GB2 (version 2.1.2)

CompoundFit: Fitting the Compound Distribution based on the GB2 by the Method of Maximum Likelihood Estimation

Description

Calculates the log-likelihood, the score functions of the log-likelihood, the weighted mean of scores, and fits the parameters of the Compound Distribution based on the GB2.

Usage

vofp.cgb2(pl)
pofv.cgb2(vl)
logl.cgb2(fac, pl, w=rep(1, dim(fac)[1]))
scores.cgb2(fac, pl, w=rep(1, dim(fac)[1]))
ml.cgb2(fac, pl0, w=rep(1, dim(fac)[1]), maxiter=100, fnscale=length(w))

Value

vofp.cgb2 returns a vector of length \(L-1\), where \(L\) is the length of the vector \(p_\ell\). pofv.cgb2 returns a vector of length \(\ell\). logl.cgb2 returns the value of the pseudo log-likelihood. scores.cgb2 returns a vector of the weighted mean of the scores of length \(L-1\). ml.cgb2 returns a list containing two objects - the vector of fitted proportions \(\hat{p_\ell}\) and the output of the BFGS fit.

Arguments

pl0

numeric; vector of initial proportions defining the number of components and the weight of each component density in the decomposition. Sums to one.

pl

numeric; vector of fitted proportions. Sums to one. If pl is equal to pl0, we obtain the GB2 distribution.

fac

numeric; matrix of Gamma factors (output of fac.cgb2.

vl

numeric; vector of parameters. Its length is equal to the length of pl - 1.

w

numeric; vector of weights of length the number of rows of the matrix fac. By default w is a vector of 1.

maxiter

numeric; maximum number of iterations to perform. By default maxiter = 100.

fnscale

numeric; an overall scaling parameter used in the function optim. By default it is equal to the length of the vector of weights w.

Author

Monique Graf and Desislava Nedyalkova

Details

There are only \(L-1\) parameters to estimate, because the probabilities \(p_\ell\) sum to 1 (L is the dimension of the vector of probailities \(p_\ell\)). Knowing this, we change the parameters \(p_\ell\) to \(v_\ell=log(p_\ell/p_L), \ \ell= 1, ..., L-1\). This calculation is done through the function vofp.cgb2. pofv.cgb2 calculates the \(p_\ell\) in function of the given \(v_\ell\). We express the log-likelihood as a weighted mean of \(log(f) = log(\sum(p_\ell f_\ell)\), evaluated at the data points, where \(f\) is the GB2 compound density. If the weights are not available, then we suppose that w \(= 1\). Analogically, the scores are obtained as weighted sums of the first derivatives of the log-likelihood, with respect to the parameters \(v_\ell, \ \ell=1, ..., L-1\), evaluated at the data points. Function ml.cgb2 performs maximum likelihood estimation through the general-purpose optimization function optim from package stats. The considered method of optimization is BFGS.

See Also

optim

Examples

if (FALSE) {
# GB2 parameters:
a <- 4
b <- 1950
p <- 0.8
q <- 0.6

# Proportions defining the component densities:
pl0 <- rep(1/3,3)

# Mixture probabilities
pl <- c(0.1,0.8,0.1)

# Random generation:
n <- 10000
set.seed(12345)
x <- rcgb2(n,a,b,p,q,pl0,pl,decomp="l")

# Factors in component densities
fac <- fg.cgb2(x,a,b,p,q, pl0,decomp="l")

# Estimate the mixture probabilities:
estim <- ml.cgb2(fac,pl0)

# estimated mixture probabilities:
estim[[1]]
#[1] 0.09724319 0.78415797 0.11859883
}