degross {degross}

Density estimation from tabulated data with given frequencies and group central moments.

Description

Estimation of a density from tabulated summary statistics evaluated within each of the big bins (or classes) partitioning the variable support. These statistics include class frequencies and central moments of orders one up to four. The log-density is modelled using a linear combination of penalized B-splines. The multinomial log-likelihood involving the frequencies adds up to a roughness penalty based on differences of neighboring B-spline coefficients and to the log of a root-n approximation of the sampling density of the observed vector of central moments within each class. The so-obtained penalized log-likelihood is maximized using the EM algorithm to get an estimation of the spline parameters and, hence, of the variable density and related quantities such as quantiles, see Lambert (2021) for details.

Usage

degross(degross.data,
       phi0 = NULL, tau0 = 1000,
       use.moments = rep(TRUE,4), freq.min = 20, diag.only=FALSE,
       penalize = TRUE,
       aa = 2, bb = 1e-06, pen.order = 3, fixed.tau = FALSE,
       plotting = FALSE, verbose = FALSE, iterlim=20)

Arguments

degross.data

A degrossData.object generated by degrossData.

phi0

Starting value for the K-vector \phi of B-spline parameters specifying the log-density. Default: NULL.

tau0

Starting value for the roughness penalty parameter. Default: 1000.

use.moments

Vector with 4 logicals indicating which tabulated sample moments to use as soft constraints. Defaults: rep(TRUE,4).

freq.min

Minimal big bin frequency required to use the corresponding observed moments as soft constraints. Default: 20.

diag.only

Logical indicating whether to ignore the off-diagonal elements of the variance-covariance matrix of the sample central moments. Default: FALSE.

penalize

Logical indicating whether a roughness penalty of order pen.order is required (with \tau \sim G(aa,bb)). Default: TRUE.

aa

Positive real giving the first parameter in the Gamma prior for tau. Default: 2.

bb

Positive real giving the second parameter in the Gamma prior for tau. Default: 1e-6.

pen.order

Integer giving the order of the roughness penalty. Default: 3.

fixed.tau

Logical indicating whether the roughness penalty parameter tau is fixed. Default: FALSE, implying its estimation.

plotting

Logical indicating whether an histogram of the data with the estimated density should be plotted. Default: FALSE.

verbose

Logical indicating whether details on the estimation progress should be displayed. Default: FALSE.

iterlim

Maximum number of iterations during the M-step. Default: 20.

Value

An object of class degross containing several components from the density estimation procedure. Details can be found in degross.object. A summary of its content can be printed using print.degross or plotted using plot.degross.

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.object, ddegross, pdegross, qdegross.

Examples

## Simulate grouped data
sim = simDegrossData(n=3500, plotting=TRUE,choice=2,J=3)
print(sim$true.density) ## Display density of the data generating mechanism

## Create a degrossData object
obj.data = with(sim, degrossData(Big.bins=Big.bins, freq.j=freq.j, m.j=m.j))
print(obj.data)

## Estimate the density underlying the grouped data
obj.fit = degross(obj.data)

## Plot the estimated density...
plot(obj.fit)
## ... and compare it with the ('target') density used to simulate the data
curve(sim$true.density(x),add=TRUE,col="red",lwd=2)
legend("topleft",
       legend=c("Observed freq.","Target density","Estimated density"),
       col=c("grey85","red","black"), lwd=c(10,2,2),
       lty=c("solid","solid","dashed"), box.lty=0, inset=.02)