Theory of Statistics - George Mason University

298 3 Basic Statistical Theory
which the sample was drawn. In the jackknife method, we compute the statistic
T using only a subset of size n − d of the given dataset; that is, we delete
a set of size d.
There are of course
C n d =
n
d
such sets.
Let T(−j) denote the estimator computed from the sample with the j th
set of observations removed; that is, T(−j) is based on a sample of size n − d.
The estimator T(−j) has properties similar to those of T. For example, if T is
unbiased, so is T(−j). If T is not unbiased, neither is T(−j); its bias, however,
is likely to be different.
The mean of the T(−j),
T (•) = 1
C n d
C n d
T(−j), (3.161)
can be used as an estimator of θ. The T(−j) may also provide some information
about the estimator T from the full sample.
For the case in which T is a linear functional of the ECDF, then T (•) =
T, so the systematic partitioning of a random sample will not provide any
additional information.
Consider the weighted differences in the estimate for the full sample and
the reduced samples:
T ∗ j = nT − (n − d)T(−j). (3.162)
The T ∗ j are called “pseudovalues”. (If T is a linear functional of the ECDF
and d = 1, then T ∗ j = T(xj); that is, it is the estimator computed from the
single observation, xj.)
We call the mean of the pseudovalues the “jackknifed” T and denote it as
J(T):
J(T) = 1
C n d
j=1
C n
d
j=1
T ∗ j
= T ∗ . (3.163)
In most applications of the jackknife, it is common to take d = 1, in which
case Cn d = n. The term “jackknife” is often reserved to refer to the case of
d = 1, and if d > 1, the term “delete d jackknife” is used. In the case of d = 1,
we can also write J(T) as
J(T) = T + (n − 1)
T − T(•)
or
Theory of Statistics c○2000–2013 James E. Gentle