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).
Is the MLE a sufficient statistic?
Proposition 5 (Relationship with sufficiency) MLE is a function of every sufficient statistic.
Is exponential family convex?
The natural parameter space of a full exponential family is a convex set.
Which distributions are exponential families?
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.
How do you calculate sufficient statistics?
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.
How do you know if a statistic is sufficient?
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”.
Is the MLE unbiased?
It is easy to check that the MLE is an unbiased estimator (E[̂θMLE(y)] = θ). To determine the CRLB, we need to calculate the Fisher information of the model. Yk) = σ2 n . (6) So CRLB equality is achieved, thus the MLE is efficient.
Is gamma an exponential family?
The gamma distribution is a two-parameter exponential family with natural parameters k − 1 and −1/θ (equivalently, α − 1 and −β), and natural statistics X and ln(X). If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential family.
Is gamma distribution an exponential family?
How do you prove sufficient?
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.