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 sufficient statistic for θ?
A sufficient statistic for θ is a statistic that captures all the information about θ contained in the sample. Formally we have the following definition. A statistic T(X) is sufficient for θ if the conditional distribution of X given T(X) = T(x) does not depend on θ.
What is complete sufficiency?
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).
Why do we need complete statistics?
In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it ensures that the distributions corresponding to different values of the parameters are distinct.
What is a sufficient statistic in statistics?
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 ˆΘ sufficient for θ?
An estimator ˆθ for θ is sufficient, if it contains all the information that we can extract from the random sample to estimate θ.
What is meant by 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”.