OutlierStatistic {probout}

Nonparametric Outlier Statistic

Description

Robust nonparametric outlier statistic for univariate or multivariate data.

Usage

OutlierStatistic( x, nproj=1000, prior=NULL, seed=NULL)

Arguments

x

A numeric vector or matrix for which the outlier statistic is to be determined.

nproj

If x is multivariate, the number of random projections to be used in computing the statistic.

prior

If x is multivariate, a prior estimate of the statistic for each observation in x, to be used as a base line for maximization relative to new random projections.

seed

An optional integer argument to set.seed for reproducible simulations. By default the current seed will be used. Reproducibility can also be achieved by calling set.seed before calling OutlierStatistic.

Value

A vector giving the maximum value of the outlier statistic for each observation over all projections.

References

W. A. Stahel, Breakdown of Covariance Estimators, doctoral thesis, Fachgruppe Fur Statistik, Eidgenossische Technische Hochshule (ETH), 1981.

D. L. Donoho, Breakdown Properties of Multivariate Location Estimators, doctoral thesis, Department of Statistics, Harvard University, 1982.

Note

Note that partition probabilities are computed from an exponential distribution fit to the outlier statistic, rather than from the empirical distribution of the outlier statistic.

See Also

partProb

Examples

 stat <- OutlierStatistic(faithful)
 q.99 <- quantile(stat,.99)
 out <- stat > q.99

 plot( faithful[,1], faithful[,2], 
       main="red : .99 quantile for outlier statistic", cex=.5)
 points( faithful[out,1], faithful[out,2], 
         pch = 4, col = "red", lwd = 1, cex = .5)

 require(mvtnorm)

 set.seed(0)
 Sigma <- crossprod(matrix(rnorm(2*2),2,2))
 x <- rmvt( 10000, sigma = Sigma, df = 2) 

 stat <- OutlierStatistic(x)
 q.95 <- quantile(stat,.95)

 hist(x, main = "gray : .95 quantile for outlier statistic", col = "black")
 abline( v = x[stat > q.95], col = "gray")
 hist(x, col = "black", add = TRUE)