# Fitting the GB2 by Minimizing the Distance Between a Set of Empirical Indicators and Their GB2 Expressions Fitting the parameters of the GB2 distribution by optimizing the squared weighted distance between a set of empirical indicators, i.e. the median, the ARPR, the RMPG, the QSR and the Gini coefficient, and the GB2 indicators using nonlinear least squares (function `nls` from package `stats`). 2.1 ```r nlsfit.gb2(med, ei4, par0=c(1/ei4[4],med,1,1), cva=1, bound1=par0[1]*max(0.2,1-2*cva), bound2=par0[1]*min(2,1+2*cva), ei4w=1/ei4) ``` ## Arguments - `med`: numeric; the empirical median. - `ei4`: numeric; the values of the empirical indicators. - `par0`: numeric; vector of initial values for the GB2 parameters $a, b, p$ and $q$. The default is to take $a$ equal to the inverse of the empirical Gini coefficient, $b$ equal to the empirical median and $p = q = 1$. - `cva`: numeric; the coefficient of variation of the ML estimate of the parameter $a$. The default value is 1. - `bound1, bound2`: numeric; the lower and upper bounds for the parameter $a$ in the algorithm. The default values are $0.2*a_{0}$ and $2*a_{0}$, where $a_{0}$ is the initial value of the parameter $a$. - `ei4w`: numeric; vector of weights of to be passed to the `nls` function. The default values are the inverse of the empirical indicators. ## Details We consider the following set of indicators $A = (median, ARPR, RMPG, QSR, Gini)$ and their corresponding GB2 expressions $A_{GB2}$. We fit the parameters of the GB2 in two consecutive steps. In the first step, we use the set of indicators, excluding the median, and their corresponding expressions in function of $a$, $ap$ and $aq$. The bounds for $a$ are defined in function of the coefficient of variation of the fitted parameter $\hat(a)$. The nonlinear model that is passed to `nls` is given by: $$ \sum_{i=1}^4 c_i (A_{empir,i}-A_{GB2,i}(a,ap,aq))^2, $$ where the weights $c_i$ take the differing scales into account and are given by the vector `ei4w`. $ap$ and $aq$ are bounded so that the constraints for the existence of the moments of the GB2 distribution and the excess for calculating the Gini coefficient are fulfilled, i.e. $ap \ge 1$ and $aq \ge 2$. In the second step, only the the parameter $b$ is estimated, optimizing the weighted square difference between the empirical median and the GB2 median in function of the already obtained NLS parameters $a, p$ and $q$. ## Returns `nlsfit.gb2` returns a list of three values: the fitted GB2 parameters, the first fitted object and the second fitted object. ## See Also `nls`, `Thomae`, `moment.gb2` ## Author(s) Monique Graf and Desislava Nedyalkova ## Examples ```r # Takes long time to run, as it makes a call to the function ml.gb2 ## Not run: library(laeken) data(eusilc) # Personal income inc <- as.vector(eusilc$eqIncome) # Sampling weights w <- eusilc$rb050 # Data set d <- data.frame(inc, w) d <- d[!is.na(d$inc),] # Truncate at 0 d <- d[d$inc > 0,] inc <- d$inc w <- d$w # ML fit, full log-likelihood fitf <- ml.gb2(inc, w)$opt1 # Estimated parameters af <- fitf$par[1] bf <- fitf$par[2] pf <- fitf$par[3] qf <- fitf$par[4] gb2.par <- c(af, bf, pf, qf) # Empirical indicators indicEMP <- main.emp(inc, w) indicEMP <- c(indicEMP[1],indicEMP[3:6]) indicE <- round(indicEMP, digits=3) # Nonlinear fit nn <- nlsfit.gb2(indicEMP[1,3:6],indicEMP[3:6]) an <- nn[[1]][1] bn <- nn[[1]][2] pn <- nn[[1]][3] qn <- nn[[1]][4] # GB2 indicators indicNLS <- c(main.gb2(0.6, an, bn, pn, qn)[1], main.gb2(0.6, an, bn, pn, qn)[3:6]) indicML <- c(main.gb2(0.6, af, bf, pf, qf)[1], main.gb2(0.6, af, bf, pf, qf)[3:6]) indicN <- round(indicNLS, digits=3) indicM <- round(indicML, digits=3) # Likelihoods nlik <- loglp.gb2(inc, an, bn, pn, qn, w) mlik <- loglp.gb2(inc, af, bf, pf, qf, w) # Results type=c("Emp. est", "NLS", "ML full") results <- data.frame(type=type, median=c(indicE[1], indicN[1], indicM[1]), ARPR=c(indicE[2], indicN[2], indicM[2]), RMPG=c(indicE[3], indicN[3], indicM[3]), QSR =c(indicE[4], indicN[4], indicM[4]), GINI=c(indicE[5], indicN[5], indicM[5]), likelihood=c(NA, nlik, mlik), a=c(NA, an, af), b=c(NA, bn, bf) ,p=c(NA, pn, pf), q=c(NA, qn, qf)) ## End(Not run) ```

NonlinearFit function