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	In ~~[[statistics]]~~, ~~'''McKay's approximation''' of the [[coefficient of variation]] is~~ a ~~statistic based on~~ a ~~sample from a [[normal distribution\|normally distributed]] population~~. ~~It was introduced~~ in ~~1932 by A~~. T. ~~McKay~~.<~~ref~~>~~McKay, A. T. (1932) Distribution of~~ the ~~coefficient~~ of ~~variation~~ and the ~~extended “t” distribution~~. ~~''Journal~~ of ~~the Royal Statistical Society'' 95~~, ~~695–698</ref> Statistical methods for~~ the ~~coefficient of variation often utilizes McKay's approximation~~.~~<ref>{{cite journal \|jstor=1267363 \|title=Comparisons of approximations to~~ the ~~percentage points~~ of the ~~sample coefficient of variation \|~~year~~=1970 \|last1=Iglevicz \|first1=Boris \|last2=Myers \|first2=Raymond \|journal=Technometrics \|volume=12 \|issue=1 \|pages=166–169}}~~<~~/ref~~><~~ref~~>~~Bennett, B~~. M. ~~(1976)~~. ~~On an approximate test for homogeneity~~ of ~~coefficients of variation~~. ~~In: Ziegler~~, ~~W. J~~. (~~ed.~~) ~~''Contributions to Applied Statistics Dedicated to A~~. ~~Linder~~. ~~Experentia Suppl.'' 22, 169-171~~<~~/ref~~><~~ref~~>~~{{cite journal \|jstor=2685039 \|title=Confidence intervals~~ for ~~a normal coefficient of variation \|year=1996 \|last1=Vangel \|first1=Mark G~~. ~~\|journal=~~The ~~American Statistician \|volume=50 \|issue=1 \|pages=21–26}}.</ref><ref>{{cite web\| url=~~http://~~pub~~.~~epsilon~~.~~slu.se~~/~~4489/1/forkman_j_110214~~.~~pdf \| title~~=~~Estimator~~ and ~~tests for common coefficients of variation in normal distributions \| accessdate=2013~~-~~09-23 \|last1=Forkman \|first1=Johannes \|journal=Communications in Statistics - Theory and Methods \|volume=38 \|pages=21–26}}~~<~~/ref~~>		In exchange for guaranteed coverage, you will be charged a higher term life insurance premium. Nevertheless, its benefits are a lot more than term-life insurance policies. Such companies have to meet strict requirements concerning their suitability for such business. 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	~~Let~~ <~~math> x_i </math~~>, ~~<math>i = 1, 2,\ldots, n</math> be <math> n </math> independent observations from~~ a ~~<math> N(\mu~~, ~~\sigma^2) </math> normal distribution~~. ~~The population coefficient~~ of ~~variation is <math> c_v = \sigma / \mu </math>~~. ~~Let <math> \bar{x} </math> and <math> s \~~,~~</math> denote~~ the ~~[[sample mean]] and~~ the ~~[[sample standard deviation]], respectively~~. ~~Then <math> \hat{c}_v = s/\bar{x} </math>~~ is the ~~sample coefficient of variation~~. ~~McKay’s approximation is~~
	:<~~math~~>
	~~K = \left( 1 + \frac{1}{c_v^2} \right) \ \frac{(n - 1) \ \hat{c}_v^2}{1 + (n - 1) \ \hat{c}_v^2/n}~~
	<~~/math~~>
	~~<!-- WHAT~~ is ~~\gamma?? -->~~
	~~Note~~ that ~~in this expression~~, the ~~first factor includes~~ the ~~population coefficient~~ of ~~variation~~, ~~which~~ is ~~usually unknown~~. ~~When <math> c_v <~~/~~math> is smaller than 1~~/~~3, then <math> K <~~/~~math> is approximately [[chi~~-~~squared distribution\|chi~~-~~square distributed]] with <math> n~~ - ~~1 </math> degrees of freedom~~. In the ~~original article by McKay~~, the ~~expression for~~ <~~math~~> K <~~/math~~> ~~looks slightly different~~, ~~since McKay defined <math> \sigma^2 </math> with denominator <math> n </math> instead of <math> n - 1 </math>. McKay~~'s ~~approximation, <math> K </math>,~~ for the ~~coefficient~~ of ~~variation is approximately chi~~-~~square distributed~~, ~~but exactly [~~[~~noncentral beta distribution\|noncentral beta distributed]]~~
	~~.<ref>{{cite web\| url=~~http://~~pub~~.~~epsilon~~.~~slu.se~~/~~3317~~/~~1/Forkman_Verrill_080610.pdf \| title~~=~~The distribution~~ of ~~McKay's approximation~~ for the ~~coefficient of variation \| accessdate=2013-09-23 \|last1=Forkman \|first1=Johannes \|last2=Verrill \|first2=Steve \|journal=Statistics & Probability Letters \|volume=78 \|pages=10–14}}</ref>~~

