Information Technology Reference
In-Depth Information
As a practical matter, we would draw an element from an
N
(0,1) popula-
tion, multiply by the sample standard deviation, and then add the sample
mean to obtain an element of our bootstrap sample. By computing the
standard deviation of each bootstrap sample, an interval estimate for the
standard deviation of the population may be derived.
IMPROVED RESULTS
In many instances, we can obtain narrower interval estimates that have a
greater probability of including the true value of the parameter by focus-
ing on sufficient statistics, pivotal statistics, and admissible statistics.
A statistic
T
is
sufficient
for a parameter if the conditional distribution
of the observations given this statistic
T
is independent of the parameter. If
the observations in a sample are exchangeable, then the order statistics of
the sample are sufficient; that is, if we know the order statistics
x
(1)
£
x
(2)
£ ...£
x
(
n
)
, then we know as much about the unknown population distrib-
ution as we would if we had the original sample in hand. If the observa-
tions are on successive independent binomial trials that end in either
success or failure, then the number of successes is sufficient to estimate the
probability of success. The minimal sufficient statistic that reduces the
observations to the fewest number of discrete values is always preferred.
A
pivotal
quantity is any function of the observations and the unknown
parameter that has a probability distribution that does not depend on the
parameter. The classic example is Student's
t
, whose distribution does not
depend on the population mean or variance when the observations come
from a normal distribution.
A decision procedure d based on a statistic
T
is
admissible
with respect
to a given loss function
L
, provided that there does not exist a second
procedure
d
* whose use would result in smaller losses whatever the
unknown population distribution.
The importance of admissible procedures is illustrated in an expected
way by Stein's paradox. The sample mean, which plays an invaluable role
as an estimator of the population mean of a normal distribution for a
single set of observations, proves to be inadmissible as an estimator when
we have three or more independent sets of observations to work with.
Specifically, if {
X
ij
} are independent observations taken from four or more
distinct normal distributions with means q
i
and variance 1, and losses are
proportional to the square of the estimation error, then the estimators
2
ˆ
k
(
[
]
)
Â
(
)
(
)
q
i
=+--
Xk
1
3
S
2
XX
-
,
where
S
2
=
XX
-
,
..
..
i
.
i
.
i
=
1
have smaller expected losses than the individual sample means, regardless
of the actual values of the population means (see Efron and Morris [1977]).
