Huber loss function

The two branches agree in value and in slope at |r| = k, both giving k^2/2 and slope k, which is what makes the loss convex and continuously differentiable.

Usage

Hubrho(r, k = 1.345)

Arguments

r

Residuals.

k

Tuning constant, positive.

Value

List with estimate (total loss), loss, psi, mean_loss, k, n, method.

Details

Formula: rho(r) = r^2/2 if |r| <= k, else k(|r| - k/2).

References

Huber (1964), Annals of Mathematical Statistics 35(1):73-101. doi:10.1214/aoms/1177703732

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Hubrho(V)
#> $estimate
#> [1] 41.24341
#> 
#> $loss
#> [1] 0.500000 1.785488 3.130488 4.475487 5.820487 7.165487 8.510487 9.855487
#> 
#> $psi
#> [1] 1.000 1.345 1.345 1.345 1.345 1.345 1.345 1.345
#> 
#> $mean_loss
#> [1] 5.155427
#> 
#> $k
#> [1] 1.345
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "rho(r) = r^2/2 for |r| <= k else k(|r| - k/2), Huber (1964)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"