Consistency of Decision in Finite and Numerable Multinomial Models
The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model <inline-formula><math xmlns="http://www.w3....
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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/2227-7390/11/11/2434 |
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| author | Isaac Akoto João T. Mexia |
| author_facet | Isaac Akoto João T. Mexia |
| author_sort | Isaac Akoto |
| collection | DOAJ |
| description | The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">M</mi><mfenced separators="" open="(" close=")"><mspace width="3.33333pt"></mspace><mo>|</mo><mi>n</mi><mo>,</mo><mi mathvariant="bold-italic">p</mi></mfenced></mrow></semantics></math></inline-formula> whose decision is indexed by a parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> and having a cost <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi mathvariant="bold-italic">p</mi></mfenced></mrow></semantics></math></inline-formula> depending on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> and on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">p</mi></semantics></math></inline-formula>, we show that, under general conditions, the probability of taking the least cost decision tends to 1 when <i>n</i> tends to <i>∞</i>, i.e., we showed that the cost decision is consistent, representing a Statistical Decision Theory approach to the concept of consistency, which is not much considered in the literature. Thus, under these conditions, we have consistency in the decision making. The key result is that the estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi mathvariant="bold-italic">p</mi><mo stretchy="false">˜</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> with components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo stretchy="false">˜</mo></mover><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>n</mi><mi>i</mi></msub><mi>n</mi></mfrac></mstyle><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="4pt"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>⋯</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>n</mi><mi>i</mi></msub></semantics></math></inline-formula> is the number of times we obtain the <i>i</i>th result when we have a sample of size <i>n</i>, is a consistent estimator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">p</mi></semantics></math></inline-formula>. This result holds both for finite and numerable models. By this result, we were able to incorporate a more general form for consistency for the cost function of a multinomial model. |
| first_indexed | 2024-03-11T03:02:24Z |
| format | Article |
| id | doaj.art-29c777c83a184f7a873152e807c1fb31 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T03:02:24Z |
| publishDate | 2023-05-01 |
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| series | Mathematics |
| spelling | doaj.art-29c777c83a184f7a873152e807c1fb312023-11-18T08:11:58ZengMDPI AGMathematics2227-73902023-05-011111243410.3390/math11112434Consistency of Decision in Finite and Numerable Multinomial ModelsIsaac Akoto0João T. Mexia1Center of Mathematics and Its Applications, NOVA School of Science and Technology, Universidade NOVA de Lisboa, Campus de Caparica, 2829-516 Caparica, PortugalCenter of Mathematics and Its Applications, NOVA School of Science and Technology, Universidade NOVA de Lisboa, Campus de Caparica, 2829-516 Caparica, PortugalThe multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">M</mi><mfenced separators="" open="(" close=")"><mspace width="3.33333pt"></mspace><mo>|</mo><mi>n</mi><mo>,</mo><mi mathvariant="bold-italic">p</mi></mfenced></mrow></semantics></math></inline-formula> whose decision is indexed by a parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> and having a cost <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi mathvariant="bold-italic">p</mi></mfenced></mrow></semantics></math></inline-formula> depending on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> and on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">p</mi></semantics></math></inline-formula>, we show that, under general conditions, the probability of taking the least cost decision tends to 1 when <i>n</i> tends to <i>∞</i>, i.e., we showed that the cost decision is consistent, representing a Statistical Decision Theory approach to the concept of consistency, which is not much considered in the literature. Thus, under these conditions, we have consistency in the decision making. The key result is that the estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi mathvariant="bold-italic">p</mi><mo stretchy="false">˜</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> with components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo stretchy="false">˜</mo></mover><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>n</mi><mi>i</mi></msub><mi>n</mi></mfrac></mstyle><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="4pt"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>⋯</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>n</mi><mi>i</mi></msub></semantics></math></inline-formula> is the number of times we obtain the <i>i</i>th result when we have a sample of size <i>n</i>, is a consistent estimator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">p</mi></semantics></math></inline-formula>. This result holds both for finite and numerable models. By this result, we were able to incorporate a more general form for consistency for the cost function of a multinomial model.https://www.mdpi.com/2227-7390/11/11/2434stochastic convergencedecision theoryestimators |
| spellingShingle | Isaac Akoto João T. Mexia Consistency of Decision in Finite and Numerable Multinomial Models Mathematics stochastic convergence decision theory estimators |
| title | Consistency of Decision in Finite and Numerable Multinomial Models |
| title_full | Consistency of Decision in Finite and Numerable Multinomial Models |
| title_fullStr | Consistency of Decision in Finite and Numerable Multinomial Models |
| title_full_unstemmed | Consistency of Decision in Finite and Numerable Multinomial Models |
| title_short | Consistency of Decision in Finite and Numerable Multinomial Models |
| title_sort | consistency of decision in finite and numerable multinomial models |
| topic | stochastic convergence decision theory estimators |
| url | https://www.mdpi.com/2227-7390/11/11/2434 |
| work_keys_str_mv | AT isaacakoto consistencyofdecisioninfiniteandnumerablemultinomialmodels AT joaotmexia consistencyofdecisioninfiniteandnumerablemultinomialmodels |