Sufficient statistic

Sufficiency in mathematical statistics, a term which may come as a property of measurable functions that map from the sample space into an arbitrary measurement space. It characterizes such illustrations as suffizient (also: exhaustive ) that transform a high-dimensional vector data in a simpler form, without losing essential information on the underlying probability distribution.

So formulated Clearly such statistics are exactly suffizient that contain all the information about the estimated parameters which are included in the sample.

Sufficiency is one next to consistency ( asymptotic ) unbiasedness and ( asymptotic ) efficiency of the four common criteria for the quality of estimators.

Definition

Formal be the sample space, any test room and a measurable map between the two spaces. It should also be a random variable on the sample space whose distribution comes from a family of probability measures. it means suffizient for the family if the distribution of does not depend on.

Neyman - characterization for dominated families of probability measures

An equivalent characterization of Suffizenz goes back to Jerzy Neyman, after which a statistic is exactly then suffizient when measurable functions and exist, so that the density can be decomposed as follows: In particular, bijective transformations suffizienter statistics are again suffizient.

Example: Binomial

A simple example of suffiency of statistics we obtained in the investigation binomialverteilter or Bernoulli - distributed random variables. The probability density of the random variable is given in this case, wherein are either 0 or 1. Therefore, one recognizes from the Neyman - characterization immediately that is suffizient for. So spoke heuristic, it is sufficient merely to know instead of the entire data vector, the number of successes in this experiment in order to obtain complete information about the unknown parameter.