Rasch’s original definition of specific objectivity insisted that measurement by items of educational tests had to be provided by estimates of person parameters of a measurement model that did not involve estimates of item parameters. It was never stated clearly, but there is no doubt that Rasch was thinking about conditional maximum likelihood (CML) estimates similar to the well-known CML estimates of item parameters. However, for technical reasons, there are only few examples of attempts to implement the CML estimates of person parameters in software applications for item analysis by Rasch models. Nevertheless, today CML estimation of person parameters in Rasch models is possible. And since the exact distribution of estimates of person parameters in Rasch models is known, it is possible to compare the performance of the specific objective CML estimates of person parameters to the performances of the other estimates of the person parameters. This paper provides one example of such an exercise. It compares five different estimators of the person parameters of a Rasch model with forty dichotomous items for a Danish cognitive test. In this paper, the point of view on the measurement quality is purely statistical. The superior measure of the latent trait has to have ignorable bias and as little standard error of measurement as possible. In this example, it turns out that the specific objective CML estimate is not the superior estimate. This motivates a slightly weaker notion of essential specific objectivity than many other estimates of person parameters in Rasch models live up to.