\( \newcommand{\gompit}{\mathrm{gompit}} \newcommand{\agompit}{\gompit^{-1}} \newcommand{\frate}{f} \newcommand{\crate}{F} \newcommand{\trate}{Y} \newcommand{\tfr}{T} \newcommand{\ofrate}{\tilde{\frate}} \newcommand{\sfrate}{\hat{\frate}} \newcommand{\ocrate}{\tilde{\crate}} \newcommand{\scrate}{\hat{\crate}} \newcommand{\otrate}{\tilde{\trate}} \newcommand{\strate}{\hat{\trate}} \newcommand{\otfr}{\tilde{\tfr}} \newcommand{\stfr}{\hat{\tfr}} \newcommand{\err}{\epsilon} \newcommand{\nobs}{n} \newcommand{\sse}{\mathit{SSE}} \)

Brass

This build method is an implementation of Brass' Gompertz relational model. The fitted cumulative fertility rate at any age \(x\) is given by the expression \begin{equation} \crate_x=\agompit(\trate_x)\,\tfr \end{equation} where \(\tfr\) is the estimated total fertility rate and \begin{equation} \trate_x = \alpha + \beta\,\strate_x . \end{equation} is the estimated gompit at age \(x\) given the standard's gompit \(\strate_x\). If \(\tfr\), \(\alpha\) and \(\beta\) are already known then the curve can be defined using a Parameters subtable with columns TFR, Alpha, Beta and Standard.

Extra Information

If the curve is specified in terms of fertility rates then the build method property must be of the form BRASS:SFertHandle where SFertHandle is the name of the standard fertility object enclosed in quotes.

FitType

The method used to estimate the Brass model from fertility rates can be specified using the FitType extrainfo property. The two options are a fit to cumulative fertility rates or a fit to fertility rates. Given a set of observed fertility rates \(\ofrate_i\), \(i=0,\ldots,\nobs-1\) for age intervals \([x_i,x_{i+1})\) then the observed cumulative fertility rate is given by \begin{equation} \ocrate_i=\sum_{0< k\le i} (x_k-x_{k-1}) \ofrate_{k-1}, \quad i=1,\ldots,\nobs \end{equation} Using "FitType=CFR" the parameter \(\tfr\) is given by \begin{equation} \tfr=\ocrate_\nobs \end{equation} and \(\alpha\) and \(\beta\) are found by solving the linear regression \begin{equation} \otrate_i = \alpha + \beta\,\strate_i+ \err_i \end{equation} where \(\otrate_i\) is the observed gompit \begin{equation} \otrate_i=\gompit(\ocrate_i/\tfr) \end{equation} Using "FitType=FR" the parameters \(\tfr\), \(\alpha\) and \(\beta\) are simultaneously estimated by minimising the sum of squared errors \begin{equation} \sse=\sum_{0\le i<\nobs}\left[\ofrate_i-\frate(x_i,x_{i+1})\right]^2 \end{equation} where \begin{equation} \frate(x,y)=\frac{\crate_y-\crate_x}{y-x} \end{equation} The default is "FitType=FR".