Is sample variance a sufficient statistic?

For example, suppose we know the sample mean and the sample variance. We say T is a sufficient statistic if the statistician who knows the value of T can do just as good a job of estimating the unknown parameter θ as the statistician who knows the entire random sample.

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.

Which is sufficient statistic for normal distribution with known mean?

– Mathematics Stack Exchange Sufficient statistic for normal distribution with known mean. The first factor depends on (x1,…,xn) only through n ∑ i=1xi. The second factor does not depend on θ. Therefore by Fisher’s factorization theorem, n ∑ i=1xi is sufficient for θ.

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:

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.

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 ).