Theory of Statistics - George Mason University

312 3 Basic Statistical Theory
an unknown parameter. In that case, we can set a confidence interval based
on the approximate distribution of g(Xn) as N(g(c), v/a 2 n).
To speak of the asymptotic distribution of an(g(Xn) − g(c)) is clear; but
to refer to “the” asymptotic distribution of g(Xn) is somewhat less so.
Because it is the useful approximate distribution resulting from asymptotic
expectations, we often say that “the asymptotic distribution” of g(Xn)
is N(g(c), v/a 2 n). You should recognize that “the” in this statement is somewhat
arbitrary. It might be better to call it “the asymptotically approximate
distribution that I’m going to use in this application”.
Again, we should distinguish “asymptotic” from “limiting”.
In the example of the delta method above, it is likely that
g(Xn) d → g(c);
that is, g(Xn) converges in distribution to the constant g(c); or the limiting
distribution of g(Xn) is degenerate at g(c). “The” asymptotic variance is 0.
**** discuss expansion of statistical functionals *** refer to Serfling
This would not be very useful in asymptotic inference. We therefore seek
“an” asymptotic variance that is more useful. In asymptotic estimation using
g(Xn), we begin with an expression of the form an(g(Xn) − g(c)) that has
a limiting distribution of the desired form (usually that means such that the
variance does not involve any unknown parameters and it does not involve n).
If this distribution is in a location-scale family, then we make the appropriate
linear transformation (which probably results in a variance that does involve
n).
We then often refer to this as the asymptotic distribution of g(Xn). Sometimes,
as mentioned above, however, the limiting distribution of g(Xn) is
degenerate.
This is not to imply that asymptotic expectations are entirely arbitrary.
Proposition 2.3 in MS2 shows that there is a certain uniqueness in the asymptotic
expectation. This proposition involves three cases regarding whether the
expectation of g(Xn) (without the an sequence) is 0. In the example above,
we have a degenerate distribution, and hence the asymptotic expectation that
defines the asymptotic variance is 0.
3.8.4 Properties of Estimators of a Variance Matrix
If the statistic is a vector, we need an estimator of the variance-covariance matrix.
Because a variance-covariance matrix is positive definite, it is reasonable
to consider only estimators that are positive definite a.s.
We have defined what it means for such an estimator to be consistent
(Definition 3.18 on page 305).
Theorem 3.16
conditions for the consistency of substitution estimators.
Theory of Statistics c○2000–2013 James E. Gentle