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Suppose a random variable X (which may be a sequence
(X1, ..., Xn) 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 X1, X2
are independent, identically
distributed random variables,
normally distributed with
expecation θ and variance 1. Then
X1 — X2 is an unbiased
estimator of zero. Therefore the pair
(X1, X2) is not a complete
statistic. On the other hand, the sum
X1 + X2
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
X1 + X2
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]\times\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-Scheffé 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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