How do you show a sufficient statistic is complete?
A statistic T is called complete if Eg(T) = 0 for all θ and some function g implies that P(g(T) = 0;θ) = 1 for all θ. 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.
What is the distribution of the sufficient statistic?
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.
Is a complete statistic also sufficient?
For some parametric families, a complete sufficient statistic does not exist (for example, see Galili and Meilijson 2016). Also, a minimal sufficient statistic need not exist. (A case in which there is no minimal sufficient statistic was shown by Bahadur in 1957.)
What is a complete statistic?
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).
What is complete sufficient statistic?
Complete Sufficient Statistic Ideally then, a statistic should ideally be complete and sufficient, which means that: The statistic isn’t missing any information about θ and. Doesn’t provide any irrelevant information (Shynk, 2012).
What is a scalar sufficient statistic?
In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if “no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter”.
What is sufficient statistic and complete sufficiency?
From Wikipedia, the free encyclopedia. In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if “no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter”.
Is function of complete statistics complete?
If a statistic is complete, then any function of that statistic is complete.
What does sufficient statistic mean in statistics?
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).