adaHuber (version 1.1)

adaHuber.cov: Adaptive Huber Covariance Estimation

Description

Adaptive Huber covariance estimator from a data sample, with robustification parameter \(\tau\) determined by a tuning-free principle.

Usage

adaHuber.cov(X, epsilon = 1e-04, iteMax = 500)

Arguments

X

An \(n\) by \(p\) data matrix.

epsilon

(optional) The tolerance level in the iterative estimation procedure. The problem is converted to mean estimation, and the stopping rule is the same as adaHuber.mean. The defalut value is 1e-4.

iteMax

(optional) Maximum number of iterations. Default is 500.

Value

A list including the following terms will be returned:

means

The Huber estimators for column means. A \(p\)-dimensional vector.

cov

The Huber estimator for covariance matrix. A \(p\) by \(p\) matrix.

Details

The observed data \(X\) is an \(n\) by \(p\) matrix. The distribution of each entry can be asymmetrix and/or heavy-tailed. The function outputs a robust estimator for the covariance matrix of \(X\). For the input matrix X, both low-dimension (\(p < n\)) and high-dimension (\(p > n\)) are allowed.

References

Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist., 35, 73<U+2013>101.

Ke, Y., Minsker, S., Ren, Z., Sun, Q. and Zhou, W.-X. (2019). User-friendly covariance estimation for heavy-tailed distributions. Statis. Sci., 34, 454-471.

See Also

adaHuber.mean for adaptive Huber mean estimation.

Examples

# NOT RUN {
n = 100
p = 5
X = matrix(rt(n * p, 3), n, p)
fit.cov = adaHuber.cov(X)
fit.cov$means
fit.cov$cov
# }