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

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:= pP₁ + (1 - p)P₂, P : = p P 1 + ( 1 - p ) P 2 , where P₁ P 1 and P₂ P 2 are arbitrary probability distributions supported on either connected or disconnected line segments, and $$(p, 1 - p)$$ ( p , 1 - p ) is any probability vector with $$0< p < 1$$ 0 < p < 1 .