Implementation of non-linear estimators using monospline

This paper presents a method for the realization of non-linear estimators based on spline interpolation. The difference of a monospline and its interpolating spline forms a monospline and then a quadrature formula is induced. When the knots of the monospline at which the conditional density is discretized are allowed to vary, a class of optimal quadrature formulas is obtained. To find the monospline with optimal knots a set of non-linear algebraic equations must be solved. If the symmetry property of the monospline is applied, the order of the non-linear equations can be reduced by about one-half. An iteration scheme of Newton type is introduced to solve the monospline. The quadrature formula associated with this monospline has the so-called positivity property which is essential in the practical implementation of non-linear recursive estimators.