Encyclopedia > Completeness (statistics)

  Article Content

Completeness (statistics)

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 X1X2 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

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]\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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Museums in England

... Hepworth Museum[?], St. Ives, Cornwall Penlee House, Penzance[?] Tate St. Ives[?] Cumbria Abbot Hall Art Gallery[?], Kendal[?] Beatrix Potter Gallery[?], ...

This page was created in 42.1 ms