Theory of Statistics - George Mason University

224 3 Basic Statistical Theory
Parametric-support families (or “truncation families”) have simple range
dependencies. A distribution in any of these families has a PDF in the general
form
fθ(x) = c(θ)g(x)IS(θ)(x).
The most useful example of distributions whose support depends on the
parameter is the uniform U(0, θ), as in Example 3.7. Many other distributions
can be transformed into this one. For example, consider X1, . . ., Xn iid
as a shifted version of the standard exponential family of distributions with
Lebesgue PDF
e −(x−α) I]α,∞[(x),
and Yi = e −Xi and θ = e −α , then Y1, . . ., Yn are iid U(0, θ); hence if we can
handle one problem, we can handle the other. We can also handle distributions
like U(θ1, θ2) a general shifted exponential, as well as some other related
distributions, such as a shifted gamma.
We can show completeness using the fact that
A
|g| dµ = 0 ⇐⇒ g = 0 a.e. on A. (3.29)
Another result we often need in going to a multiparameter problem is Fubini’s
theorem.
The sufficient statistic in the simple univariate case where S(θ) = (θ1, θ2)
is T(X) = (X(1), X(n)), as we can see from the the factorization theorem by
writing the joint density of a sample as
c(θ)g(x)I]x (1),x (n)[(x).
For example, for a distribution such as U(0, θ) we see that X(n) is sufficient
by writing the joint density of a sample as
1
θ I]0,x (n)[.
Example 3.8 complete sufficient statistics in a two-parameter exponential
distribution
In Examples 1.11 and 1.18, we considered a shifted version of the exponential
family of distributions, called the two-parameter exponential with parameter
(α, θ). The Lebesgue PDF is
θ −1 e −(x−α)/θ I]α,∞[(x)
Suppose we have observations X1, X2, . . ., Xn.
In Examples 1.11 and 1.18, we worked out the distributions of X(1) and
Xi − nX(1). Now, we want to show that T = (X(1), Xi − nX(1)) is
sufficient and complete for (α, θ).
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Theory of Statistics c○2000–2013 James E. Gentle