Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data
A new class of distribution called the Fréchet binomial (FB) distribution is proposed. The new suggested model is very flexible because its probability density function can be unimodal, decreasing and skewed to the right. Furthermore, the hazard rate function can be increasing, decreasing, up-side-d...
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MDPI AG
2022-08-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/8/389 |
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| author | Salem A. Alyami Mohammed Elgarhy Ibrahim Elbatal Ehab M. Almetwally Naif Alotaibi Ahmed R. El-Saeed |
| author_facet | Salem A. Alyami Mohammed Elgarhy Ibrahim Elbatal Ehab M. Almetwally Naif Alotaibi Ahmed R. El-Saeed |
| author_sort | Salem A. Alyami |
| collection | DOAJ |
| description | A new class of distribution called the Fréchet binomial (FB) distribution is proposed. The new suggested model is very flexible because its probability density function can be unimodal, decreasing and skewed to the right. Furthermore, the hazard rate function can be increasing, decreasing, up-side-down and reversed-J form. Important mixture representations of the probability density function (pdf) and cumulative distribution function (cdf) are computed. Numerous sub-models of the FB distribution are explored. Numerous statistical and mathematical features of the FB distribution such as the quantile function (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>U</mi><mi>N</mi></mrow></msub><mi>F</mi></mrow></semantics></math></inline-formula>); moments (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula>); incomplete <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><msub><mi>M</mi><mi>O</mi></msub></mrow></semantics></math></inline-formula>); conditional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>M</mi><mi>O</mi></msub></mrow></semantics></math></inline-formula>); <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> generating function (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>O</mi></msub><mi>G</mi><mi>F</mi></mrow></semantics></math></inline-formula>); probability weighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>W</mi><msub><mi>M</mi><mi>O</mi></msub></mrow></semantics></math></inline-formula>); order statistics; and entropy are computed. When the life test is shortened at a certain time, acceptance sampling (ACS) plans for the new proposed distribution, FB distribution, are produced. The truncation time is supposed to be the median lifetime of the FB distribution multiplied by a set of parameters. The smallest sample size required ensures that the specified life test is obtained at a particular consumer’s risk. The numerical results for a particular consumer’s risk, FB distribution parameters and truncation time are generated. We discuss the method of maximum likelihood to estimate the model parameters. A simulation study was performed to assess the behavior of the estimates. Three real datasets are used to illustrate the importance and flexibility of the proposed model. |
| first_indexed | 2024-03-09T10:02:03Z |
| format | Article |
| id | doaj.art-f2d114e5276d47939e7239c37dab28a1 |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T10:02:03Z |
| publishDate | 2022-08-01 |
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| spelling | doaj.art-f2d114e5276d47939e7239c37dab28a12023-12-01T23:24:35ZengMDPI AGAxioms2075-16802022-08-0111838910.3390/axioms11080389Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime DataSalem A. Alyami0Mohammed Elgarhy1Ibrahim Elbatal2Ehab M. Almetwally3Naif Alotaibi4Ahmed R. El-Saeed5Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi ArabiaThe Higher Institute of Commercial Sciences, Al Mahalla Al Kubra 31951, EgyptDepartment of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi ArabiaFaculty of Business Administration, Delta University of Science and Technology, Gamasa 11152, EgyptDepartment of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi ArabiaDepartment of Basic Sciences, Obour High Institute for Management & Informatics, Obour 11848, EgyptA new class of distribution called the Fréchet binomial (FB) distribution is proposed. The new suggested model is very flexible because its probability density function can be unimodal, decreasing and skewed to the right. Furthermore, the hazard rate function can be increasing, decreasing, up-side-down and reversed-J form. Important mixture representations of the probability density function (pdf) and cumulative distribution function (cdf) are computed. Numerous sub-models of the FB distribution are explored. Numerous statistical and mathematical features of the FB distribution such as the quantile function (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>U</mi><mi>N</mi></mrow></msub><mi>F</mi></mrow></semantics></math></inline-formula>); moments (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula>); incomplete <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><msub><mi>M</mi><mi>O</mi></msub></mrow></semantics></math></inline-formula>); conditional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>M</mi><mi>O</mi></msub></mrow></semantics></math></inline-formula>); <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> generating function (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>O</mi></msub><mi>G</mi><mi>F</mi></mrow></semantics></math></inline-formula>); probability weighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mi>O</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>W</mi><msub><mi>M</mi><mi>O</mi></msub></mrow></semantics></math></inline-formula>); order statistics; and entropy are computed. When the life test is shortened at a certain time, acceptance sampling (ACS) plans for the new proposed distribution, FB distribution, are produced. The truncation time is supposed to be the median lifetime of the FB distribution multiplied by a set of parameters. The smallest sample size required ensures that the specified life test is obtained at a particular consumer’s risk. The numerical results for a particular consumer’s risk, FB distribution parameters and truncation time are generated. We discuss the method of maximum likelihood to estimate the model parameters. A simulation study was performed to assess the behavior of the estimates. Three real datasets are used to illustrate the importance and flexibility of the proposed model.https://www.mdpi.com/2075-1680/11/8/389Fréchet distributionbinomial distributionmomentsentropyacceptance sampling planmaximum likelihood approach |
| spellingShingle | Salem A. Alyami Mohammed Elgarhy Ibrahim Elbatal Ehab M. Almetwally Naif Alotaibi Ahmed R. El-Saeed Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data Axioms Fréchet distribution binomial distribution moments entropy acceptance sampling plan maximum likelihood approach |
| title | Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data |
| title_full | Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data |
| title_fullStr | Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data |
| title_full_unstemmed | Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data |
| title_short | Fréchet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data |
| title_sort | frechet binomial distribution statistical properties acceptance sampling plan statistical inference and applications to lifetime data |
| topic | Fréchet distribution binomial distribution moments entropy acceptance sampling plan maximum likelihood approach |
| url | https://www.mdpi.com/2075-1680/11/8/389 |
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