This build method fits fertility rates to Bermudez, Blanquero, Hernandez and Planelles' Weibull mixture model \begin{equation} \begin{array}{lcll} \frate(x)&=&R_1\alpha_1\beta_1 (\alpha_1 x)^{\beta_1-1} \exp\left[-(\alpha_1 x)^{\beta_1}\right] &\eqnote{non-marital}\\ \\ &&+R_2\alpha_2\beta_2 (\alpha_2 x)^{\beta_2-1} \exp\left[-(\alpha_2 x)^{\beta_2}\right] & \eqnote{marital} \end{array} \end{equation}
Fitting is done using a nonlinear optimisation routine. Its default behaviour can be changed by appending an extrainfo string to the buildmethod value, WEIBULL_MIX:ExtraInfo. Here ExtraInfo is a comma-separated list of quoted strings of the form "Parameter=Value".
The fitting routine proceeds iteratively from a set of starting values for the parameters. The user can set the starting value in the extrainfo string. For example, "Alpha2=0.03" sets the starting value of \(\alpha_2\) to 0.03.
Fit flags can be used to specify whether a parameter is adjusted to fit the fertility probabilities or held fixed at its starting value. For example, "FitR1=0" specifies \(R_1\) to be fixed at its starting value, and "FitR1=1" specifies \(R_1\) to be adjusted. The default behaviour is to fit all parameters. Entire components can be held fixed using the parameters FitNonMarital and FitMarital. For example the extrainfo "FitMarital=0","R1=0" can be used to remove the entire marital component from the fit.
Given a set of observed fertility rates \({_n\ofrate_x}\) the build method searches for parameters that minimise the objective function \begin{equation} \sse=\sum_x \FitError({_n\ofrate_x},{_n\frate_x}) . \end{equation} The form of the function \(\FitError(\ofrate,\frate)\) is controlled by the ErrType extrainfo property \begin{equation} \FitError(\ofrate,\frate)=\left\{% \begin{array}{ll} (\frate - \ofrate)^2 & \eqnote{ABS}\\ \weight(\frate - \ofrate\log\frate) & \eqnote{POISSON}\\ \end{array}\right. \end{equation} The value of the the weight \(\weight\) is controlled by the PCurve extrainfo property. If this is not specified then it defaults to \(\weight=1\). The default error type is "ErrType=ABS".
The error type POISSON is for use when the observed fertilities \(_n\ofrate_x\) display appreciable sample noise. The weight \(\weight\) then corresponds to the population in age group \([x,x+n)\) at risk. An extrainfo string of the form "PCurve=PCurveHandle" specifies that these weights are to be taken from the PCurve object PCurveHandle.