On sufficient conditions for implementing the functional approach
Seminar Room 1, Newton Institute
Let us consider the general nonlinear regression model under standard assumptions on the experimental errors. Let also the following assumptions be fulfilled: (i) the regression function depends on a scalar variable belonging to the design interval, (ii) the derivatives of the function with respect to the parameters generate an extended Chebyshev system on the design interval, (iii) the matrix of second derivatives of the optimality criterion with respect to the different information matrix elements is positive definite. Then under non-restrictive assumptions it can be proved that the Jacobi matrix of the system of differential equations that defines implicitly support points and weight coefficients of the optimal design is invertible. This allows us to implement the Implicit Function Theorem for representing the points and the weights by a Taylor series. The corresponding theorems as well as particular examples of nonlinear models are elaborated. The results are generalisations of those given in the monograph published recently by the author.