How do you show a sufficient statistic is minimal?

How do you show a sufficient statistic is minimal?

Definition 1 (Minimal Sufficiency). A sufficient statistic T is minimal if for every sufficient statistic T and for every x, y ∈ X, T(x) = T(y) whenever T (x) = T (y). In other words, T is a function of T (there exists f such that T(x) = f(T (x)) for any x ∈ X).

Are all complete sufficient statistics minimal?

A complete statistic is boundedly complete. If T is complete (or boundedly complete) and S = ψ(T) for a measurable ψ, then S is complete (or boundedly complete). It can be shown that a complete and sufficient statistic is minimal sufficient (Theorem 6.2. 28).

How do you know if an estimator is sufficient?

The mathematical definition is as follows. A statistic T = r(X1,X2,··· ,Xn) is a sufficient statistic if for each t, the conditional distribution of X1,X2, ···,Xn given T = t and θ does not depend on θ.

What does it mean for an estimator to be sufficient?

An estimator of a parameter θ which gives as much information about θ as is possible from the sample at hand is called a sufficient estimator. Sufficient estimators exist when one can reduce the dimensionality of the observed data without loss of information.

Is a function sufficient statistic sufficient?

. Typically, the sufficient statistic is a simple function of the data, e.g. the sum of all the data points. The concept is equivalent to the statement that, conditional on the value of a sufficient statistic for a parameter, the joint probability distribution of the data does not depend on that parameter.

Does a sufficient statistic always exist?

Hence, a sufficient statistic always exists. We can compute the density of the sufficient statistics. Many statistical problems can be phrased in the language of decision theory. Suppose as usual that we have data X whose distribution depend on a parameter Θ.

What is a minimal sufficient statistic?

A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. In other words, S(X) is minimal sufficient if and only if. S(X) is sufficient, and. if T(X) is sufficient, then there exists a function f such that S(X) = f(T(X)).

How do you prove a necessary and sufficient condition?

The assertion that a statement is a “necessary and sufficient” condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false.

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. If we have a sufficient statistic, then the Rao- Blackwell theorem gives a procedure for finding the unbiased estimator with the smallest variance.

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.

Is a sufficient condition?

A sufficient condition is a condition or set of conditions that will produce the event. A necessary condition must be there, but it alone does not provide sufficient cause for the occurrence of the event. Only the sufficient grounds can do this. In other words, all of the necessary elements must be there.

How do you prove something is sufficient?

Recall that an exponential family of random variables has its density of the form fX(x|θ) = c(θ)h(x) exp(ν(θ)T(x)). Thus, the sufficient statistic is sum of the observations T(x) = x1 + ··· + xn and the natural parameter ν(θ) = ln(θ/(1 − θ)), the log-odds, Example 6 (Gamma random variables).

When do you need to use a sufficient estimator?

If sufficient estimator exists, no other estimator from the sample can provide additional information about the population being estimated. If there is a sufficient estimator, then there is no need to consider any of the non-sufficient estimators. A good estimator is a function of sufficient statistics.

Is there such a thing as a minimal sufficient statistic?

If there exists a minimal sufficient statistic, and this is usually the case, then every complete sufficient statistic is necessarily minimal sufficient (note that this statement does not exclude the option of a pathological case in which a complete sufficient exists while there is no minimal sufficient statistic).

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.

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.