What does sufficient statistic mean in statistics?

What does sufficient statistic mean in statistics?

From Wikipedia, the free encyclopedia. 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”.

Why are sufficient statistics useful?

Sufficiency is ‘sought out’ because, along with other conditions (unbiasedness and completeness), it helps to identify estimators that have the smallest variance. The intuitive idea is that for purposes of estimating the parameter the sufficient statistic contains all relevant information.

How do you find the sufficient statistic for a normal distribution?

know the MLE for θ is ¯X; Example 4 shows a sufficient statistic for µ in normal distribution is ¯X, and this is the MLE for µ; Example 6 shows a sufficient statistic of θ for the uniform distribution on (0,θ) is max(X1,···,Xn), and this is the MLE for θ.

How do you prove minimal sufficient statistics?

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).

What is meant by minimal sufficient statistics?

Are minimal sufficient statistics complete?

Theorem 12 (Bahadur’s theorem). If T is a finite-dimensional boundedly complete sufficient statistic, then it is minimal sufficient. As we have seen a sufficient statistic contains all the information about the parameter. The opposite is when a statistic does not contain any information about the parameter.

Why do we complete statistics?

In essence, it (completeness is a property of a statistic) is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the distributions corresponding to different values of the parameters are distinct.

Are complete statistics sufficient?

For some parametric families, a complete sufficient statistic does not exist (for example, see Galili and Meilijson 2016). Also, a minimal sufficient statistic need not exist. (A case in which there is no minimal sufficient statistic was shown by Bahadur in 1957.)

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 ).

When to use practical significance in statistical analysis?

In closing, statistical significance indicates that your sample provides sufficient evidence to conclude that the effect exists in the population. Practical significance asks whether that effect is large enough to care about. Use statistical analyses to determine statistical significance and subject-area expertise to assess practical significance.

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 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.