Maximum Likelihood Estimation of the General Nonlinear Functional Relationship with Replicated Observations and Correlated Errors

The problem considered is that of estimating a p-parameter functional relationship η = η(ξ;α1, …, αp given replicated observations at each of n unknown values of the independent variable ξ. The errors of observation at (ξ, η) are assumed to have a bivariate normal distribution involving parameters σ2i, pi and r2i and to be independent of those at any other true point. An iterative method for obtaining maximum likelihood estimates of the structural and incidental parameters is proposed. Closed form expressions in terms of the ξi are obtained for the asymptotic covariance matrix of estimates of the structural parameters αv and the scale incidental parameters σv, pi and τ2i. An illustrative application to simulated data and a discussion of convergence are included.