How do you know if a statistic is sufficient?

A sufficient statistic summarizes all of the information in a sample about a chosen parameter. For example, the sample mean, x̄, estimates the population mean, μ. x̄ is a sufficient statistic if it retains all of the information about the population mean that was contained in the original data points.

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

Answer The probability density function of a normal random variable with mean and variance can be written in exponential form as: Therefore, the statistics Y 1 = ∑ i = 1 n X i 2 and Y 2 = ∑ i = 1 n X i are joint sufficient statistics for and .

What is a sufficient statistic for θ?

Formally, a statistic T(X1,···,Xn) is said to be sufficient for θ if the conditional distribution of X1,···,Xn, given T = t, does not depend on θ for any value of t. In other words, given the value of T, we can gain no more knowledge about θ from knowing more about the probability distribution of X1,···,Xn.

How do you prove a statistic is minimal sufficient?

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).

Why do we need sufficient statistics?

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.

Is a sufficient statistic unbiased?

The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity.

Is the MLE a sufficient statistic?

If the MLE is itself sufficient, it is minimal sufficient. Rao–Blackwellization is never needed for the MLE. A statistic A = a(X) is ancillary if the distribution of A does not depend on θ. Intuitively, A then carries no information about θ.

What is the sufficient statistic for θ is sample is arises from a beta distribution in which α β θ 0?

3. (7.2. 8) What is the sufficient statistic for θ if the sample arises from a beta distribution in which α = β = θ > 0. and so by the factorization theorem ∏ Xi(1 − Xi) is a sufficient statistic for θ.

Is a function sufficient statistic sufficient?

This depends on context. If X (possibly a vector) is an observation from some statistical model, and T=T(X) is sufficient, then any one-to-one function of T is also sufficient, see Function of a sufficient statistic. But in some cases a function of T which is not one-to-one might also be sufficient.

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.

Are 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. For if h1 and h2 were two such estimators, we would have Eθ{h1(T) − h2(T)} = 0 for all θ, and hence h1 = h2.

Are sufficient statistics unique?

Sufficient statistic always exists and it is not unique. The complete sample X is a sufficient statistic.

Why are sufficient statistics important?

Does MLE always exist?

Maximum likelihood is a common parameter estimation method used for species distribution models. Maximum likelihood estimates, however, do not always exist for a commonly used species distribution model – the Poisson point process.

What are regularity conditions for MLE?

The regularity conditions include the following: the true parameter value θ must be interior to the parameter space, the log-likelihood function must be thrice differentiable, and the third derivatives must be bounded.

Does sufficient statistics always exist?

What is complete sufficient statistic?

Complete Sufficient Statistic Ideally then, a statistic should ideally be complete and sufficient, which means that: The statistic isn’t missing any information about θ and. Doesn’t provide any irrelevant information (Shynk, 2012).

Are all sufficient statistics unbiased?

Unbiasedness is only a relative notion in terms of optimality, useful to fnd a “best” estimator by restricting the class of estimators. There exist sufficient estimators that are terribly biased and unbiased estimators that are far from sufficient.

What is the connection between a sufficient statistic and an MLE?

A theorem relating the two concepts indicates that if a maximum likelihood estimate (MLE) for a parameter is unique, then it is a function of every sufficient statistic.

Are 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).