# Huber's Least Favourable Distribution Family Function M-estimation of the two parameters of Huber's least favourable distribution. The one parameter case is also implemented. ```r huber1(llocation = "identitylink", k = 0.862, imethod = 1) huber2(llocation = "identitylink", lscale = "loglink", k = 0.862, imethod = 1, zero = "scale") ``` ## Arguments - `llocation, lscale`: Link functions applied to the location and scale parameters. See `Links` for more choices. - `k`: Tuning constant. See `rhuber` for more information. - `imethod, zero`: See `CommonVGAMffArguments` for information. The default value of `zero` means the scale parameter is modelled as intercept-only. ## Details Huber's least favourable distribution family function is popular for resistant/robust regression. The center of the distribution is normal and its tails are double exponential. By default, the mean is the first linear/additive predictor (returned as the fitted values; this is the location parameter), and the log of the scale parameter is the second linear/additive predictor. The Fisher information matrix is diagonal; Fisher scoring is implemented. The `VGAM` family function `huber1()` estimates only the location parameter. It assumes a scale parameter of unit value. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. ## References Huber, P. J. and Ronchetti, E. (2009). **Robust Statistics**, 2nd ed. New York: Wiley. ## Author(s) T. W. Yee. Help was given by Arash Ardalan. ## Note Warning: actually, `huber2()` may be erroneous since the first derivative is not continuous when there are two parameters to estimate. `huber1()` is fine in this respect. The response should be univariate. ## See Also `rhuber`, `uninormal`, `laplace`, `CommonVGAMffArguments`. ## Examples ```r set.seed(1231); NN <- 30; coef1 <- 1; coef2 <- 10 hdata <- data.frame(x2 = sort(runif(NN))) hdata <- transform(hdata, y = rhuber(NN, mu = coef1 + coef2 * x2)) hdata$x2[1] <- 0.0 # Add an outlier hdata$y[1] <- 10 fit.huber2 <- vglm(y ~ x2, huber2(imethod = 3), hdata, trace = TRUE) fit.huber1 <- vglm(y ~ x2, huber1(imethod = 3), hdata, trace = TRUE) coef(fit.huber2, matrix = TRUE) summary(fit.huber2) ## Not run: # Plot the results plot(y ~ x2, data = hdata, col = "blue", las = 1) lines(fitted(fit.huber2) ~ x2, data = hdata, col = "darkgreen", lwd = 2) fit.lm <- lm(y ~ x2, hdata) # Compare to a LM: lines(fitted(fit.lm) ~ x2, data = hdata, col = "lavender", lwd = 3) # Compare to truth: lines(coef1 + coef2 * x2 ~ x2, data = hdata, col = "orange", lwd = 2, lty = "dashed") legend("bottomright", legend = c("truth", "huber", "lm"), col = c("orange", "darkgreen", "lavender"), lty = c("dashed", "solid", "solid"), lwd = c(2, 2, 3)) ## End(Not run) ```

huber function