VGAM (version 1.1-9)

huber2: Huber's Least Favourable Distribution Family Function

Description

M-estimation of the two parameters of Huber's least favourable distribution. The one parameter case is also implemented.

Usage

huber1(llocation = "identitylink", k = 0.862, imethod = 1)
huber2(llocation = "identitylink", lscale = "loglink",
       k = 0.862, imethod = 1, zero = "scale")

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

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.

Author

T. W. Yee. Help was given by Arash Ardalan.

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.

References

Huber, P. J. and Ronchetti, E. (2009). Robust Statistics, 2nd ed. New York: Wiley.

See Also

rhuber, uninormal, laplace, CommonVGAMffArguments.

Examples

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)


if (FALSE)  # 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))