Quantization for the mixtures of uniform distributions on connected and disconnected line segments

Abstract

In this paper, we have studied various mixed distributions generated by two uniform distributions: first, where the supports are two connected line segments, and second, where the supports are two disconnected line segments. For these mixed distributions, we have determined the optimal sets of $n$-means and the corresponding $n$th quantization errors for all positive integers $n $. The methods developed in this paper can be applied more generally to investigate optimal quantization for any mixed distribution $P := pP_1 + (1 - p)P_2,$ where $P_1$ and $P_2$ are arbitrary probability distributions supported on either connected or disconnected line segments, and $(p, 1 - p)$ is any probability vector with $0 < p < 1$.