Suppose a
random variable X (which may be a sequence
(
X_{1}, ...,
X_{n}) of
scalar-valued
random variables), has a probability distribution belonging to a known
family of probability distributions, parametrized by θ, which
may be either
vector- or scalar-valued. A function
g(
X)
is an
unbiased estimator of zero if the expectation
E(
g(
X)) remains zero regardless of the value of the
parameter θ. Then
X is a
complete statistic
precisely if it admits no such unbiased
estimator of
zero.
For example, suppose X_{1}, X_{2}
are independent, identically
distributed random variables,
normally distributed with
expecation θ and variance 1. Then
X_{1} — X_{2} is an unbiased
estimator of zero. Therefore the pair
(X_{1}, X_{2}) is not a complete
statistic. On the other hand, the sum
X_{1} + X_{2}
can be shown to be a complete statistic. That means that there
is no non-zero function g such that
- <math>E(g(X_1+X_2))</math>
remains zero regardless of changes in the value of θ.
That fact may be seen as follows.
The probability distribution of
X_{1} +
X_{2}
is normal with expectation 2θ and variance 2.
Its probability density function is therefore
- <math>{\rm constant}\cdot\exp\left(-(x-2\theta)^2/4\right).</math>
The expectation above would therefore be a constant times
- <math>\int_{-\infty}^\infty g(x)\exp\left(-(x-2\theta)^2/4\right)\,dx.</math>
A bit of algebra reduces this to
- <math>[{\rm a\ nowhere\ zero\ function\ of\ }\theta]\cdot\int_{-\infty}^\infty
h(x)\,e^{x\theta}\,dx{\rm\ where\ }h(x)=g(x)\,e^{-x^2/4}.</math>
As a function of θ this is a two-sided
Laplace transform
of
h(
x), and cannot be identically zero unless
h(
x)
zero almost everywhere.
One reason for the importance of the concept is the Lehmann-Scheffe theorem[?],
which states that a statistic that is complete, sufficient, and unbiased is the best unbiased estimator, i.e., the one that has a smaller mean squared error than any other unbiased estimator, or, more generally, a smaller expected loss, for any convex loss function.
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