An object returned by the degross function is a list containing several components resulting from the density estimation procedure.
A degross object is a list containing, after convergence of the EM algorithm :
lpost & lpost.ni: value of the log-posterior for the complete data based on the expected small bin frequencies n.i at convergence of the EM algorithm.
lpost.mj : value of the log-posterior for the observed data based on the big bin frequencies freq.j.
llik.ni: log-likelihood for the complete data based on the estimated small bin frequencies n.i.
llik.mj : log-likelihood for the observed data based on the big bin frequencies freq.j.
moments.penalty : log of the joint (asymptotic) density for the observed sample moments.
penalty : \(\log p(\phi|\tau) + \log p(\tau)\).
Score & Score.mj: score (w.r.t. \(\phi\)) of the log of the observed joint posterior function.
Score.ni: score (w.r.t. \(\phi\)) of the log-posterior for the complete data based on the expected small bin frequencies n.i at convergence of the EM algorithm.
Fisher & Fisher.ni: information matrix (w.r.t. \(\phi\)) based on the log-posterior for the complete data based on the expected small bin frequencies n.i at convergence of the EM algorithm.
Fisher.mj : information matrix (w.r.t. \(\phi\)) based on the log of the observed joint posterior function.
M.j : theoretical moments of the fitted density within a big bin.
pi.i : small bin probabilities (at convergence).
ui : small bin midpoints.
delta : width of the small bins.
gamma.j : big bin probabilities (at convergence).
tau : value of the roughness penalty parameter \(\tau\) (tau0 if fixed.tau=TRUE, estimated otherwise).
phi : vector with the spline parameters (at convergence).
n.i : small bin frequencies under the estimated density (at convergence).
edf : the effective degrees of freedom (or effective number of spline parameters) (at convergence).
aic : -2*(llik.mj + moments.penalty) + 2edf.
bic : -2(llik.mj + moments.penalty) + \(\log(n)\)*edf.
log.evidence : approximation to the log of \(p(\hat{\phi}_\tau,\hat{\tau} | D)\) \(|\Sigma_\phi|^{(1/2)}\).
degross.data : the degrossData object from which density estimation proceeded.
use.moments : vector of 4 logicals indicating which tabulated sample moments were used as soft constraints during estimation.
diag.only : logical indicating whether the off-diagonal elements of the variance-covariance matrix of the sample central moments were ignored. Default: FALSE.
logNormCst : log of the normalizing constant when evaluating the density.
Lambert, P. (2021) Moment-based density and risk estimation from grouped summary statistics. arXiv:2107.03883.