The Leader Property in Quasi Unidimensional Cases
The following problem was studied: let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bol...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/22/4199 |
| _version_ | 1797464676460658688 |
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| author | Anișoara Maria Răducan Gheorghiță Zbăganu |
| author_facet | Anișoara Maria Răducan Gheorghiță Zbăganu |
| author_sort | Anișoara Maria Răducan |
| collection | DOAJ |
| description | The following problem was studied: let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></mfenced><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> be a sequence of i.i.d. <i>d</i>-dimensional random vectors. Let <i>F</i> be their probability distribution and for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> consider the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">S</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><msub><mi mathvariant="bold">Z</mi><mn>1</mn></msub><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mi>n</mi></msub><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> was called a “leader” in the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≥</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mspace width="0.166667em"></mspace><mspace width="4pt"></mspace></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> was an “anti-leader” if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≤</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>. The comparison of two vectors was the usual one: if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>=</mo><mfenced separators="" open="(" close=")"><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mn>1</mn></mfenced></msubsup><mo>,</mo><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mn>2</mn></mfenced></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mi>d</mi></mfenced></msubsup></mfenced><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≥</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> means <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>≥</mo><msubsup><mi>Z</mi><mrow><mi>k</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>,</mo></mrow></semantics></math></inline-formula> while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≤</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> means <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>≤</mo><msubsup><mi>Z</mi><mrow><mi>k</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has a leader, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has an anti-leader and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has both a leader and an anti-leader. Sometimes these probabilities can be computed or estimated, for instance in the case when <i>F</i> is discrete or absolutely continuous. The limits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mo movablelimits="true" form="prefix">lim</mo><mo movablelimits="true" form="prefix">inf</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>=</mo><mo movablelimits="true" form="prefix">lim</mo><mo movablelimits="true" form="prefix">inf</mo><msub><mi>b</mi><mi>n</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>=</mo><mo movablelimits="true" form="prefix">lim</mo><mo movablelimits="true" form="prefix">inf</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> were considered. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> it was said that <i>F</i> has the leader property, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> they said that <i>F</i> has the anti-leader property and if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> then <i>F</i> has the order property. In this paper we study an in-between case: here the vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">Z</mi></semantics></math></inline-formula> has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">Z</mi><mo>=</mo><mi mathvariant="bold-italic">f</mi><mfenced open="(" close=")"><mi>X</mi></mfenced></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">f</mi><mo>=</mo><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>f</mi><mi>d</mi></msub></mfenced><mo>:</mo><mfenced separators="" open="[" close="]"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula> and <i>X</i> is a random variable. The aim is to find conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">f</mi></semantics></math></inline-formula> in order that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. The most examples will focus on a more particular case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">Z</mi><mo>=</mo><mfenced separators="" open="(" close=")"><mi>X</mi><mo>,</mo><mspace width="4.pt"></mspace><msub><mi>f</mi><mn>2</mn></msub><mfenced open="(" close=")"><mi>X</mi></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><mspace width="4.pt"></mspace><msub><mi>f</mi><mi>d</mi></msub><mfenced open="(" close=")"><mi>X</mi></mfenced></mfenced></mrow></semantics></math></inline-formula> with <i>X</i> uniformly distributed on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T18:10:38Z |
| format | Article |
