Biweight Location

This function computes the biweight location, a robust measure of central tendency for a numeric vector. The biweight location is less sensitive to outliers than the sample mean.

Usage

biweight_location(
  x,
  loc = NULL,
  c = 6,
  tol = 1e-06,
  max_iter = 50,
  drop.na = FALSE
)

Arguments

x

A numeric vector.

loc

Initial guess for the location (default: median of x).

c

A numeric value specifying the tuning constant for the biweight estimator (c = 6 by default).

tol

Convergence tolerance for the iterative computation (default: 1e-6).

max_iter

Maximum number of iterations (default: 50).

drop.na

A logical value indicating whether to remove missing values (NA) from the calculations. If TRUE, missing values will be removed. If FALSE (the default), the result is NA when x contains missing values.

Value

The biweight location of x.

Details

Starting from the initial location \(M\) (the median by default), the biweight location is $$\zeta = M + \frac{\sum_{|u_i|<1} (x_i - M)(1 - u_i^2)^2}{\sum_{|u_i|<1} (1 - u_i^2)^2}, \quad u_i = \frac{x_i - M}{c \cdot \mathrm{MAD}}$$ where MAD is the (unscaled) median absolute deviation. The estimate is iterated, replacing \(M\) by \(\zeta\), until the change is smaller than tol or max_iter iterations are reached.

References

• Mosteller, F., and Tukey, J. W. (1977). Data Analysis and Regression: A Second Course in Statistics. Addison-Wesley, pp. 203-209.

• Beers, T.C., Flynn, K., Gebhardt, K., (1990). Measures of location and scale for velocities in clusters of galaxies - A robust approach. The Astronomical Journal, 100:32-46.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
# iron contents (mg/kg): a few samples are far above the others
fe <- forageLIBS$Fe[!is.na(forageLIBS$Fe)]
c(mean = mean(fe), median = stats::median(fe), biweight = biweight_location(fe))
#>     mean   median biweight 
#> 205.5288 137.0000 139.1975