How do you show that a statistic is sufficient?
How do you show that a statistic is sufficient?
A statistic T(X) is sufficient for θ if the conditional distribution of X given T(X) = T(x) does not depend on θ. The sufficiency depends on the parameter of interest. If X is discrete, then so is T(X) and sufficiency means that P(X = x|T(X) = T(x)) is known, i.e., it does not depend on any unknown quantity.
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”.
Are all sufficient statistics unbiased?
Any estimator of the form U = h(T) of a complete and sufficient statistic T is the unique unbiased estimator based on T of its expectation. Hence, if T is complete and sufficient, U = h(T) is the MVUE of its expectation.
How do you show that a statistic is not sufficient?
If you want to show a statistic is not a sufficient statistic , you can compare it with minimal sufficient statistic. Use the fact that a minimal sufficient statistic is a function of any sufficient statistic.
Is complete sufficient statistic unique?
Let Y be a complete sufficient statis- tic. If there are unbiased estimators, then there exists a unique MVUE. The MVUE can also be characterized as the unique unbiased function T = ϕ(Y ) of the complete sufficient statistic Y .
What is sufficiency principle?
The Sufficiency Principle, S, (or Birnbaum’s S) allows us to potentially reduce our data footprint and eliminate extra, non-informative data. The data reduction method summarizes the data while retaining all the information about a particular parameter, θ.
What is sufficiency sample?
A sufficient statistic summarizes all of the information in a sample about a chosen parameter. For example, the sample mean, x̄, estimates the population mean, μ. x̄ is a sufficient statistic if it retains all of the information about the population mean that was contained in the original data points.
Does sufficiency imply Unbiasedness?
Are sufficient statistics unique?
Sufficient statistic always exists and it is not unique. The complete sample X is a sufficient statistic.