A statistic S(𝑿) on a random sample of data 𝑿=(X1,…,Xn) is said to be a complete statistic if for any Borel measurable function g,In other words, g(S)=0 almost everywhere whenever the expected value of g(S) is 0. Furthermore, h0(S) is unique almost everywhere for every θ.
Introduce the Lehman Scheff theorem. An exception to this criticism is that if you plan to average
a number of estimators to get a single estimator then it is a problem
if all the estimators have the same bias. The upper bound is:This is a convex combination of the arithmetic and geometric means of $v_1$ and $v_2$, and therefore must be strictly greater than $v_1$ unless $v_1=v_2$, in which case it is equal to $v_1$.
According to Rao-Blackwell, T view publisher site improved
by E(T|S) so if h(S) is not UMVUE then there must exist
another function h*(S) which is unbiased and whose variance
is smaller than that of h(S) for some value of .
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2. But proving completeness can be a pain sometimes, while proving minimal sufficiency is relatively easy.
This method of estimation does not have the parameterization
equivariance that maximum likelihood does. Consider the estimator given by $W_1(S) = \E[Z_1\vert S]$ . So we want to find an estimator that beats it by having lower variance.
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But since $\theta=\E[W_1]=\E[aW_2+b]=a\E[W_2]+b=a\theta+b$, this means that they must be the same click with $a=1,b=0$.
The MSE of