McKay's approximation for the coefficient of variation

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

$K = \left( 1 + \frac{1}{c_v^2} \right) \ \frac{(n - 1) \ \hat{c}_v^2}{1 + (n - 1) \ \hat{c}_v^2/n}$

Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When $c_v$ is smaller than 1/3, then $K$ is approximately chi-square distributed with $n - 1$ degrees of freedom. In the original article by McKay, the expression for $K$ looks slightly different, since McKay defined $\sigma^2$ with denominator $n$ instead of $n - 1$. McKay's approximation, $K$, 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 38. 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 78. pp. 10–14. doi:10.1016/j.spl.2007.04.018. Retrieved 2013-09-23.