Equations (3) and (4) in the preprint version (3.1 and 3.2 in the published JCGS journal version) pertain to the first order conditions to the maximization problem formulated in Equation (1) (or 2.1). If you obtain the parameter estimates by maximum likelihood (ML) for example, then Equation (4) is the score function corresponding to the log-likelihood function. This is sometimes also called the estimating function. The gradient of the summed objective function sums these score contributions and this needs to be zero at the optimal parameter estimates (Equation 3).
A simple example is the estimation of a constant mean (either via OLS or ML). The score function then simply corresponds to the residuals. And the sum of the residuals is zero at the estimated parameter (= empirical mean).
The empirical score contributions can thus be seen as measures of model deviation and the idea of the subsequently discussed parameter instability tests in Section 3.2 is the following: If the model fits well to the current subsample, then the scores should fluctuate randomly around zero. If there is a systematic change in parameters, then there should be a shift in the scores as well. This can be captured by looking at cumulative sums of the scores (Equation 5 or 3.3) and then brought out by suitable test statistics (Equations 6/7 or 3.4/3.5).
More details are given in the references cited in the paper. Also some of our follow-up work has more details, e.g.:
Carolin Strobl, Florian Wickelmaier, Achim Zeileis (2011). Accounting for Individual Differences in Bradley-Terry Models by Means of Recursive Partitioning. Journal of Educational and Behavioral Statistics, 36(2), 135–153.
http://dx.doi.org/10.3102/1076998609359791
Edgar C. Merkle, Achim Zeileis (2013). Tests of Measurement Invariance without Subgroups: A Generalization of Classical Methods. Psychometrika, 78(1), 59–82. http://dx.doi.org/10.1007/s11336-012-9302-4
Ting Wang, Edgar C. Merkle, Achim Zeileis (2014). Score-Based Tests of Measurement Invariance: Use in Practice. Frontiers in Psychology, 5(438). http://dx.doi.org/10.3389/fpsyg.2014.00438
