CompoundVarest: Variance Estimation of the Compound GB2 Distribution

In GB2: Generalized Beta Distribution of the Second Kind: Properties, Likelihood, Estimation

Variance Estimation of the Compound GB2 Distribution

Description

Calculation of variance estimates of the parameters of the compound GB2 distribution and of the estimated compound GB2 indicators under cluster sampling.

Usage

scoreU.cgb2(fac, pl)
varscore.cgb2(U, w=rep(1,dim(U)[1]))
desvar.cgb2(data=data, U=U, ids=NULL, probs=NULL, strata = NULL, variables = NULL, 
fpc=NULL, nest = FALSE, check.strata = !nest, weights=NULL, pps=FALSE, 
variance=c("HT","YG"))
hess.cgb2(U, pl, w=rep(1,dim(U)[1]))
vepar.cgb2(ml, Vsc, hess)
derivind.cgb2(shape1, scale, shape2, shape3, pl0, pl, prop=0.6, decomp="r")
veind.cgb2(Vpar, shape1, scale, shape2, shape3, pl0, pl, decomp="r")

Arguments

fac	numeric; matrix of Gamma factors (output of fac.cgb2.
pl	numeric; a vector of fitted mixture probabilities. Sums to one. If pl is equal to pl0, we obtain the GB2 distribution.
U	numeric; vector of scores. Output of the scoreU.cgb2 function.
w	numeric; vector of some extrapolation weights. By default w is a vector of 1.
data	dataset containing the design information per unit.
ids, probs, strata, variables, fpc, nest, check.strata, weights, pps, variance	parameters of svydesign.
ml	numeric; output of the ml.cgb2 function. A list with two components. First component: estimated mixture probabilities. Second component: list containing the output of optim.
Vsc	numeric; 4 by 4 matrix.
hess	numeric; Hessian (bread) for the sandwich variance estimate.
shape1, scale ,shape2, shape3	numeric; positive parameters of the GB2 distribution.
pl0	numeric; a vector of initial proportions defining the number of components and the weight of each component density in the decomposition. Sums to one.
prop	numeric; proportion (in general is set to 0.6).
decomp	string; specifying if the decomposition of the GB2 is done with respect to the right tail ("r") or the left tail ("l") of the distribution.
Vpar	numeric; 4 by 4 matrix. Output of the function vepar.cgb2.

Details

Function scoreU.cgb2 calculates the N \times (L-1) matrix of scores U is defined as

U(k,\ell)=p_{\ell} ≤ft( \frac{F(k,\ell)}{∑_{j=1}^L p_{j}\, F(k,j)} - 1\right),

where p_\ell, \ell=1,..,L is the vector of fitted mixture probabilities and F is the N \times L matrix of gamma factors, output of fg.cgb2. The linearized scores are the columns of U. They serve to compute the linearization approximation of the covariance matrix of the parameters v_\ell=\log(p_\ell/p_L), \ell=1,...,L-1. Function varscore.cgb2 calculates the middle term of the sandwich variance estimator, that is the ((L-1) \times (L-1)) estimated variance-covariance matrix of the (L-1) weighted sums of the columns of U, without design information. desvar.cgb2 calculates the design-based variance-covariance matrix of the (L-1) weighted sums of the columns of U, invoking svydesign and svytotal of package survey. hess.cgb2 calculates the Hessian ((L-1) \times (L-1)) matrix of second derivatives of the pseudo-log-likelihood with respect to the parameters v_\ell). It should be negative definite. If not, the maximum likelihood estimates are spurious. vepar.cgb2 calculates the sandwich covariance matrix estimate of the vector of parameters v. veind.cgb2 calculates estimates, standard error, covariance and correlation matrices of the indicators under the compound GB2.

Value

scoreU.cgb2 returns a N \times (L-1) matrix of scores <codeU. varscore.cgb2 returns the variance-covariance estimate of the weighted sums of scores U, given by weighted cross products. desvar.cgb2 returns a list of two elements. The first is the output of svytotal and the second is the design-based variance-covariance matrix of the weighted sums of the scores. hess.cgb2 returns the matrix of second derivatives of the likelihood with respect to the parameters (bread for the sandwich variance estimate). vepar.cgb2 returns a list of five elements - [["type"]] with value "parameter", [["estimate"]] estimated parameters, [["stderr"]] corresponding standard errors, [["Vcov"]] variance -covariance matrix and [["Vcor"]] - correlation matrix. veind.cgb2 returns a list of five elements: [["type"]] with value "indicator", [["estimate"]] estimated indicators under the compound GB2, [["stderr"]] corresponding standard errors, [["Vcov"]] variance -covariance matrix and [["Vcor"]] - correlation matrix.

Author(s)

Monique Graf and Desislava Nedyalkova

References

Davison, A. (2003), Statistical Models. Cambridge University Press.

Freedman, D. A. (2006), On The So-Called "Huber Sandwich Estimator" and "Robust Standard Errors". The American Statistician, 60, 299–302.

Graf, M., Nedyalkova, D., Muennich, R., Seger, J. and Zins, S. (2011) AMELI Deliverable 2.1: Parametric Estimation of Income Distributions and Indicators of Poverty and Social Exclusion. Technical report, AMELI-Project.

Pfeffermann, D. and Sverchkov, M. Yu. (2003), Fitting Generalized Linear Models under Informative Sampling. In, Skinner, C.J. and Chambers, R.L. (eds.). Analysis of Survey Data, chapter 12, 175–195. Wiley, New York.

Examples

## Not run: 
# Example (following of example in CompoundFit)

# Estimated mixture probabilities:
(pl.hat <- estim[[1]])

# scores per unit
U <- scoreU.cgb2(fac, pl.hat)  

# Conditional variances given a,b,p,q:

# 1. Variance of sum of scores:
(Vsc <- t(U) 
(Vsc <- varscore.cgb2(U))        

# 2. sandwich variance-covariance matrix estimate of (v_1,v_2):
(hess <-  hess.cgb2(U,pl.hat))
(Parameters <- vepar.cgb2(estim, Vsc, hess))

# 3. Theoretical indicators  (with mixture prob pl)
decomp <- "r"
(theoretical <- main.cgb2( 0.6,a,b,p,q,pl0, pl,decomp=decomp))

# Estimated indicators and conditional variances : takes a long time!
(Indic <- veind.cgb2(Parameters,a,b,p,q, pl0, pl.hat, decomp="r") )


## End(Not run)

GB2 documentation built on June 22, 2022, 9:07 a.m.