Ancillary statistic

Ancillary statistic

In statistics, an ancillary statistic is a statistic whose probability distribution does not depend on which of the probability distributions among those being considered is the distribution of the statistical population from which the data were taken. This concept was introduced by the great statistical geneticist Sir Ronald Fisher.

Example

Suppose "X"1, ..., "X""n" are independent and identically distributed, and are normally distributed with expected value "μ" and variance 1. Let:overline{X}_n=(X_1+,cdots,+X_n)/nbe the sample mean.

The following statistical measures of dispersion of the sample
*Range: max("X"1, ..., "Xn") − min("X"1, ..., "Xn")
*Interquartile range: "Q"3 − "Q"1
*Sample variance::: hat{sigma}^2:=,frac{sum left(X_i-overline{X} ight)^2}{n} are all "ancillary statistics", because their probability distributions do not change as "μ" changes.

Ancillary complement

Given a statistic "T" that is not sufficient, an ancillary complement is a statistic "U" that is ancillary to "T" and such that (T,U) is sufficient. [ [http://www.utstat.toronto.edu/dfraser/documents/237.pdf Ancillary Statistics: A Review] by M. Ghosh, N. Reid and D.A.S. Fraser] Intuitively, an ancillary complement "adds the missing information" (without duplicating any).

The statistic is particularly useful if one takes "T" to be a Maximum Likelihood Estimator, which in generally will not be sufficient; then one can ask for an ancillary complement. In this case, Fisher argues that one must condition on an ancillary complement to determine information content: one should consider the Fisher information content of "T" to not be the marginal of "T", but the conditional distribution of "T", given "U": how much information does "T" "add"? This is not possible in general, as no ancillary complement need exist, and if one exists, it need not be unique, nor does a maximum ancillary complement exist.

Example

In baseball, suppose a scout observes a batter in "N" at-bats. Suppose (unrealistically) that the number "N" is chosen by some random process that is independent of the batter's ability -- say a coin is tossed after each at-bat and the result determines whether the scout will stay to watch the batter's next at-bat. The eventual data are the number "N" of at-bats and the number "X" of hits: the data (X,N) are a sufficient statistic. The observed batting average "X"/"N" fails to convey all of the information available in the data because it fails to report the number "N" of at-bats (e.g., a batting average of .400, which is very high, based on only five at-bats does not inspire anywhere near as much confidence in the player's ability than a 0.400 average based on 100 at-bats). The number "N" of at-bats is an ancillary statistic because
* It is a part of the observable data (it is a "statistic"), and
* Its probability distribution does not depend on the batter's ability, since it was chosen by a random process independent of the batter's ability.This ancillary statistic is an ancillary complement to the observed batting average "X"/"N", i.e., the batting average "X"/"N" is not a sufficient statistic, in that it conveys less than all of the relevant information in the data, but conjoined with "N", it becomes sufficient.

Notes

ee also

* Basu's theorem


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Ancillary — *Ancillary administration *Ancillary jurisdiction *Ancillary statistic *Ancillary data *Ancillary relief *Ancillary doctrine …   Wikipedia

  • Sufficient statistic — In statistics, a sufficient statistic is a statistic which has the property of sufficiency with respect to a statistical model and its associated unknown parameter, meaning that no other statistic which can be calculated from the same sample… …   Wikipedia

  • Basu's theorem — In statistics, Basu s theorem states that any complete sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu.It is often used in statistics as a tool to prove independence of two statistics, by… …   Wikipedia

  • Completeness (statistics) — In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • Bootstrapping (statistics) — In statistics, bootstrapping is a modern, computer intensive, general purpose approach to statistical inference, falling within a broader class of resampling methods.Bootstrapping is the practice of estimating properties of an estimator (such as… …   Wikipedia

  • Normalization (statistics) — For other uses, see Standard score and Normalizing constant. In one usage in statistics, normalization is the process of isolating statistical error in repeated measured data. A normalization is sometimes based on a property. Quantile… …   Wikipedia

  • Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function …   Wikipedia

  • Robust statistics — provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions. Contents 1 Introduction 2 Examples of robust and non robust… …   Wikipedia

  • Debabrata Basu — (Bengali: দেবব্রত বসু) (5 July 1924 – 24 March 2001) was a mathematical statistician who made fundamental contributions to the foundations of statistics. Basu invented simple examples that displayed some difficulties of likelihood based… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”