# MLfullGB2: Maximum Likelihood Estimation of the GB2 Based on the Full... In GB2: Generalized Beta Distribution of the Second Kind: Properties, Likelihood, Estimation

 MLfullGB2 R Documentation

## Maximum Likelihood Estimation of the GB2 Based on the Full Log-likelihood

### Description

Performs maximum pseudo-likelihood estimation through the general-purpose optimisation function `optim` from package `stats`. Two methods of optimization are considered: BFGS and L-BFGS-B (see `optim` documentation for more details). Initial values of the parameters to be optimized over (a, b, p and q) are given from the Fisk distribution and p=q=1. The function to be maximized by `optim` is the negative of the full log-likelihood and the gradient is equal to the negative of the scores, respectively for the case of a sample of persons and a sample of households.

### Usage

```ml.gb2(z, w=rep(1, length(z)), method=1, hess=FALSE)
mlh.gb2(z, w=rep(1, length(z)), hs=rep(1, length(z)), method=1, hess = FALSE)
```

### Arguments

 `z` numeric; vector of data values. `w` numeric; vector of weights. Must have the same length as `z`. By default `w` is a vector of 1. `hs` numeric; vector of household sizes. Must have the same length as `z`. By default `hs` is a vector of 1. `method` numeric; the method to be used by `optim`. By default, codemethod = 1 and the used method is BFGS. If `method = `2, method L-BFGS-B is used. `hess` logical; By default, `hess = FALSE`, the hessian matrix is not calculated.

### Details

Function `ml.gb2` performs maximum likelihood estimation through the general-purpose optimization function `optim` from package `stats`, based on the full log-likelihood calculated in a sample of persons. Function `mlh.gb2` performs maximum likelihood estimation through the general-purpose optimization function `optim` from package `stats`, based on the full log-likelihood calculated in a sample of households.

### Value

`ml.gb2` and `mlh.gb2` return a list with 1 argument: `opt1` for the output of the BFGS fit or `opt2` for the output of the L-BFGS fit. Further values are given by the values of `optim`.

Monique Graf

### References

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.

`optim` for the general-purpose optimization and `fisk` for the Fisk distribution.

### Examples

```## Not run:
library(laeken)
data(eusilc)

# Income
inc <- as.vector(eusilc\$eqIncome)

# Weights
w <- eusilc\$rb050

# Data set
d <- data.frame(inc, w)
d <- d[!is.na(d\$inc),]

# Truncate at 0
inc <- d\$inc[d\$inc > 0]
w   <- d\$w[d\$inc > 0]

# Fit using the full log-likelihood
fitf <- ml.gb2(inc, w)

# Fitted GB2 parameters
af <- fitf\$par
bf <- fitf\$par
pf <- fitf\$par
qf <- fitf\$par

# Likelihood
flik <- fitf\$value

# If we want to compare the indicators

# GB2 indicators
indicf <- round(main.gb2(0.6,af,bf,pf,qf), digits=3)
# Empirical indicators
indice <- round(main.emp(inc,w), digits=3)

# Plots
plotsML.gb2(inc,af,bf,pf,qf,w)

## End(Not run)
```

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