Sufficient Statistic - Exponential Family

Exponential Family

According to the Pitman–Koopman–Darmois theorem, among families of probability distributions whose domain does not vary with the parameter being estimated, only in exponential families is there a sufficient statistic whose dimension remains bounded as sample size increases. Less tersely, suppose are independent identically distributed random variables whose distribution is known to be in some family of probability distributions. Only if that family is an exponential family is there a (possibly vector-valued) sufficient statistic whose number of scalar components does not increase as the sample size n increases.

This theorem shows that sufficiency (or rather, the existence of a scalar or vector-valued of bounded dimension sufficient statistic) sharply restricts the possible forms of the distribution.