MKmisc (version 1.9)

meanAD: The Mean Absolute Deviation

Description

Computes (standardized) mean absolute deviation.

Usage

meanAD(x, na.rm = FALSE, constant = sqrt(pi/2))

Arguments

x

a numeric vector.

na.rm

logical. Should missing values be removed?

constant

standardizing contant; see details below.

Author

Matthias Kohl

Details

The mean absolute deviation is a consistent estimator of \(\sqrt{2/\pi}\sigma\) for the standard deviation of a normal distribution. Under minor deviations of the normal distributions its asymptotic variance is smaller than that of the sample standard deviation (Tukey (1960)).

It works well under the assumption of symmetric, where mean and median coincide. Under the normal distribution it's about 18% more efficient (asymptotic relative efficiency) than the median absolute deviation ((1/qnorm(0.75))/sqrt(pi/2)) and about 12% less efficient than the sample standard deviation (Tukey (1960)).

References

Tukey, J. W. (1960). A survey of sampling from contaminated distribution. In Olink, I., editor, Contributions to Probablity and Statistics. Essays in Honor of H. Hotelling., pages 448-485. Stanford University Press.

See Also

sd, mad, sIQR.

Examples

## right skewed data
## mean absolute deviation
meanAD(rivers)
## standardized IQR
sIQR(rivers)
## median absolute deviation
mad(rivers)
## sample standard deviation
sd(rivers)

## for normal data
x <- rnorm(100)
sd(x)
sIQR(x)
mad(x)
meanAD(x)

## Asymptotic relative efficiency for Tukey's symmetric gross-error model
## (1-eps)*Norm(mean, sd = sigma) + eps*Norm(mean, sd = 3*sigma)
eps <- seq(from = 0, to = 1, by = 0.001)
ARE <- function(eps){
  0.25*((3*(1+80*eps))/((1+8*eps)^2)-1)/(pi*(1+8*eps)/(2*(1+2*eps)^2)-1)
}
plot(eps, ARE(eps), type = "l", xlab = "Proportion of gross-errors",
     ylab = "Asymptotic relative efficiency", 
     main = "ARE of mean absolute deviation w.r.t. sample standard deviation")
abline(h = 1.0, col = "red")
text(x = 0.5, y = 1.5, "Mean absolute deviation is better", col = "red", 
    cex = 1, font = 1)
## lower bound of interval
uniroot(function(x){ ARE(x)-1 }, interval = c(0, 0.002))
## upper bound of interval
uniroot(function(x){ ARE(x)-1 }, interval = c(0.5, 0.55))
## worst case
optimize(ARE, interval = c(0,1), maximum = TRUE)