LogLikelihood: Full Log-likelihood of the GB2 Distribution

Description

Calculates the log-likelihood, the score functions of the log-likelihood and the Fisher information matrix based on all four parameters of the GB2 distribution.

Usage

loglp.gb2(x, shape1, scale, shape2, shape3, w=1)
loglh.gb2(x, shape1, scale, shape2, shape3, w=1, hs)
scoresp.gb2(x, shape1, scale, shape2, shape3, w=1)
scoresh.gb2(x, shape1, scale, shape2, shape3, w=1, hs)
info.gb2(shape1, scale, shape2, shape3)

Arguments

vector of data values.

shape1

positive parameter.

scale

positive parameter.

shape2, shape3

positive parameters of the Beta distribution.

vector of weights.

vector of household sizes.

Details

We express the log-likelihood as a weighted mean of $log(f)$, evaluated at the data points, where $f$ is the GB2 density with parameters shape1 $= a$, scale $= b$, shape2 $= p$ and shape3 $= q$. If the weights are not available, then we suppose that w $= 1$. loglp.gb2 calculates the log-likelihood in the case where the data is a sample of persons and loglh.gb2 is adapted for a sample of households. Idem for the scores, which are obtained as weighted sums of the first derivatives of $log(f)$ with respect to the GB2 parameters, evaluated at the data points. The Fisher information matrix of the GB2 was obtained by Brazauskas (2002) and is expressed in terms of the second derivatives of the log-likelihood with respect to the parameters of the GB2.

References

Brazauskas, V. (2002) Fisher information matrix for the Feller-Pareto distribution. Statistics & Probability Letters, 59, 159--167. Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, chapter 6. Wiley, Ney York.