| id | doaj.art-32025e997d87465e92ee48e4d4fd3d39 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:10:38Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-32025e997d87465e92ee48e4d4fd3d392023-11-24T09:07:35ZengMDPI AGMathematics2227-73902022-11-011022419910.3390/math10224199The Leader Property in Quasi Unidimensional CasesAnișoara Maria Răducan0Gheorghiță Zbăganu1“Gheorghe Mihoc—Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania“Gheorghe Mihoc—Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, RomaniaThe following problem was studied: let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></mfenced><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> be a sequence of i.i.d. <i>d</i>-dimensional random vectors. Let <i>F</i> be their probability distribution and for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> consider the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">S</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><msub><mi mathvariant="bold">Z</mi><mn>1</mn></msub><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mi>n</mi></msub><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> was called a “leader” in the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≥</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mspace width="0.166667em"></mspace><mspace width="4pt"></mspace></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> was an “anti-leader” if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≤</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>. The comparison of two vectors was the usual one: if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>=</mo><mfenced separators="" open="(" close=")"><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mn>1</mn></mfenced></msubsup><mo>,</mo><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mn>2</mn></mfenced></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mi>d</mi></mfenced></msubsup></mfenced><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≥</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> means <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>≥</mo><msubsup><mi>Z</mi><mrow><mi>k</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>,</mo></mrow></semantics></math></inline-formula> while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≤</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> means <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Z</mi><mrow><mi>j</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>≤</mo><msubsup><mi>Z</mi><mrow><mi>k</mi></mrow><mfenced open="(" close=")"><mi>i</mi></mfenced></msubsup><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has a leader, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has an anti-leader and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has both a leader and an anti-leader. Sometimes these probabilities can be computed or estimated, for instance in the case when <i>F</i> is discrete or absolutely continuous. The limits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mo movablelimits="true" form="prefix">lim</mo><mo movablelimits="true" form="prefix">inf</mo><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>=</mo><mo movablelimits="true" form="prefix">lim</mo><mo movablelimits="true" form="prefix">inf</mo><msub><mi>b</mi><mi>n</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>=</mo><mo movablelimits="true" form="prefix">lim</mo><mo movablelimits="true" form="prefix">inf</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> were considered. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> it was said that <i>F</i> has the leader property, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> they said that <i>F</i> has the anti-leader property and if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> then <i>F</i> has the order property. In this paper we study an in-between case: here the vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">Z</mi></semantics></math></inline-formula> has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">Z</mi><mo>=</mo><mi mathvariant="bold-italic">f</mi><mfenced open="(" close=")"><mi>X</mi></mfenced></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">f</mi><mo>=</mo><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>f</mi><mi>d</mi></msub></mfenced><mo>:</mo><mfenced separators="" open="[" close="]"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula> and <i>X</i> is a random variable. The aim is to find conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">f</mi></semantics></math></inline-formula> in order that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. The most examples will focus on a more particular case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">Z</mi><mo>=</mo><mfenced separators="" open="(" close=")"><mi>X</mi><mo>,</mo><mspace width="4.pt"></mspace><msub><mi>f</mi><mn>2</mn></msub><mfenced open="(" close=")"><mi>X</mi></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><mspace width="4.pt"></mspace><msub><mi>f</mi><mi>d</mi></msub><mfenced open="(" close=")"><mi>X</mi></mfenced></mfenced></mrow></semantics></math></inline-formula> with <i>X</i> uniformly distributed on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/10/22/4199stochastic orderrandom vectormultivariate distributions |
| spellingShingle | Anișoara Maria Răducan Gheorghiță Zbăganu The Leader Property in Quasi Unidimensional Cases Mathematics stochastic order random vector multivariate distributions |
| title | The Leader Property in Quasi Unidimensional Cases |
| title_full | The Leader Property in Quasi Unidimensional Cases |
| title_fullStr | The Leader Property in Quasi Unidimensional Cases |
| title_full_unstemmed | The Leader Property in Quasi Unidimensional Cases |
| title_short | The Leader Property in Quasi Unidimensional Cases |
| title_sort | leader property in quasi unidimensional cases |
| topic | stochastic order random vector multivariate distributions |
| url | https://www.mdpi.com/2227-7390/10/22/4199 |
| work_keys_str_mv | AT anisoaramariaraducan theleaderpropertyinquasiunidimensionalcases AT gheorghitazbaganu theleaderpropertyinquasiunidimensionalcases AT anisoaramariaraducan leaderpropertyinquasiunidimensionalcases AT gheorghitazbaganu leaderpropertyinquasiunidimensionalcases |