|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[statistics]], an '''ancillary statistic''' is a [[statistic]] whose [[sampling distribution]] does not depend on the parameters of the model. An ancillary statistic is a [[pivotal quantity]] that is also a statistic. Ancillary statistics can be used to construct [[prediction interval]]s.
| |
|
| |
|
| This concept was introduced by the statistical geneticist Sir [[Ronald Fisher]].
| |
|
| |
|
| ==Example==
| | Hello and welcome. I'm Shemeka Lohman. Dispatching is generate profits make cash but I plan on changing the. Kansas is where I've always been living but my husband wants us to movement. Her husband doesn't like it means she does but what she really likes doing is bird keeping but she's thinking on starting something mroe challenging. See what's new on my website here: http://floorsteamcleaners.23boards.info/hoover-vacuums/hoover-spirit-vacuum.html<br><br>my web blog ... [http://floorsteamcleaners.23boards.info/hoover-vacuums/hoover-spirit-vacuum.html best floor] |
| | |
| Suppose ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[Independent identically-distributed random variables|independent and identically distributed]], and are [[normal distribution|normally distributed]] with unknown [[expected value]] ''μ'' and known [[variance]] 1. Let
| |
| | |
| :<math>\overline{X}_n = \frac{X_1+\,\cdots\,+X_n}{n}</math>
| |
| | |
| be the [[Arithmetic mean|sample mean]].
| |
| | |
| The following statistical measures of dispersion of the sample
| |
| *[[Range (statistics)|Range]]: max(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) − min(''X''<sub>1</sub>, ..., ''X<sub>n</sub>'')
| |
| *[[Interquartile range]]: ''Q''<sub>3</sub> − ''Q''<sub>1</sub>
| |
| *[[Sample variance]]:
| |
| :: <math>\hat{\sigma}^2:=\,\frac{\sum \left(X_i-\overline{X}\right)^2}{n}</math>
| |
| are all ''ancillary statistics'', because their sampling distributions do not change as ''μ'' changes. Computationally, this is because in the formulas, the ''μ'' terms cancel – adding a constant number to a distribution (and all samples) changes its sample maximum and minimum by the same amount, so it does not change their difference, and likewise for others: these measures of dispersion do not depend on location.
| |
| | |
| Conversely, given i.i.d. normal variables with known mean 1 and unknown variance ''σ''<sup>2</sup>, the sample mean <math>\overline{X}</math> is ''not'' an ancillary statistic of the variance, as the sampling distribution of the sample mean is ''N''(1, ''σ''<sup>2</sup>/''n''), which does depend on ''σ'' <sup>2</sup> – this measure of location (specifically, its [[standard error]]) depends on dispersion.
| |
| | |
| ==Ancillary complement==
| |
| Given a statistic ''T'' that is not [[Sufficiency (statistics)|sufficient]], an '''ancillary complement''' is a statistic ''U'' that is ancillary and such that (''T'', ''U'') is sufficient.<ref>[http://www.utstat.toronto.edu/dfraser/documents/237.pdf Ancillary Statistics: A Review] by M. Ghosh, N. Reid and D.A.S. Fraser</ref> 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 general 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 [[statistical independence|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 [[List of Major League Baseball batting champions|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 [[sufficiency (statistics)|sufficient statistic]], in that it conveys less than all of the relevant information in the data, but conjoined with ''N'', it becomes sufficient.
| |
| | |
| ==See also==
| |
| | |
| * [[Basu's theorem]]
| |
| * [[Prediction interval]]
| |
| * [[Group family]]
| |
| * [[Conditionality principle]]
| |
| | |
| {{More footnotes|date=November 2009}}
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| {{DEFAULTSORT:Ancillary Statistic}}
| |
| [[Category:Statistical theory]]
| |
Hello and welcome. I'm Shemeka Lohman. Dispatching is generate profits make cash but I plan on changing the. Kansas is where I've always been living but my husband wants us to movement. Her husband doesn't like it means she does but what she really likes doing is bird keeping but she's thinking on starting something mroe challenging. See what's new on my website here: http://floorsteamcleaners.23boards.info/hoover-vacuums/hoover-spirit-vacuum.html
my web blog ... best floor