What is a sufficient statistic used for?
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 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.
How do you show ancillary statistics?
A statistics is ancillary if its distribution does not depend on θ. More precisely, a statistic S(X) is ancillary for Θ it its distribution is the same for all θ ∈ Θ. That is, Pθ(S(X) ∈ A) is constant for θ ∈ Θ for any set A. (Xi − ¯X)2.
Is Normal distribution Exponential family?
The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families. Some distributions are exponential families only if some of their parameters are held fixed.
Which is a sufficient statistic for the parameter θ?
A statistic t = T ( X) is sufficient for underlying parameter θ precisely if the conditional probability distribution of the data X, given the statistic t = T ( X ), does not depend on the parameter θ. As an example, the sample mean is sufficient for the mean ( μ) of a normal distribution with known variance.
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 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 ).
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.