# Fitting the Compound Distribution based on the GB2 by the Method of Maximum Likelihood Estimation 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. 2.1 ```r 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)) ``` ## 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`. ## 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. ## Returns `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. ## See Also `optim` ## Author(s) Monique Graf and Desislava Nedyalkova ## Examples ```r ## Not run: # 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 ## End(Not run) ```

CompoundFit function