degross.object {degross}

Object resulting from the estimation of a density from grouped (tabulated) summary statistics

Description

An object returned by the degross function is a list containing several components resulting from the density estimation procedure.

Value

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.

Author(s)

Philippe Lambert [e-mail]

References

Lambert, P. (2021) Moment-based density and risk estimation from grouped summary statistics. arXiv:2107.03883.

See Also

degross, print.degross, plot.degross.