Theory of Statistics - George Mason University

204 3 Basic Statistical Theory
“confidence” and “significance”. The formulation of the problem of statistical
inference as in equations (3.3) and (3.4) avoids a direct confrontation of the
question of how likely the decision is to be correct or incorrect. The conclusion
in that kind of setup is the simple statement that the conditional distribution
of Θ is QH, given that its marginal distribution is Q0 and the conditional
distribution of the data is PΘ.
Examples
As an example of the approach indicated in equations (3.1) and (3.2), assume
that a given sample y1, . . ., yn is taken in some prescribed manner from some
member of a family of distributions
P = {N(µ, σ 2 ) | µ ∈ IR, σ 2 ∈ IR+}.
Statistical inference in this situation may lead us to place the population
giving rise to the observed sample in the family of distributions
PH = {N(µ, σ 2 ) | µ ∈ [µ1, µ2], σ 2 ∈ IR+}
(think confidence intervals!). The process of identifying the subfamily may be
associated with various auxiliary statements (such as level of “confidence”).
As another example, we assume that a given sample y1, . . ., yn is taken
independently from some member of a family of distributions
P = {P | P ≪ λ},
where λ is the Lebesgue measure, and our inferential process may lead us to
decide that the sample arose from the family
t
PH = P | P ≪ λ and dP = .5 ⇒ t ≥ 0
(think hypothesis tests concerning the median!).
Notice that “P” in the example above is used to denote both a population
and the associated probability measure; this is a notational convention that
we adopted in Chapter 1 and which we use throughout this book.
Statistical inference following the setup of equations (3.3) and (3.4) can
be illustrated by observable data that follows a Bernoulli distribution with a
parameter Π which, in turn, has a beta marginal distribution with parameters
α and β. (That is, in equation (3.3), Θ is Π, PΠ is a Bernoulli distribution with
parameter Π, and Q0 is a beta distribution with parameters α and β.) Given
the single observation X = x, we can work out the conditional distribution of
Π to be a beta with parameters x + α and 1 − x + β.
It is interesting to note the evolutionary nature of this latter example.
Suppose that we began with Q0 (of equation (3.3)) being a beta with parameters
x1 + α and 1 − x1 + β, where x1 is the observed value as described
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Theory of Statistics c○2000–2013 James E. Gentle