huber: Huber's Least Favourable Distribution Family Function

In VGAM: Vector Generalized Linear and Additive Models

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")

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.

Value

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

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.

Author(s)

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

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)


## 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)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.