roblm: Huber's Regression M-Estimator
Description Usage Arguments Value References See Also
View source: R/robustregression.R
Description
This uses Huber's (1964) loss function to estimate a robust least squares fit.
This is a simple M-estimator. The standard errors are based on the modified-HC4 method of
computing robust standard errors. While this estimator is resistant to outliers in the
response variable (leverage points), it is not robust to outliers in the design matrix.
The modified HC4 standard errors afford some robustness to leverage points, however, as well
as robustness to heteroskedasticity. Huber-White standard errors originate with Huber (1967)
and White (1980), and the modified HC4 method is dued to Cribari-Neto and Da Silva (2011)
as an improvement to Cribari-Neto's (2004) HC4 method.
Let ε be the squared residuals, (y - yhat)^2. Huber's loss function is a function
of the squared errors, ρ(ε) = if(ε < k) then ε; else 2k*√(ε) - k^2.
k is a constant specified by the analyst. A common default choice is 1.345, which yields an asymptotic
efficiency of 95% that of the OLS estimate under Gaussian errors. Note that if errors deviate in
the slighest from Gaussian, the asymptotic relative efficiency of Huber's M-Estimate may exceed
that of OLS.
The modified HC4 standard errors are calculated as follows:
Let xtxi = (X'X)^-1, the Fisher Information Matrix. Let h be the diagonal of the hat matrix. Let δ = min(1, N * h/P) + min(1.5, N * h/P). Let ε be the squared errors, and ω be the diagonal matrix of ε^2 / (1-h)^{δ}. Next, let Ω = X'ωX. Finally, the variance-covariance matrix is given by vcov = (xtxi)Ω(xtxi). The standard errors are the square root of the diagonal of vcov.
Alternatively, one may use the roblm.boot function to compute bootstrap estimates
of standard errors.
Usage
1 |
Arguments
formula |
model formula |
data |
data frame |
k |
tuning constant for outlier downweighting. defaults to 1.345. |
Value
a list
References
#'
Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101.
Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability. 221–233.
White, H. (1980). A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity. Econometrica. 48 (4): 817–838. CiteSeerX 10.1.1.11.7646. doi:10.2307/1912934.
Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics and Data Analysis 45, 215–233. MR2045469
Cribari-Neto, F. & Da Silva, W.B. (2011), A New Heteroskedasticity-Consistent Covariance Matrix Estimator for the Linear Regression Model. Advances in Statistical Analysis, 95(2), 129–146.
See Also
roblm.boot
