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Which is a sufficient statistic for an exponential family?
T is a sufficient statistic. Condition on T, the conditional distribution is g ( x) (up to a normalization constant), which is independent of the parameter θ. This is the definition of sufficiency. In fact, for the exponential family it is independent of T.
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 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:
When is a statistic sufficient for a family of probability distributions?
Sufficient statistic. In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.
T(x) is a sufficient statistic of the distribution. For exponential families, the sufficient statistic is a function of the data that holds all information the data x {\\displaystyle x} provides with regard to the unknown parameter values.
Are there any exponential families with fixed parameters?
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. Some distributions are exponential families only if some of their parameters are held fixed.
When is the exponential family said to be in canonical form?
As in the scalar valued case, the exponential family is said to be in canonical form if A vector exponential family is said to be curved if the dimension of is less than the dimension of the vector
When is an exponential family said to be curved?
A vector exponential family is said to be curved if the dimension of. is less than the dimension of the vector. That is, if the dimension of the parameter vector is less than the number of functions of the parameter vector in the above representation of the probability density function.