Ancillary Statistic

Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantuanna didagoan pikeun narjamahkeun.

Konsep ieu mimiti dikenalkeun ku ahli statistik génétik Sir Ronald Fisher.

• Anggap X₁, ..., X_n mangrupa bebas sarta kasebar identik, sarta kasebar normal mibanda nilai ekspektasi μ jeung varian 1. (The use as an example, of this particular parametrized family of probability distributions, all having the same variance, is unréalistic, in that it amounts to a situation in which the statistician somehow knows the exact value of the polpulation variance, but can only estimate the population méan by using the observed values of the data X₁, ..., X_n.) Let

${\displaystyle {\overline {X}}_{n}=(X_{1}+\,\cdots \,+X_{n})/n}$ 

be the sample mean. Variabel acak

${\displaystyle {\overline {X}}_{n}-\mu }$ 

is not an ancillary statistic, even though its probability distribution does not depend on μ That is because it is not a statistic, since its value depends on the unobservable population mean μ

The random variable

${\displaystyle \max\{\,X_{1},\dots ,X_{n}\,\}-\min\{\,X_{1},\dots ,X_{n}\,\}}$ 

is an ancillary statistic, because
• Its probability distribution does not change as μ changes, and
• it depends only on the data X₁, ..., X_n and not on the unobservable parameter μ, i.e., it is a statistic.

• In baseball, suppose a scout observes a batter in N at-bats. Suppose (unréalistically) that the number N is chosen by some random process that is independent of the batter's ability—say a coin is tossed after éach 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 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 0.400, which is very high, based on only five at-bats does not inspire anywhere néar 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.