  In this paper, we focus on examination results when examinees selectively
skip examinations, to compare the difficulty levels of these examinations. We
call the resultant data 'selectively omitted examination data' Examples of this
type of examination are university entrance examinations, certification
examinations, and the outcome of students' job-hunting activities. We can learn
the number of students accepted for each examination and organization but not
the examinees' identity. No research has focused on this type of data. When we
know the difficulty level of these examinations, we can obtain a new index to
assess organization ability, how many students pass, and the difficulty of the
examinations. This index would reflect the outcomes of their education
corresponding to perspectives on examinations. Therefore, we propose a novel
method, Peak Shift Estimation, to estimate the difficulty level of an
examination based on selectively omitted examination data. First, we apply Peak
Shift Estimation to the simulation data and demonstrate that Peak Shift
Estimation estimates the rank order of the difficulty level of university
entrance examinations very robustly. Peak Shift Estimation is also suitable for
estimating a multi-level scale for universities, that is, A, B, C, and D rank
university entrance examinations. We apply Peak Shift Estimation to real data
of the Tokyo metropolitan area and demonstrate that the rank correlation
coefficient between difficulty level ranking and true ranking is 0.844 and that
the difference between 80 percent of universities is within 25 ranks. The
accuracy of Peak Shift Estimation is thus low and must be improved; however,
this is the first study to focus on ranking selectively omitted examination
data, and therefore, one of our contributions is to shed light on this method.
