Theoretical basis of the iterative process of the joint assessment of difficulties in tasks and levels of training students

In this paper, we study the iterative process of the joint numerical assessment of levels of training students and difficulties in tasks of diagnostic tools using the dichotomous response matrix $A$ of $N\times M$ size, with allowance for the contribution of tasks of different difficulty to the assessments obtained. It is shown that not for any matrix $A$ there exist infinite iterative sequences, and in the case of their existence, they do not always converge. A wide range of sufficient conditions for their convergence have been obtained, which are based on the following: 1) matrix $A$ contains at least three different columns; 2) if one places the columns of $A$ in non-decreasing order of column sums, then for any position of the vertical dividing line between the columns there exists a row, which has, at least, one unity to the left of the line, and at least one zero to the right of the line. It is established that the response matrix $A$ obtained as a result of testing reliability satisfies these two conditions. The properties of such matrices have been studied. In particular, the equivalence of the above conditions of primitiveness of the square matrix $B$ of order $M$ with the entries $b_{ij}=\sum^N_{\ell=1}(1-a_{\ell i})a_{\ell j}$ has been proved. Using the matrix analysis we have proved that the primitiveness of the matrix $B$ ensures the convergence of iterative sequences, as well as independence of their limits of the choice of the initial approximation. We have estimated the rate of convergence of these sequences and found their limits.