Joint convergence of parameter and nuisance — morie_joint

\(\sqrt n(\hat\theta_n - \theta_0, \hat\eta_n - \eta_0) \rightsquigarrow -\dot\Psi_0^{-1}Z\) (Cor. 3.2, p. 47), under the no-bias condition and stochastic equicontinuity.

Usage

morie_joint_convergence(psi_dot, scores, n = NULL)

Arguments

psi_dot: the derivative operator, square and invertible.
scores: n by d stacked influence contributions.
n: sample size; from scores when NULL.

Value

list: avar, se, correlation, jointly, operator_invertible, conditions, warning, n, d, method.

Details

JOINTLY, not separately. The two estimates solve the same estimating equation and are correlated, so a valid region for a function of both needs the joint law; combining marginal limits as if independent understates the variability. One operator inverse gives both blocks, which is why their dependence is determined rather than assumed. Mirrors morie.fn.ksr073.

References

Kosorok, Cor. 3.2, p. 47.

Examples

set.seed(1)
morie_joint_convergence(diag(2), matrix(rnorm(200), ncol = 2))$jointly
#> [1] TRUE