A necessary and sufficient criteria for the existence of the least squares estimate for a 3-parametric exponential function. (Q1412578)

To let the 3-parametric exponential model \(y(t;a,b,c)=a+be^{ct}\) fit the given data \(\{(t_n,y_n):n=1,\ldots,m\}\) for given weights \(p_n\), the sum \(\sum_{n=1}^m p_n [y_n-y(t_n;a,b,c)]^2\) is minimized over \(a,b,c\) with \(bc\neq0\). The minimizing solution may be unique, may not exist, or there may be multiple solutions. It is proved in this paper that no solution exists if and only if the data points are on a line \(y=kt+l\), \(k\neq0\) or \(y_2=\cdots=y_m\) or \(y_1=\cdots=y_{m-1}\). Also a suggestion is made for a good starting point \((a_0,b_0,c_0)\) for the optimization procedure.