Statistical Estimation of Quantization for Probability Distributions: Best Equivariant Estimator of Principal Points

Quantization gives a discrete approximation (with a finite set of points called quantizer) for a probability distribution. When the approximation is optimal with respect to a loss function, it is called optimal quantization (and its set of points is called an optimal quantizer), which has been studied and applied in various areas. Especially in statistics, an optimal quantizer under a quadratic loss function (optimal quantizer of order 2) has been widely investigated and is often called a set of principal points (or simply, principal points) for a probability distribution. In practice, however, the values of the parameters of the probability distribution are sometimes unknown, and hence we have to estimate principal points based on a random sample. A common method for estimating principal points is using principal points of the empirical distribution obtained by a random sample, which can be viewed as a nonparametric estimator of principal points. Several papers discussed statistical parametric estimation of principal points based on maximum likelihood estimators of the parameters. In this paper, a class of equivariant estimators, which includes previous parametric estimators of principal points is considered, and the best equivariant estimator of principal points is derived. It turns out that, under some condition, the best equivariant estimator coincides with a previously obtained parametric estimator. However, it is also shown that, for some probability distributions not satisfying the condition, the best equivariant estimator may not be equivalent to previous estimators.