Complete Statistic Pdf For Free

A sufficient statistic T is called a complete statistic if no function of it has zero expected value for all distributions concerned unless this function itself is zero for all possible distributions concerned (except possibly a set of measure zero).

The statistic s is said to be complete for the distribution of X if for every measurable function g (which must be independent of parameter ?) the following implication holds: E?[g(s(X))]=0,?? implies that P?(g(s(X))=0)=1,??.

A statistic T is called complete if Eg(T) = 0 for all ? and some function g implies that P(g(T) = 0;?) = 1 for all ?.

Completeness of a statistic is also related to minimal sufficiency. Theorem 12 (Bahadur's theorem). If T is a finite-dimensional boundedly complete sufficient statistic, then it is minimal sufficient. Xi(u) = E?[Si(T) | U = u], Yi(t) = E?[Xi(U) | T = t].

A sufficient statistic T is called a complete statistic if no function of it has zero expected value for all distributions concerned unless this function itself is zero for all possible distributions concerned (except possibly a set of measure zero).

If a statistic is complete, then any function of that statistic is complete.

A statistic T is called complete if Eg(T) = 0 for all ? and some function g implies that P(g(T) = 0;?) = 1 for all ?. This use of the word complete is analogous to calling a set of vectors v1,...,vn complete if they span the whole space, that is, any v can be written as a linear combination v = ?ajvj of these vectors.

A complete statistic is boundedly complete. If T is complete (or boundedly complete) and S = ?(T) for a measurable ?, then S is complete (or boundedly complete). It can be shown that a complete and sufficient statistic is minimal sufficient (Theorem 6.2. 28).