  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.
