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

TL;DR

This paper investigates optimal quantization for mixed distributions formed by two uniform distributions supported on either connected or disconnected line segments, providing explicit solutions for all positive integers n.

Contribution

It determines the optimal n-means and quantization errors for mixed uniform distributions on line segments, extending methods to more general mixed distributions.

Findings

Explicit optimal n-means for mixed uniform distributions on line segments.

Quantization errors computed for all positive integers n.

Method applicable to general mixed distributions supported on 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 $0 < p < 1$.