Chain-Ladder as Maximum Likelihood Revisited

ABSTRACTIt has long been known that maximum likelihood estimation in a Poisson model reproduces the chain-ladder technique. We revisit this model. A new canonical parametrisation is proposed to circumvent the inherent identification problem in the parametrisation. The maximum likelihood estimators for the canonical parameter are simple, interpretable and easy to derive. The boundary problem where all observations in one particular development year or on particular underwriting year is zero is also analysed.

Some remarks on modular regular types

Here we consider some problems concerning regular types. In the first place we consider a strongly minimal set D. One can ask what is the strength of the assumption that D has (full) elimination of imaginaries (namely, every definable set X over D has as canonical parameter some tuple from D). We show that D cannot be locally modular. Nontriviality of D is immediate. However, to exclude the locally modular nontrivial case one has to understand structures of the form G/E, where G is a modular strongly minimal group and E is a definable equivalence relation on G with finite classes. We show that the quotient structure G/E can be obtained in two steps. First quotient by a finite subgroup K of G to obtain a strongly minimal group H. Now let Γ be a finite subgroup of the group Aff(H) of definable affine automorphisms of H (namely maps of the form x → αx + a, where α is a definable automorphism of H and a ∈ H), and quotient H by Γ (namely form the orbit space of H under Γ). It can clearly be arranged that Γ contains no nontrivial subgroup of translations.In the second place we look at a nontrivial modular regular type p whose pregeometry is actually a geometry. The geometry is then known to be (infinite-dimensional) projective geometry over a division ring F. We ask whether F is definable (internally to p). If F is finite, this is clear. In fact in this case p must have U-rank 1. So we assume F to be infinite. We are only able to show definability of F in the case where F is a field, using some results on 2-transitive subgroups of PGL [V]. Moreover in the superstable case we also observe that p is isolated.

Analytical Properties of Poincaré Halfmaps in a Class of Piecewise-Linear Dynamical Systems

A piecewise-linear nerve conduction equation is investigated further. The theory of Poincare halfmaps induced by the flow of a three-dimensional linear saddle-focus is developed. Using a description of the dynamical system in diagonalized coordinates, a canonical formulation of two-dimensional halfmaps is found. This leads for each halfmap to an implicit scalar equation plus a side condition. The effect of the halfmaps on different types of invariant curves occurring is investigated. Thereby the capacity of the halfmaps to separate adjacent points (such that the images acquire a finite distance) is shown. Two of three possible mechanisms for separating points are investigated in detail. The regions in the canonical parameter space where the different separating mechanisms appear are indicated analytically. The possible appearance of chaotic solutions, at least in the neighborhood of homoclinic trajectories in state space, is demonstrated. The underlying separation mechanism is present also in regions of state space far from a homoclinic orbit.