	~~== References==~~
	~~<references/>~~

	~~[[Category:Statistical deviation and dispersion]]~~

217.168.41.242: /* Why languages containing cross-serial dependencies are non-context-free */

2013-10-25T05:59:06Z

Why languages containing cross-serial dependencies are non-context-free

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In [[statistics]], '''McKay's approximation''' of the [[coefficient of variation]] is a statistic based on a sample from a [[normal distribution|normally distributed]] population. It was introduced in 1932 by A. T. McKay.<ref>McKay, A. T. (1932) Distribution of the coefficient of variation and the extended “t” distribution. ''Journal of the Royal Statistical Society'' 95, 695–698</ref> Statistical methods for the coefficient of variation often utilizes McKay's approximation.<ref>{{cite journal |jstor=1267363 |title=Comparisons of approximations to the percentage points of the sample coefficient of variation |year=1970 |last1=Iglevicz |first1=Boris |last2=Myers |first2=Raymond |journal=Technometrics |volume=12 |issue=1 |pages=166–169}}</ref><ref>Bennett, B. M. (1976). On an approximate test for homogeneity of coefficients of variation. In: Ziegler, W. J. (ed.) ''Contributions to Applied Statistics Dedicated to A. Linder. Experentia Suppl.'' 22, 169-171</ref><ref>{{cite journal |jstor=2685039 |title=Confidence intervals for a normal coefficient of variation |year=1996 |last1=Vangel |first1=Mark G. |journal=The American Statistician |volume=50 |issue=1 |pages=21–26}}.</ref><ref>{{cite web| url=http://pub.epsilon.slu.se/4489/1/forkman_j_110214.pdf | title=Estimator and tests for common coefficients of variation in normal distributions | accessdate=2013-09-23 |last1=Forkman |first1=Johannes |journal=Communications in Statistics - Theory and Methods |volume=38 |pages=21–26}}</ref>

Let <math> x_i </math>, <math>i = 1, 2,\ldots, n</math> be <math> n </math> independent observations from a <math> N(\mu, \sigma^2) </math> normal distribution. The population coefficient of variation is <math> c_v = \sigma / \mu </math>. Let <math> \bar{x} </math> and <math> s \,</math> denote the [[sample mean]] and the [[sample standard deviation]], respectively. Then <math> \hat{c}_v = s/\bar{x} </math> is the sample coefficient of variation. McKay’s approximation is
:<math>
K = \left( 1 + \frac{1}{c_v^2} \right) \ \frac{(n - 1) \ \hat{c}_v^2}{1 + (n - 1) \ \hat{c}_v^2/n}
</math>

Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When <math> c_v </math> is smaller than 1/3, then <math> K </math> is approximately [[chi-squared distribution|chi-square distributed]] with <math> n - 1 </math> degrees of freedom. In the original article by McKay, the expression for <math> K </math> looks slightly different, since McKay defined <math> \sigma^2 </math> with denominator <math> n </math> instead of <math> n - 1 </math>. McKay's approximation, <math> K </math>, for the coefficient of variation is approximately chi-square distributed, but exactly [[noncentral beta distribution|noncentral beta distributed]]
.<ref>{{cite web| url=http://pub.epsilon.slu.se/3317/1/Forkman_Verrill_080610.pdf | title=The distribution of McKay's approximation for the coefficient of variation | accessdate=2013-09-23 |last1=Forkman |first1=Johannes |last2=Verrill |first2=Steve |journal=Statistics & Probability Letters |volume=78 |pages=10–14}}</ref>

== References==
<references/>

[[Category:Statistical deviation and dispersion]]

Cross-serial dependency - Revision history

en>AnomieBOT: Dating maintenance tags: {{Clarify}}

217.168.41.242: /* Why languages containing cross-serial dependencies are non-context-free */