# A convexity‐constrained parameterization of the random effects generalized partial credit model

## TL;DR

This paper introduces a new mathematical approach to improve the estimation of parameters in a statistical model used for scoring items.

## Contribution

The novel contribution is a convexity-constrained parameterization that ensures proper maximum likelihood estimation.

## Key findings

- A closed-form expression using a cumulant generating function is introduced.
- The convexity constraints ensure a single global maximum in the likelihood function.
- The method is demonstrated with an example and includes goodness-of-fit testing.

## Abstract

An alternative closed‐form expression for the marginal joint probability distribution of item scores under the random effects generalized partial credit model is presented. The closed‐form expression involves a cumulant generating function and is therefore subjected to convexity constraints. As a consequence, complicated moment inequalities are taken into account in maximum likelihood estimation of the parameters of the model, so that the estimation solution is always proper. Another important favorable consequence is that the likelihood function has a single local extreme point, the global maximum. Furthermore, attention is paid to expected a posteriori person parameter estimation, generalizations of the model, and testing the goodness‐of‐fit of the model. Procedures proposed are demonstrated in an illustrative example.