Which is an example of a sufficiently 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 ).

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 a statistic sufficient for a family of probability distributions?

Sufficient statistic. In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.

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.

How to figure out the statistic F ( x1 x2 xn )?

A statistic T = r(X1,X2,···,Xn) is sufficient if and only if the joint density can be factored as follows: f(x1,x2,···,xn|θ) = u(x1,x2,···,xn)v(r(x1,x2,···,xn),θ) (2) where u and v are non-negative functions.

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:

Is the median of the sample sufficient for the mean?

On the other hand, for an arbitrary distribution the median is not sufficient for the mean: even if the median of the sample is known, knowing the sample itself would provide further information about the population mean.

When is a statistic sufficient for the underlying parameter?

Both the statistic and the underlying parameter can be vectors. 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.