Monotone Majorizable Functionals

Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus define MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already Z.Z(1 — sg) is not MMF. It is proved that extensionality is not Dialectica-realizable by MMF, and a simple example of a MF which is not hereditarily majorizable is given.     

Nonparametric item response theory axioms and properties under nonlinearity and their exemplification with knowledge space theory

This paper investigates the dichotomous Mokken nonparametric item response theory (IRT) axioms and properties under incomparabilities among latent trait values and items. Generalized equivalents of the unidimensional nonparametric IRT axioms and properties are formulated for nonlinear (quasi-ordered) person and indicator spaces. It is shown that monotone likelihood ratio (MLR) for the total score variable and nonlinear latent trait implies stochastic ordering (SO) of the total score variable, but may fail to imply SO of the nonlinear latent trait. The reason for this and conditions under which the implication holds are specified, based on a new, simpler proof of the fact that in the unidimensional case MLR implies SO. The approach is applied in knowledge space theory (KST), a combinatorial test theory. This leads to a (tentative) Mokken-type nonparametric axiomatization in the currently parametric theory of knowledge spaces. The nonparametric axiomatization is compared with the assumptions of the parametric basic local independence model which is fundamental in KST. It is concluded that this paper may provide a first step toward a basis for a possible fusion of the two split directions of psychological test theories IRT and KST.