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How do you find a minimal sufficient statistic?
Definition 1 (Minimal Sufficiency). A sufficient statistic T is minimal if for every sufficient statistic T and for every x, y ∈ X, T(x) = T(y) whenever T (x) = T (y). In other words, T is a function of T (there exists f such that T(x) = f(T (x)) for any x ∈ X).
Does minimal sufficient statistic always exist?
Sufficient statistic always exists and it is not unique. The complete sample X is a sufficient statistic. (Factorize as f(X|θ)=f(T(x)|θ)h(x) f ( X | θ ) = f ( T ( x ) | θ ) h ( x ) , where T(x)=x T ( x ) = x and h(x)=1 h ( x ) = 1 .)
How do you show statistic is sufficient?
The mathematical definition is as follows. A statistic T = r(X1,X2,··· ,Xn) is a sufficient statistic if for each t, the conditional distribution of X1,X2, ···,Xn given T = t and θ does not depend on θ.
Which is an example of a sufficiently sufficient statistic?
For example, for a Gaussian distribution with unknown mean and variance, the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, the sample mean and sample variance ).
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.
What do you call a jointly sufficient statistic?
In such a case, the sufficient statistic may be a set of functions, called a jointly sufficient statistic. Typically, there are as many functions as there are parameters.
When is an unknown parameter a sufficient statistic?
More generally, the “unknown parameter” may represent a vector of unknown quantities or may represent everything about the model that is unknown or not fully specified. In such a case, the sufficient statistic may be a set of functions, called a jointly sufficient statistic.