A Compound Poisson Perspective of Ewens–Pitman Sampling Model
The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/21/2820 |
| _version_ | 1797512104177041408 |
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| author | Emanuele Dolera Stefano Favaro |
| author_facet | Emanuele Dolera Stefano Favaro |
| author_sort | Emanuele Dolera |
| collection | DOAJ |
| description | The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, which is indexed by real parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> such that either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>></mo><mo>−</mo><mi>α</mi></mrow></semantics></math></inline-formula>, or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mi>m</mi><mi>α</mi></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large <i>n</i> asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case. |
| first_indexed | 2024-03-10T05:57:10Z |
| format | Article |
| id | doaj.art-c29a9a698205469799d8bd91f482efb0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:57:10Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c29a9a698205469799d8bd91f482efb02023-11-22T21:19:20ZengMDPI AGMathematics2227-73902021-11-01921282010.3390/math9212820A Compound Poisson Perspective of Ewens–Pitman Sampling ModelEmanuele Dolera0Stefano Favaro1Department of Mathematics, University of Pavia, Via Adolfo Ferrata 5, 27100 Pavia, ItalyCollegio Carlo Alberto, Piazza V. Arbarello 8, 10122 Torino, ItalyThe Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, which is indexed by real parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> such that either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>></mo><mo>−</mo><mi>α</mi></mrow></semantics></math></inline-formula>, or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mi>m</mi><mi>α</mi></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large <i>n</i> asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.https://www.mdpi.com/2227-7390/9/21/2820Berry–Esseen type theoremEwens–Pitman sampling modelexchangeable random partitionslog-series compound poisson sampling modelMittag–Leffler distribution functionnegative binomial compound poisson sampling model |
| spellingShingle | Emanuele Dolera Stefano Favaro A Compound Poisson Perspective of Ewens–Pitman Sampling Model Mathematics Berry–Esseen type theorem Ewens–Pitman sampling model exchangeable random partitions log-series compound poisson sampling model Mittag–Leffler distribution function negative binomial compound poisson sampling model |
| title | A Compound Poisson Perspective of Ewens–Pitman Sampling Model |
| title_full | A Compound Poisson Perspective of Ewens–Pitman Sampling Model |
| title_fullStr | A Compound Poisson Perspective of Ewens–Pitman Sampling Model |
| title_full_unstemmed | A Compound Poisson Perspective of Ewens–Pitman Sampling Model |
| title_short | A Compound Poisson Perspective of Ewens–Pitman Sampling Model |
| title_sort | compound poisson perspective of ewens pitman sampling model |
| topic | Berry–Esseen type theorem Ewens–Pitman sampling model exchangeable random partitions log-series compound poisson sampling model Mittag–Leffler distribution function negative binomial compound poisson sampling model |
| url | https://www.mdpi.com/2227-7390/9/21/2820 |
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