Diagonals-parameter symmetry model and its property for square contingency tables with ordinal categories

Previously, the diagonals-parameter symmetry model based on $f$-divergence (denoted by DPS[$f$]) was reported to be equivalent to the diagonals-parameter symmetry model regardless of the function $f$, but the proof was omitted. Here, we derive the DPS[$f$] model and the proof of the relation between the two models. We can obtain various interpretations of the diagonals-parameter symmetry model from the result. Additionally, the necessary and sufficient conditions for symmetry and property between test statistics for goodness of fit are discussed.