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inductive
generalization about a population based upon our analysis of a
sample, we are on shakier ground. It is one thing to assert that if an
observation comes from a normal distribution with mean zero, the proba-
bility is one-half that it is positive. It is quite another if, on observing that
half the observations in the sample are positive, we assert that half of all
the possible observations that might be drawn from that population will
be positive also.
Newton's Law of gravitation provided an almost exact fit (apart from
measurement error) to observed astronomical data for several centuries;
consequently, there was general agreement that Newton's generalization
from observation was an accurate description of the real world. Later, as
improvements in astronomical measuring instruments extended the range
of the observable universe, scientists realized that Newton's Law was only
a generalization and not a property of the universe at all. Einstein's
Theory of Relativity gives a much closer fit to the data, a fit that has not
been contradicted by any observations in the century since its formulation.
But this still does not mean that relativity provides us with a complete,
correct, and comprehensive view of the universe.
In our research efforts, the only statements we can make with God-like
certainty are of the form “our conclusions fit the data.” The true nature of
the real world is unknowable. We can speculate, but never conclude.
LOSSES
In our first advanced course in statistics, we read in the first chapter of
Lehmann [1986] that the “optimal” statistical procedure would depend
on the losses associated with the various possible decisions. But on day
one of our venture into the real world of practical applications, we were
taught to ignore this principle.
At that time, the only computationally feasible statistical procedures
were based on losses that were proportional to the square of the difference
between estimated and actual values. No matter that the losses really
might be proportional to the absolute value of those differences, or the
cube, or the maximum over a certain range. Our options were limited by
our ability to compute.
Computer technology has made a series of major advances in the past
half century. What required days or weeks to calculate 40 years ago takes
only milliseconds today. We can now pay serious attention to this long
neglected facet of decision theory: the losses associated with the varying
types of decision.
Suppose we are investigating a new drug: We gather data, perform a
statistical analysis, and draw a conclusion. If chance alone is at work yield-
ing exceptional values and we opt in favor of the new drug, we've made
