Calculation of variance estimates of the estimated GB2 parameters and the estimated GB2 indicators under cluster sampling.
varscore.gb2(x, shape1, scale, shape2, shape3, w=rep(1,length(x)), hs=rep(1,length(x)))
vepar.gb2(x, Vsc, shape1, scale, shape2, shape3, w=rep(1,length(x)), hs=rep(1,length(x)))
derivind.gb2(shape1, scale, shape2, shape3)
veind.gb2(Vpar, shape1, scale, shape2, shape3)varscore.gb2 calculates the middle term of the sandwich variance estimator under simple random cluster sampling. vepar.gb2 returns a list of two elements:
the estimated variance-covariance matrix of the estimated GB2 parameters and the second-order partial derivative of the pseudo log-likelihood function.
The function veind.gb2 returns the estimated variance-covariance matrix of the estimated GB2 indicators. derivind.gb2 calculates the numerical derivatives of the GB2 indicators and is for internal use only.
numeric; vector of data values.
numeric; 4 by 4 matrix.
numeric; positive parameter.
numeric; positive parameter.
numeric; positive parameters of the Beta distribution.
numeric; vector of weights. Must have the same length as x. By default w is a vector of 1.
numeric; vector of household sizes. Must have the same length as x. By default w is a vector of 1.
numeric; 4 by 4 matrix.
Monique Graf and Desislava Nedyalkova
Knowing the first and second derivatives of \(log(f)\), and using the sandwich variance estimator (see Freedman (2006)), the calculation of the variance estimates of the GB2
parameters is straightforward. Vsc is a square matrix of size the number of parameters, e.g. the estimated design variance-covariance matrix of the estimated parameters. We know that the GB2 estimates of the Laeken indicators are functions of the GB2 parameters. In this case, the variance estimates of the fitted indicators are obtained
using the delta method. The function veind.gb2 uses Vpar, the sandwich variance estimator of the vector of parameters, in order to obtain the sandwich variance estimator of the indicators. More details can be found in Graf and Nedyalkova (2011).
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.