Ancillary Statistic

Examples

• Suppose X₁, ..., X_n are independent and identically distributed, and are normally distributed with expected value μ and variance 1. (The use as an example, of this particular parametrized family of probability distributions, all having the same variance, is unrealistic, in that it amounts to a situation in which the statistician somehow knows the exact value of the population variance, but can only estimate the population mean by using the observed values of the data X₁, ..., X_n.) Let

$\backslash overline\{X\}\_n=(X\_1+\backslash ,\backslash cdots\backslash ,+X\_n)/n$

be the sample mean. The random variable

$\backslash overline\{X\}\_n-\backslash mu$

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

$\backslash max\backslash \{\backslash ,X\_1,\backslash dots,X\_n\backslash ,\backslash \}-\backslash min\backslash \{\backslash ,X\_1,\backslash dots,X\_n\backslash ,\backslash \}$

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 (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 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 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.

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