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How are sufficient, complete and ancillary statistics used?
Sufficient, Complete, and Ancillary Statistics Consider again the basic statistical model, in which we have a random experiment with an observable random variable X taking values in a set S. Once again, the experiment is typically to sample n objects from a population and record one or more measurements for each item.
Why do we use minimal sufficient and complete statistics?
Minimal sufficient and complete statistics We introduced the notion of sufficient statistics in order to have a function of the data that contains all information about the parameter. However, a sufficient statistic does not have to be any simpler than the data itself.
When is y a sufficient statistic for P?
The definition of sufficiency tells us that if the conditional distribution of X 1, X 2, …, X n, given the statistic Y, does not depend on p, then Y is a sufficient statistic for p. The conditional distribution of X 1, X 2, …, X n, given Y, is by definition:
Which is a sufficiently sufficient statistic for θ?
If U and V are equivalent statistics and U is minimally sufficient for θ then V is minimally sufficient for θ. Sufficiency is related to several of the methods of constructing estimators that we have studied. Suppose that U is sufficient for θ and that there exists a maximum likelihood estimator of θ.
When is X a special case in statistics?
As usual, the most important special case is when X is a sequence of independent, identically distributed random variables. In this case X is a random sample from the common distribution.
When is U a minimally sufficient statistic for θ?
Let fθ denote the probability density function of X corresponding to the parameter value θ ∈ T and suppose that U = u(X) is a statistic taking values in R. Then U is minimally sufficient for θ if the following condition holds: for x ∈ S and y ∈ S fθ(x) fθ(y) is independent of θ if and only if u(x) = u(y)