Theorem 4.5: strong uniform consistency on a bounded interval

Stronger than pointwise convergence, and what licenses using the whole estimated curve – a maximum, a crossing point – rather than one pre-chosen t. The interval must be BOUNDED; the proof runs by monotonicity plus pointwise convergence on a finite grid, as in Glivenko-Cantelli.

Usage

morie_fauzi_theorem_4_5(mrl_hat, mrl_true, t_grid, interval = NULL)

Arguments

mrl_hat

estimated curve.

mrl_true

true curve, same length.

t_grid

the grid both are evaluated on.

interval

optional bounded c(lo, hi) to restrict to.

Value

list: sup_error, argmax_t, interval, mode, requires_bounded_B, stronger_than_pointwise, proof_device, licenses, method.

References

Fauzi and Maesono (2023), Theorem 4.5.

Examples

tg <- seq(0, 2, length.out = 21)
morie_fauzi_theorem_4_5(rep(1.1, 21), rep(1, 21), tg)$sup_error
#> [1] 0.1