One example is estimates from ordinary least squares regression when there is collinearity. They are unbiased but have huge variance. Ridge regression on the same problem yields estimates that are biased but have much lower variance.

One example is estimates from ordinary least squares regression when there is collinearity. They are unbiased but have huge variance. Ridge regression on the same problem yields estimates that are biased but have much lower variance. E.g.

install.packages("ridge")
library(ridge)
set.seed(831)

data(GenCont)
ridgemod <- linearRidge(Phenotypes ~ ., data = as.data.frame(GenCont))
summary(ridgemod)
linmod <- lm(Phenotypes ~ ., data = as.data.frame(GenCont))
summary(linmod)

The t values are much larger for ridge regression than linear regression. The bias is fairly small.