Why do we use minimal sufficient and complete statistics?

Why do we use minimal sufficient and complete statistics?

Minimal sufficient and complete statistics We introduced the notion of sufficient statistics in order to have a function of the data that contains all information about the parameter. However, a sufficient statistic does not have to be any simpler than the data itself.

Which is an example of a minimal sufficiency statistic?

An example based on the uniform distribution is given in (38). Suppose that a statistic U is sufficient for θ. Then U is minimally sufficient if U is a function of any other statistic V that is sufficient for θ. Once again, the definition precisely captures the notion of minimal sufficiency, but is hard to apply.

When is a statistic U sufficient for θ?

Let U = u(X) be a statistic taking values in a set R. Intuitively, U is sufficient for θ if U contains all of the information about θ that is available in the entire data variable X. Here is the formal definition: A statistic U is sufficient for θ if the conditional distribution of X given U does not depend on θ ∈ T.

Which is sufficient for the coefficient parameter β?

This shows that the statistic T ≡ (T1, T2) is sufficient for the coefficient parameter β. There is no requirement that the design matrix be of full rank for sufficiency, but if it is not of full rank then these statistics are not minimal sufficient (and you obtain a minimal sufficient statistic by reducing the design matrix to full rank).

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 ).

Can you multiply a sufficient statistic by a nonzero constant?

In particular we can multiply a sufficient statistic by a nonzero constant and get another sufficient statistic. An implication of the theorem is that when using likelihood-based inference, two sets of data yielding the same value for the sufficient statistic T ( X) will always yield the same inferences about θ.

Is the k th order statistic equal to the smallest value?

In statistics, the k th order statistic of a statistical sample is equal to its k th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference .

Which is an example of a minimal sufficient statistic?

minimal sufficient statistic is unique in the sense that two statistics that are functions of each other can be treated as one statistic. For example, if T is minimal sufficient, then so is (T;eT), but no one is going to use (T;eT). If the range of X is Rk, then there exists a minimal sufficient statistic. Example 6.2.15.

Is the nth order statistic the maximum or minimum?

Similarly, for a sample of size n, the nth order statistic (or largest order statistic) is the maximum, that is, The sample range is the difference between the maximum and minimum. It is a function of the order statistics:

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:

How to determine the sufficiency of a statistic?

To learn how to apply the Exponential Criterion to identify a sufficient statistic. To extend the definition of sufficiency for one parameter to two (or more) parameters. Sufficiency is the kind of topic in which it is probably best to just jump right in and state its definition.

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.

Which is a sufficiently sufficient statistic for θ?

If U and V are equivalent statistics and U is minimally sufficient for θ then V is minimally sufficient for θ. Sufficiency is related to several of the methods of constructing estimators that we have studied. Suppose that U is sufficient for θ and that there exists a maximum likelihood estimator of θ.

Which is sufficient for the data variable θ?

If U and V are equivalent statistics and U is sufficient for θ then V is sufficient for θ. The entire data variable X is trivially sufficient for θ. However, as noted above, there usually exists a statistic U that is sufficient for θ and has smaller dimension, so that we can achieve real data reduction.

How are sufficient, complete and ancillary statistics used?

Sufficient, Complete, and Ancillary Statistics Consider again the basic statistical model, in which we have a random experiment with an observable random variable X taking values in a set S. Once again, the experiment is typically to sample n objects from a population and record one or more measurements for each item.

Which is sufficient for the statistic U ( X )?

Let fθ denote the probability density function of X and suppose that U = u(X) is a statistic taking values in R. Then U is sufficient for θ if and only if there exists G: R × T → [0, ∞) and r: S → [0, ∞) such that fθ(x) = G[u(x), θ]r(x); x ∈ S, θ ∈ T Let h θ denote the PDF of U for θ ∈ T .

How is V a function of sufficient statistics?

By the previous result, V is a function of the sufficient statistics U . That is, E(Θ ∣ X) = E(Θ ∣ U) . The next result is the Rao-Blackwell theorem, named for CR Rao and David Blackwell. The theorem shows how a sufficient statistic can be used to improve an unbiased estimator.

How to find a sufficient statistic for θ?

From here, you can apply the factorization theorem to find a (two-dimensional) sufficient statistic, ( T 1, T 2) for θ. To show minimal sufficiency, suppose X’ and Y’ were another sample from the same distribution.

Can a pair be a minimal sufficient statistic?

And the latter, the pair, cannot be a minimal sufficient statistic if the former is sufficient because if you know the maximum you don’t have enough information to find the pair, and the pair is sufficient.

Which is minimal sufficient, x ( 1 ) or x ( n )?

( − θ, θ), but only max { − X ( 1), X ( n) } is minimal sufficient, as stated here. Can this be explained by the fact that while both can describe the data adequately, max { − X ( 1), X ( n) } is of lesser dimension and is thus minimal sufficient?

What does it mean when a model is sufficient?

Being sufficient does not mean it gives enough information to describe the data; rather it means it gives all information in the data that is relevant to inference about θ, given that the proposed model is right. The model is that the observations come from a uniform distribution on an interval symmetric about 0.