lmomco (version 2.4.14)

sen.mean: Sen Weighted Mean Statistic

Description

The Sen weighted mean statistic \(\mathcal{S}_{n,k}\) is a robust estimator of the mean of a distribution $$\mathcal{S}_{n,k} = {n \choose 2k+1}^{-1} \sum_{i=1}^n {i - 1 \choose k} {n - i \choose k } x_{i:n}\mbox{,}$$ where \(x_{i:n}\) are the sample order statistics and \(k\) is a weighting or trimming parameter. If \(k = 2\), then the \(\mathcal{S}_{n,2}\) is the first symmetrical TL-moment (trim = 1). Note that \(\mathcal{S}_{n,0} = \mu = \overline{X}_n\) or the arithmetic mean and \(\mathcal{S}_{n,k}\) is the sample median if either \(n\) is even and \(k = (n/2) - 1\) or \(n\) is odd and \(k = (n-1)/2\).

Usage

sen.mean(x, k=0)

Value

An R

list is returned.

sen

The sen mean \(\mathcal{S}_{n,k}\).

source

An attribute identifying the computational source: “sen.mean”.

Arguments

x

A vector of data values that will be reduced to non-missing values.

k

A weighting or trimming parameter \(0 < k < (n-1)/2\).

Author

W.H. Asquith

References

Jurečková, J., and Picek, J., 2006, Robust statistical methods with R: Boca Raton, Fla., Chapman and Hall/CRC, ISBN 1--58488--454--1, 197 p.

Sen, P.K., 1964, On some properties of the rank-weighted means: Journal Indian Society of Agricultural Statistics: v. 16, pp. 51--61.

See Also

TLmoms, gini.mean.diff

Examples

fake.dat <- c(123, 34, 4, 654, 37, 78)
sen.mean(fake.dat); mean(fake.dat) # These should be the same values

sen.mean(fake.dat, k=(length(fake.dat)/2) - 1); median(fake.dat)
# Again, same values

# Finally, the sen.mean() is like a symmetrically trimmed TL-moment
# Let us demonstrate by computed a two sample trimming for each side
# for a Normal distribution having a mean of 100.
fake.dat <- rnorm(20, mean=100)
lmr <- TLmoms(fake.dat, trim=2)
sen <- sen.mean(fake.dat, k=2)

print(abs(lmr$lambdas[1] - sen$sen)) # zero is returned