McKay's approximation for the coefficient of variation

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In statistics, McKay's approximation of the coefficient of variation is a statistic based on a sample from a normally distributed population. It was introduced in 1932 by A. T. McKay.[1] Statistical methods for the coefficient of variation often utilizes McKay's approximation.[2][3][4][5]

Let x i {\displaystyle x_{i}} {\displaystyle x_{i}}, i = 1 , 2 , … , n {\displaystyle i=1,2,\ldots ,n} {\displaystyle i=1,2,\ldots ,n} be n {\displaystyle n} {\displaystyle n} independent observations from a N ( μ , σ 2 ) {\displaystyle N(\mu ,\sigma ^{2})} {\displaystyle N(\mu ,\sigma ^{2})} normal distribution. The population coefficient of variation is c v = σ / μ {\displaystyle c_{v}=\sigma /\mu } {\displaystyle c_{v}=\sigma /\mu }. Let x ¯ {\displaystyle {\bar {x}}} {\displaystyle {\bar {x}}} and s {\displaystyle s\,} {\displaystyle s\,} denote the sample mean and the sample standard deviation, respectively. Then c ^ v = s / x ¯ {\displaystyle {\hat {c}}_{v}=s/{\bar {x}}} {\displaystyle {\hat {c}}_{v}=s/{\bar {x}}} is the sample coefficient of variation. McKay's approximation is

K = ( 1 + 1 c v 2 )   ( n − 1 )   c ^ v 2 1 + ( n − 1 )   c ^ v 2 / n {\displaystyle K=\left(1+{\frac {1}{c_{v}^{2}}}\right)\ {\frac {(n-1)\ {\hat {c}}_{v}^{2}}{1+(n-1)\ {\hat {c}}_{v}^{2}/n}}} {\displaystyle K=\left(1+{\frac {1}{c_{v}^{2}}}\right)\ {\frac {(n-1)\ {\hat {c}}_{v}^{2}}{1+(n-1)\ {\hat {c}}_{v}^{2}/n}}}

Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When c v {\displaystyle c_{v}} {\displaystyle c_{v}} is smaller than 1/3, then K {\displaystyle K} {\displaystyle K} is approximately chi-square distributed with n − 1 {\displaystyle n-1} {\displaystyle n-1} degrees of freedom. In the original article by McKay, the expression for K {\displaystyle K} {\displaystyle K} looks slightly different, since McKay defined σ 2 {\displaystyle \sigma ^{2}} {\displaystyle \sigma ^{2}} with denominator n {\displaystyle n} {\displaystyle n} instead of n − 1 {\displaystyle n-1} {\displaystyle n-1}. McKay's approximation, K {\displaystyle K} {\displaystyle K}, for the coefficient of variation is approximately chi-square distributed, but exactly noncentral beta distributed .[6]

References

  1. McKay, A. T. (1932). "Distribution of the coefficient of variation and the extended "t" distribution". Journal of the Royal Statistical Society. 95: 695–698. doi:10.2307/2342041.
  2. Iglevicz, Boris; Myers, Raymond (1970). "Comparisons of approximations to the percentage points of the sample coefficient of variation". Technometrics. 12 (1): 166–169. doi:10.2307/1267363. JSTOR 1267363.
  3. Bennett, B. M. (1976). "On an approximate test for homogeneity of coefficients of variation". Contributions to Applied Statistics Dedicated to A. Linder. Experentia Suppl. 22: 169–171.
  4. Vangel, Mark G. (1996). "Confidence intervals for a normal coefficient of variation". The American Statistician. 50 (1): 21–26. doi:10.1080/00031305.1996.10473537. JSTOR 2685039..
  5. Forkman, Johannes. "Estimator and tests for common coefficients of variation in normal distributions" (PDF). Communications in Statistics - Theory and Methods. pp. 21–26. doi:10.1080/03610920802187448. Retrieved 2013-09-23.
  6. Forkman, Johannes; Verrill, Steve. "The distribution of McKay's approximation for the coefficient of variation" (PDF). Statistics & Probability Letters. pp. 10–14. doi:10.1016/j.spl.2007.04.018. Retrieved 2013-09-23.