n Let �:=

3.2 Likelihood equations and errors-in-variables regression; Solari's example.. Here is a case, noted by Solari (1969), where the likelihood equations (3.1.1) have solutions, none of which are maximum likelihood estimates. be observed points in the plane. Suppose we want to do a form of " errors in variables " regression, in other words to fit the data by a straight line, assuming normal errors in both variables, so that X i 2 and τ 2. Let c := σ 2 and h := τ 2. Then the joint density is n (ch) −n/2 (2π) −n exp − i=1 (x i − a i) 2 /(2c) + (y i − ba i) 2 /(2h). n Let � := i=1. Taking logarithms, the likelihood equations are equivalent to the vanishing of the gradient (with respect to all n + 3 parameters) of −(n/2) log(ch) − [(X i − a i) 2 /(2c) + (Y i − ba i) 2 /(2h)].