Sal Ly - Academia.edu (original) (raw)

Papers by Sal Ly

Social Science Research Network, 2019

Proceedings of the 1st International Conference on Applied Mathematics in Engineering and Reliability (Ho Chi Minh City, Vietnam, 4-6 May 2016), 2016

Journal of Advanced Engineering and Computation, 2019

When doing research on scientific issues, it is very significant if our research issues are close... more When doing research on scientific issues, it is very significant if our research issues are closely connected to real applications. In reality, when analyzing data in practice, there are frequently several models that can appropriate to the survey data. Hence, it is necessary to have a standard criterion to choose the most ecient model. In this article, our primary interest is to compare and discuss about the criteria for selecting a model and its applications. The authors provide approaches and procedures of these methods and apply to the traffic violation data where we look for the most appropriate model among Poisson regression, Zero-inflated Poisson regression and Negative binomial regression to capture between number of violated speed regulations and some factors including distance covered, motorcycle engine and age of respondents by using AIC, BIC and Vuong's test. Based on results on the training, validation and test data set, we find that the criteria AIC and BIC are mor...

Risks, 2019

Determining distributions of the functions of random variables is one of the most important probl... more Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distr...

Journal of Risk and Financial Management, 2019

Determining distributions of the functions of random variables is a very important problem with a... more Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are ...

SSRN Electronic Journal, 2019

Determining distributions of the functions of random variables is a very important problem with w... more Determining distributions of the functions of random variables is a very important problem with wide applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient <i>Y = X₁/X ₂</i> and the ratio of one variable over the sum of two variables <i>Z = X₁/X₁+X₂</i> of two dependent or independent random variables <i>X₁</i> and <i>X₂</i> by using copulas to capture the structures between <i>X₁</i> and <i>X₂</i>. We then extend the theory by establishing the density and distribution functions for the quotients <i>Y = X₁/ X₂ and <i>Z = X₁/X₁+X₂ of two dependent normal random variables <i>X₁</i> and <i>X₂</i> in case of Gaussian copulas. Thereafter, we develop the theory on the median for the ratios of both <i>Y</i> and <i>Z</i> on two normal random variables <i>X₁</i> and <i>X₂</i>. Furthermore, the result of median for <i>Z</i> is also extended to a larger family of symmetric distributions and symmetric copulas of <i>X₁</i> and <i>X₂</i>. Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent random variables. Since the densities and distributions of the ratios of both <i>Y</i> and <i>Z</i> are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall's coefficient. Our findings are useful for academics, practitioners, and policy makers.</i></i>

Social Science Research Network, 2019

Proceedings of the 1st International Conference on Applied Mathematics in Engineering and Reliability (Ho Chi Minh City, Vietnam, 4-6 May 2016), 2016

Journal of Advanced Engineering and Computation, 2019

When doing research on scientific issues, it is very significant if our research issues are close... more When doing research on scientific issues, it is very significant if our research issues are closely connected to real applications. In reality, when analyzing data in practice, there are frequently several models that can appropriate to the survey data. Hence, it is necessary to have a standard criterion to choose the most ecient model. In this article, our primary interest is to compare and discuss about the criteria for selecting a model and its applications. The authors provide approaches and procedures of these methods and apply to the traffic violation data where we look for the most appropriate model among Poisson regression, Zero-inflated Poisson regression and Negative binomial regression to capture between number of violated speed regulations and some factors including distance covered, motorcycle engine and age of respondents by using AIC, BIC and Vuong's test. Based on results on the training, validation and test data set, we find that the criteria AIC and BIC are mor...

Risks, 2019

Determining distributions of the functions of random variables is one of the most important probl... more Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distr...

Journal of Risk and Financial Management, 2019

Determining distributions of the functions of random variables is a very important problem with a... more Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are ...

SSRN Electronic Journal, 2019

Determining distributions of the functions of random variables is a very important problem with w... more Determining distributions of the functions of random variables is a very important problem with wide applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient <i>Y = X₁/X ₂</i> and the ratio of one variable over the sum of two variables <i>Z = X₁/X₁+X₂</i> of two dependent or independent random variables <i>X₁</i> and <i>X₂</i> by using copulas to capture the structures between <i>X₁</i> and <i>X₂</i>. We then extend the theory by establishing the density and distribution functions for the quotients <i>Y = X₁/ X₂ and <i>Z = X₁/X₁+X₂ of two dependent normal random variables <i>X₁</i> and <i>X₂</i> in case of Gaussian copulas. Thereafter, we develop the theory on the median for the ratios of both <i>Y</i> and <i>Z</i> on two normal random variables <i>X₁</i> and <i>X₂</i>. Furthermore, the result of median for <i>Z</i> is also extended to a larger family of symmetric distributions and symmetric copulas of <i>X₁</i> and <i>X₂</i>. Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent random variables. Since the densities and distributions of the ratios of both <i>Y</i> and <i>Z</i> are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall's coefficient. Our findings are useful for academics, practitioners, and policy makers.</i></i>