Expression of the parameters shape2 \(=p\) and shape3 \(=q\) of the GB2 distribution as functions of shape1 \(=a\) and scale \(=b\),
profile log-likelihood of the GB2 distribution, scores of the profile log-likelihood.
prof.gb2(x, shape1, scale, w=rep(1, length(x)))
proflogl.gb2(x, shape1, scale, w=rep(1, length(x)))
profscores.gb2(x, shape1, scale, w=rep(1, length(x)))numeric; vector of data values.
numeric; positive parameter.
numeric; positive parameter.
numeric; vector of weights. Must have the same length as x. By default w is a vector of 1.
prof returns a vector containing the values of \(r\), \(s\), \(p\), \(q\) as well as two other parameters used in the calculation of the profile log-likelihood and its first derivatives.
proflogl.gb2 returns the value of the profile log-likelihood and profscores.gb2 returns the vector of the first derivatives of the profile log-likelihhod with respect to \(a\) and \(b\).
Using the full log-likelihood equations for the GB2 distribution, the parameters \(p\) and \(q\) can be estimated as functions of \(a\) and \(b\). These functions are plugged into the log-likelihood expression, which becomes a function of \(a\) and \(b\) only. This is obtained by reparametrizing the GB2, i.e. we set \(r=\frac{p}{p+q}\) and \(s=p+q\). More details can be found in Graf (2009).
Graf, M. (2009) The Log-Likelihood of the Generalized Beta Distribution of the Second Kind. working paper, SFSO.