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 nth 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_557_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </InlineMediaObject> <EquationSource Format="TEX">$ P:= pP_1 + (1 - p)P_2, $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>:</mo> <mo>=</mo> <mi>p</mi> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_557_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$P_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_557_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$P_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are arbitrary probability distributions supported on either connected or disconnected line segments, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_557_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$(p, 1 - p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mn>1</mn> <mo>-</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is any probability vector with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_557_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$0 <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

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

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2025年9月13日

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Asha Barua,
Gustavo Fernandez,
Ashley Gomez,
Ogla Lopez,
Mrinal Kanti Roychowdhury

摘要

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 nth 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, $ $P : = p P_{1} + (1 - p) P_{2},$ where $P_1$ $P_{1}$ and $P_2$ $P_{2}$ are arbitrary probability distributions supported on either connected or disconnected line segments, and $(p, 1 - p)$ $(p, 1 - p)$ is any probability vector with $0< p < 1$ $0 < p < 1$ .