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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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).
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Tarter
 Parametric equation, Mathematical Expression, curve properties

Click to edit the document details