# Data driven robust methods Robust estimation often relies on a dispersion function that is more slowly varying at large values than the squared function. However, the choice of tuning constant in dispersion function may impact the estimation efficiency to a great extent. For a given family of dispersion functions, we suggest obtaining the `best' tuning constant from the data so that the asymptotic efficiency is maximized. This library provides a robust linear regression with a tuning parameter being automatically chosen to provide the necessary resistance against outliers. The robust (loss) functions include the Huber, Tukey bisquare and the exponential loss. ```r rlmDD(yy, xx, beta0, betaR, method, plot) ``` ## Arguments - `yy`: Vector representing the response variable - `xx`: Design matrix of the covariates excluding the intercept in the first column - `beta0`: Initial parameter estimate using `lm` - `betaR`: Robust estimate of beta with a fixed tuning constant using `rlm` - `method`: Huber, Bisquare or Exponential - `plot`: "Y" gives a plot: the efficiency factor versus a range of tunning parameter values. ## Returns The function returns a list including - **esti**: Value of the robust estimate - **Std.Error**: Standard error of the robust estimate - **tunning**: Optimum tunning parameter - **R2**: R-squared value ## References Wang, Y-G., Lin, X., Zhu, M., & Bai, Z. (2007). Robust estimation using the Huber function with a data-dependent tuning constant. Journal of Computational and Graphical Statistics, 16(2), 468-481. Wang, X., Jiang, Y., Huang, M., & Zhang, H. (2013). Robust variable selection with exponential squared loss. Journal of the American Statistical Association, 108, 632-643. Wang, N., Wang, Y-G., Hu, S., Hu, Z. H., Xu, J., Tang, H., & Jin, G. (2018). Robust Regression with Data-Dependent Regularization Parameters and Autoregressive Temporal Correlations. Environmental Modeling & Assessment, in press. ## Author(s) You-Gan Wang, Na Wang ## See Also `rlm` function from package `MASS` ## Examples ```r library(MASS) data(stackloss) LS <- lm(stack.loss ~ stack.x) RB <- rlm(stack.loss ~ stack.x, psi = psi.huber, k = 1.345) DD1 <- rlmDD(stack.loss, stack.x, LS$coef, RB$coef, method = "Huber", plot = "Y") LS <- lm(stack.loss ~ stack.x) RB <- rlm(stack.loss ~ stack.x, psi = psi.bisquare, c = 4.685) DD2 <- rlmDD(stack.loss, stack.x, LS$coef, RB$coef, method = "Bisquare", plot = "Y") LS <- lm(stack.loss ~ stack.x) RB <- rlm(stack.loss ~ stack.x, psi = psi.huber, k = 1.345) DD3 <- rlmDD(stack.loss, stack.x, LS$coef, RB$coef, method = "Exponential", plot = "Y") ## Plasma dataset data(plasma) y <- plasma$y x <- cbind(plasma$calories, plasma$dietary) LS <- lm(y ~ x) RB <- rlm(y ~ x, psi = psi.huber, k = 1.345) DD.h <- rlmDD(y, x, LS$coef, RB$coef, method = "Huber", plot = "Y") LS <- lm(y ~ x) RB <- rlm(y ~ x, psi = psi.bisquare, c = 4.685) DD.b <- rlmDD(y, x, LS$coef, RB$coef, method = "Bisquare", plot = "Y") LS <- lm(y ~ x) RB <- rlm(y ~ x, psi = psi.huber, k = 1.345) DD.e <- rlmDD(y, x, LS$coef, RB$coef, method = "Exponential", plot = "Y") ```

rlmDD function