\((Tg)(w) = \int g(x)f_{X|W}(x|w)\,dx\), represented on a sieve
basis as \(\hat T_{jk} = \hat E[p_k(X)q_j(W)]\). The operator's
singular values decay to zero – that decay IS the ill-posedness,
and how fast it decays determines whether the problem is mildly or
severely ill-posed. The singular values are returned so that is
visible rather than an abstraction. Mirrors
morie.fn.hrznpivt.
Usage
morie_npiv_operator(X, W, K = 5L, kind = "poly")
Arguments
- X
numeric endogenous regressor.
- W
numeric instrument, same length as X.
- K
sieve dimension for both bases.
- kind
basis type, "poly" or "fourier".
Value
list: T, singular_values, decay_ratio, severity, K, n, method.
References
Horowitz, Ch. 5, Sec. 5.3.
Examples
set.seed(1)
z <- rnorm(200)
morie_npiv_operator(z + rnorm(200), z)$severity
#> [1] "severe"
