Maximal point-polyserial correlation for non-normal random distributions

We consider the problem of determining the maximum value of the point-polyserial correlation between a random variable with an assigned continuous distribution and an ordinal random variable with $k$ categories, which are assigned the first $k$ natural values $1,2,\text{…},k$ , and arbitrary probabilities $p_i$ . For different parametric distributions, we derive a closed-form formula for the maximal point-polyserial correlation as a function of the $p_i$ and of the distribution's parameters; we devise an algorithm for obtaining its maximum value numerically for any given $k$ . These maximum values and the features of the corresponding $k$ -point discrete random variables are discussed with respect to the underlying continuous distribution. Furthermore, we prove that if we do not assign the values of the ordinal random variable a priori but instead include them in the optimization problem, this latter approach is equivalent to the optimal quantization problem. In some circumstances, it leads to a significant increase in the maximum value of the point-polyserial correlation. An application to real data exemplifies the main findings. A comparison between the discretization leading to the maximum point-polyserial correlation and those obtained from optimal quantization and moment matching is sketched.