\(\sqrt n(\hat\theta_n - \theta_0) \rightsquigarrow -V^{-1}Z\)
(Thm. 2.13, p. 29). The limit is a SANDWICH
\(V^{-1}\Sigma V^{-1}\): V is the curvature of the criterion and
\(\Sigma = P\dot m\dot m'\) the variability of its gradient.
They coincide only for a correctly specified log-likelihood – the
information equality – and assuming they do elsewhere is the
standard way to get standard errors wrong. Mirrors
morie.fn.ksr057.
Usage
morie_m_normality(m_dot_scores, V = NULL)
Arguments
- m_dot_scores
n by p gradient at theta_0.
- V
curvature matrix; Sigma is used when NULL, which ASSUMES
the information equality.
Value
list: Sigma, V, avar, se, information_equality_assumed,
information_equality_holds, n, p, method.
References
Kosorok, Thm. 2.13, p. 29.
Examples
set.seed(1)
morie_m_normality(matrix(rnorm(200), ncol = 2))$information_equality_assumed
#> [1] TRUE
