What is a minimal exponential family?

We say that an exponential family is minimal if the functions c(qq) and the statistics T(x) each are linearly independent.

How do you find the sufficient statistic for an exponential distribution?

We shall show that T = max(X1,···,Xn) is a sufficient statistic for θ. It can be seen that if xi < 0 for at least one value of i (i = 1,···,n), then fn(x|θ) = 0 for every value of θ > 0. Therefore it is only necessary to consider the factorization of fn(x|θ) for values of xi ≥ 0 (i = 1,···,n).

What is a minimal sufficient statistic?

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). 3-1. Page 2.

Is exponential family complete?

In an exponential family, it turns out that not only is the statistic М minimal sufficient, it is complete. minimal sufficient statistic М()=(М½() М×())’, then М is complete.

What is curved exponential family?

Curved exponential families are a subset of the class of exponential families where the dimension of the parameter space does not match the dimension of the exponential family.

What is a full rank exponential family?

If the parameter space for an exponential family contains an s- dimensional open set, then it is called full rank. An exponential family that is not full rank is generally called a curved exponential family, as typically the parameter space is a curve in Rs of dimension less than s.

Does a minimal sufficient statistic always exist?

Relation to sufficient statistics Under mild conditions, a minimal sufficient statistic does always exist. In particular, these conditions always hold if the random variables (associated with Pθ ) are all discrete or are all continuous.

How can you find a sufficient statistic?

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.

Is minimal sufficient statistics complete?

A minimal sufficient statistic is not necessarily complete. In this example, X1,…,Xn is a random sample from uniform(θ,θ +1), θ ∈ 勿, and we showed that T = (X(1),X(n)) is the minimal sufficient statistic for θ.

Is Poisson exponential family?

The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families.

Is beta distribution exponential family?

The family of beta(α,β) distributions is an exponential family.

Is a minimal sufficient statistic unique?

If both T and S are minimal sufficient statistics, then by definition there is a one-to-one function ψ such that T = ψ(S); hence, the minimal sufficient statistic is unique in the sense that two statistics that are functions of each other can be treated as one statistic.