes a more general concept than the principle of quantum mechanics that earned W. K. Heisenberg a 1932 Nobel Prize. But what exactly is statistical uncertainty? One of the many possible answers to this question concerns what Savage calls “quantitative expressions.” Curves per se and quantitative expressions for curves can be matched well or matched poorly. For this reason we place much emphasis on the distinctions and connections between model parameters and curve properties. Parameters are symbols within some mathematical expression that may vary between problems, but not within a given context or problem. In this sense they are representational devices that are intermediates between constants and variables. An early recorded application of a parameter was published by the seventeenth century mathematician and philosopher René Descartes (Bell, 1937). Descartes realized that the concept diameter , when applied to a circle, resembled the para meter of a para bola, the curve property "latus rectum." Then, as now, the notational linkage of two different entities, the first a symbol and the second a curve property like the diameter or latus rectum of a particular curve, was a major advantage of the parametric approach. The word parameter sounds like the word perimeter , yet the role played by any parameter is contrary to that played by any perimeter. Rather than defining boundaries, parameters extend flexibility. When a mathematical expression does not contain any parameters, it is completely specified to the degree that it can be tabled. On the other hand, when such an expression contains unspecified parameters, it corresponds to a whole spectrum or family of tables, one for each distinct set of parameter values. DESCARTES
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For any statistical analysis, it was once considered essential to reduce the problem under consideration to the estimation or testing of a handful of parameters or curve properties. Today, because modern computational procedures can provide thousands of times more representational scope than was available merely a few years ago, this reduction is less important than it once was. There now is a rich palette of distinct model, parameter, and curve property combinations. As is natural when there are many strategies possible, some may be more appropriate or more effective than others. In addition, some may be more difficult than others from a computational or conceptual viewpoint. To choose among strategies, one can call upon the venerable fourteenth century approach called the law of parsimony : adopt “...the simplest assumption in the formulation of a theory or in the interpretation of data, especially in accordance with the rule of Ockham's razor” ( Compton’s Interactive Encyclopedia , 1993, 1994).
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