Conditional Quantization for Some Discrete Distributions
Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the finite support are preselected, then the quantization is called a condit...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/11/1717 |
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| Summary: | Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the finite support are preselected, then the quantization is called a conditional quantization. In this paper, we have determined the conditional quantization, first for two different finite discrete distributions with a same conditional set, and for a finite discrete distribution with two different conditional sets. Next, we have determined the conditional and unconditional quantization for an infinite discrete distribution with support <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mstyle><mo>:</mo><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>}</mo></mrow></semantics></math></inline-formula>. We have also investigated the conditional quantization for an infinite discrete distribution with support <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><mo>:</mo><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>}</mo></mrow></semantics></math></inline-formula>. At the end of the paper, we have given a conjecture and discussed about some open problems based on the conjecture. |
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| ISSN: | 2227-7390 |