Sample Coefficient of Variation.
Compute the sample coefficient of variation.
cv(x, method = "moments", sd.method = "sqrt.unbiased",
l.moment.method = "unbiased", plot.pos.cons = c(a = 0.35, b = 0),
na.rm = FALSE)x |
numeric vector of observations. |
method |
character string specifying what method to use to compute the sample coefficient
of variation. The possible values are |
sd.method |
character string specifying what method to use to compute the sample standard
deviation when |
l.moment.method |
character string specifying what method to use to compute the
L-moments when |
plot.pos.cons |
numeric vector of length 2 specifying the constants used in the formula for
the plotting positions when |
na.rm |
logical scalar indicating whether to remove missing values from |
Let \underline{x} denote a random sample of n observations from some distribution with mean μ and standard deviation σ.
Product Moment Coefficient of Variation (method="moments")
The coefficient of variation (sometimes denoted CV) of a distribution is
defined as the ratio of the standard deviation to the mean. That is:
CV = \frac{σ}{μ} \;\;\;\;\;\; (1)
The coefficient of variation measures how spread out the distribution is relative to the size of the mean. It is usually used to characterize positive, right-skewed distributions such as the lognormal distribution.
When sd.method="sqrt.unbiased", the coefficient of variation is estimated
using the sample mean and the square root of the unbaised estimator of variance:
\widehat{CV} = \frac{s}{\bar{x}} \;\;\;\;\;\; (2)
where
\bar{x} = \frac{1}{n} ∑_{i=1}^n x_i \;\;\;\;\;\; (3)
s = [\frac{1}{n-1} ∑_{i=1}^n (x_i - \bar{x})^2]^{1/2} \;\;\;\;\;\; (4)
Note that the estimator of standard deviation in equation (4) is not unbiased.
When sd.method="moments", the coefficient of variation is estimated using
the sample mean and the square root of the method of moments estimator of variance:
\widehat{CV} = \frac{s_m}{\bar{x}} \;\;\;\;\;\; (5)
s = [\frac{1}{n} ∑_{i=1}^n (x_i - \bar{x})^2]^{1/2} \;\;\;\;\;\; (6)
L-Moment Coefficient of Variation (method="l.moments")
Hosking (1990) defines an L-moment analog of the
coefficient of variation (denoted the L-CV) as:
τ = \frac{l_2}{l_1} \;\;\;\;\;\; (7)
that is, the second L-moment divided by the first L-moment. He shows that for a positive-valued random variable, the L-CV lies in the interval (0, 1).
When l.moment.method="unbiased", the L-CV is estimated by:
t = \frac{l_2}{l_1} \;\;\;\;\;\; (8)
that is, the unbiased estimator of the second L-moment divided by the unbiased estimator of the first L-moment.
When l.moment.method="plotting.position", the L-CV is estimated by:
\tilde{t} = \frac{\tilde{l_2}}{\tilde{l_1}} \;\;\;\;\;\; (9)
that is, the plotting-position estimator of the second L-moment divided by the plotting-position estimator of the first L-moment.
See the help file for lMoment for more information on
estimating L-moments.
A numeric scalar – the sample coefficient of variation.
Traditionally, the coefficient of variation has been estimated using product moment estimators. Hosking (1990) introduced the idea of L-moments and the L-CV. Vogel and Fennessey (1993) argue that L-moment ratios should replace product moment ratios because of their superior performance (they are nearly unbiased and better for discriminating between distributions).
Steven P. Millard (EnvStats@ProbStatInfo.com)
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Taylor, J.K. (1990). Statistical Techniques for Data Analysis. Lewis Publishers, Boca Raton, FL.
Vogel, R.M., and N.M. Fennessey. (1993). L Moment Diagrams Should Replace Product Moment Diagrams. Water Resources Research 29(6), 1745–1752.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.
# Generate 20 observations from a lognormal distribution with # parameters mean=10 and cv=1, and estimate the coefficient of variation. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(250) dat <- rlnormAlt(20, mean = 10, cv = 1) cv(dat) #[1] 0.5077981 cv(dat, sd.method = "moments") #[1] 0.4949403 cv(dat, method = "l.moments") #[1] 0.2804148 #---------- # Clean up rm(dat)
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