Truncated Moments for Heavy-Tailed and Related Distribution Classes
Suppose that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ξ</mi><mo>+</mo></msup></semantics></math></inline-formula> is the positive part of a random...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/9/2172 |
| _version_ | 1827742865515610112 |
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| author | Saulius Paukštys Jonas Šiaulys Remigijus Leipus |
| author_facet | Saulius Paukštys Jonas Šiaulys Remigijus Leipus |
| author_sort | Saulius Paukštys |
| collection | DOAJ |
| description | Suppose that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ξ</mi><mo>+</mo></msup></semantics></math></inline-formula> is the positive part of a random variable defined on the probability space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="sans-serif">Ω</mi><mo>,</mo><mi mathvariant="script">F</mi><mo>,</mo><mi mathvariant="double-struck">P</mi><mo>)</mo></mrow></semantics></math></inline-formula> with the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula>. When the moment <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">E</mi><msup><mfenced open="(" close=")"><msup><mi>ξ</mi><mo>+</mo></msup></mfenced><mi>p</mi></msup></mrow></semantics></math></inline-formula> of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is finite, then the truncated moment <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover><mi>F</mi><mo>¯</mo></mover><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mfenced separators="" open="{" close="}"><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">E</mi><mfenced separators="" open="(" close=")"><msup><mi>ξ</mi><mi>p</mi></msup><mn>1</mn><mspace width="-2.9pt"></mspace><msub><mi mathvariant="normal">I</mi><mrow><mo>{</mo><mi>ξ</mi><mo>></mo><mi>x</mi><mo>}</mo></mrow></msub></mfenced></mfenced></mrow></semantics></math></inline-formula>, defined for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>⩾</mo><mn>0</mn></mrow></semantics></math></inline-formula>, is the survival function or, in other words, the distribution tail of the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula>. In this paper, we examine which regularity properties transfer from the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> to the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> and which properties transfer from the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> to the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula>. The construction of the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> describes the truncated moment transformation of the initial distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula>. Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> belongs to one of these classes, it follows that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes. |
| first_indexed | 2024-03-11T04:13:25Z |
| format | Article |
| id | doaj.art-a5ca532ce704429ba39a2cbd6dec418c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:13:25Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a5ca532ce704429ba39a2cbd6dec418c2023-11-17T23:20:56ZengMDPI AGMathematics2227-73902023-05-01119217210.3390/math11092172Truncated Moments for Heavy-Tailed and Related Distribution ClassesSaulius Paukštys0Jonas Šiaulys1Remigijus Leipus2Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaInstitute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaInstitute of Applied Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaSuppose that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ξ</mi><mo>+</mo></msup></semantics></math></inline-formula> is the positive part of a random variable defined on the probability space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="sans-serif">Ω</mi><mo>,</mo><mi mathvariant="script">F</mi><mo>,</mo><mi mathvariant="double-struck">P</mi><mo>)</mo></mrow></semantics></math></inline-formula> with the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula>. When the moment <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">E</mi><msup><mfenced open="(" close=")"><msup><mi>ξ</mi><mo>+</mo></msup></mfenced><mi>p</mi></msup></mrow></semantics></math></inline-formula> of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is finite, then the truncated moment <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover><mi>F</mi><mo>¯</mo></mover><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mfenced separators="" open="{" close="}"><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">E</mi><mfenced separators="" open="(" close=")"><msup><mi>ξ</mi><mi>p</mi></msup><mn>1</mn><mspace width="-2.9pt"></mspace><msub><mi mathvariant="normal">I</mi><mrow><mo>{</mo><mi>ξ</mi><mo>></mo><mi>x</mi><mo>}</mo></mrow></msub></mfenced></mfenced></mrow></semantics></math></inline-formula>, defined for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>⩾</mo><mn>0</mn></mrow></semantics></math></inline-formula>, is the survival function or, in other words, the distribution tail of the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula>. In this paper, we examine which regularity properties transfer from the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> to the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> and which properties transfer from the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> to the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula>. The construction of the distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> describes the truncated moment transformation of the initial distribution function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula>. Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mrow><mi>ξ</mi><mo>,</mo><mi>p</mi></mrow></msub></semantics></math></inline-formula> belongs to one of these classes, it follows that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes.https://www.mdpi.com/2227-7390/11/9/2172truncated momentalternative moment formulaheavy-tailed distributiondominated variationconsistent variationregular variation |
| spellingShingle | Saulius Paukštys Jonas Šiaulys Remigijus Leipus Truncated Moments for Heavy-Tailed and Related Distribution Classes Mathematics truncated moment alternative moment formula heavy-tailed distribution dominated variation consistent variation regular variation |
| title | Truncated Moments for Heavy-Tailed and Related Distribution Classes |
| title_full | Truncated Moments for Heavy-Tailed and Related Distribution Classes |
| title_fullStr | Truncated Moments for Heavy-Tailed and Related Distribution Classes |
| title_full_unstemmed | Truncated Moments for Heavy-Tailed and Related Distribution Classes |
| title_short | Truncated Moments for Heavy-Tailed and Related Distribution Classes |
| title_sort | truncated moments for heavy tailed and related distribution classes |
| topic | truncated moment alternative moment formula heavy-tailed distribution dominated variation consistent variation regular variation |
| url | https://www.mdpi.com/2227-7390/11/9/2172 |
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