Editing Sufficient statistic (section)

==Exponential family==
{{main|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 family|exponential families]] is there a sufficient statistic whose dimension remains bounded as sample size increases. Intuitively, this states that nonexponential families of distributions on the real line require [[nonparametric statistics]] to fully capture the information in the data.

Less tersely, suppose <math>X_n, n = 1, 2, 3, \dots</math> are [[independent identically distributed]] '''real''' random variables whose distribution is known to be in some family of probability distributions, parametrized by <math>\theta</math>, satisfying certain technical regularity conditions, then that family is an ''exponential'' family if and only if there is a <math>\R^m</math>-valued sufficient statistic <math>T(X_1, \dots, X_n)</math> whose number of scalar components <math>m</math> does not increase as the sample size ''n'' increases.<ref>{{Cite journal |last1=Tikochinsky |first1=Y. |last2=Tishby |first2=N. Z. |last3=Levine |first3=R. D. |date=1984-11-01 |title=Alternative approach to maximum-entropy inference |url=http://dx.doi.org/10.1103/physreva.30.2638 |journal=Physical Review A |volume=30 |issue=5 |pages=2638–2644 |doi=10.1103/physreva.30.2638 |bibcode=1984PhRvA..30.2638T |issn=0556-2791}}</ref>

This theorem shows that the existence of a finite-dimensional, real-vector-valued sufficient statistics sharply restricts the possible forms of a family of distributions on the '''real line'''. 

When the parameters or the random variables are no longer real-valued, the situation is more complex.<ref>{{Cite journal |last=Andersen |first=Erling Bernhard |date=September 1970 |title=Sufficiency and Exponential Families for Discrete Sample Spaces |url=http://dx.doi.org/10.1080/01621459.1970.10481160 |journal=Journal of the American Statistical Association |volume=65 |issue=331 |pages=1248–1255 |doi=10.1080/01621459.1970.10481160 |issn=0162-1459}}</ref>