From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case
Abstract
:1. Introduction
2. State of the Art
3. Pareto Distribution
3.1. Product of Two Pareto Distributed Random Variables
3.2. Product of Gaussian and Pareto Distributed Random Variables
3.3. Product of Log-Normal and Pareto Distributed Random Variables
4. Student’s t Distribution
4.1. Product of Two Student’s t Distributed Random Variables
4.2. Product of Gaussian and Student’s t Distributed Random Variables
4.3. Product of Log-Normal and Student’s t Distributed Random Variables
5. Parameters Estimation–Simulation Study
6. Real Data Application—Distribution of Electricity Transaction Values
7. Conclusions
8. Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Gaussian and Log-Normal Distributions
Appendix A.1. Product of Two Gaussian Distributed Random Variables
Appendix A.2. Product of Two Log-Normally Distributed Random Variables
Appendix B. Proofs
Appendix B.1. Proof of Lemma 2
Appendix B.2. Proof of Lemma 3
References
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20 December | ||||
Variable | Distribution | KS statistic | RMSE | KLD |
Volume | log-normal | 0.0810 | 0.0359 | 0.1367 |
Pareto | 0.2263 | 0.1350 | 0.5717 | |
Price | log-normal | 0.1719 | 0.0813 | 3.1737 |
Gaussian | 0.1305 | 0.0560 | 2.5211 | |
Student’s t | 0.0523 | 0.0206 | 2.5304 | |
Value | log-normal (x, y) | 0.0501 | 0.0247 | 0.0819 |
log-normal xy | 0.0498 | 0.0247 | 0.0208 | |
log-normal—Student’s t (x, y) | 0.0487 | 0.0242 | 0.0214 | |
log-normal—Student’s t xy | 0.0500 | 0.0245 | 0.0214 | |
15 March | ||||
Volume | log-normal | 0.0952 | 0.0501 | 0.2868 |
Pareto | 0.2881 | 0.1702 | 0.9211 | |
Price | Gaussian | 0.1014 | 0.0333 | 1.4262 |
Student’s t | 0.0891 | 0.0308 | 1.4707 | |
Value | log-normal—Student’s t (x, y) | 0.0414 | 0.0167 | 1.4370 |
log-normal—Student’s t xy | 0.0369 | 0.0148 | 1.4202 |
Multivariate Pareto |
Independent Pareto |
Gaussian-Pareto |
Log-normal-Pareto |
Multivariate Student’s t |
Independent Student’s t |
Gaussian-Student’s t |
Log-normal-Student’s t |
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Adamska, J.; Bielak, Ł.; Janczura, J.; Wyłomańska, A. From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case. Mathematics 2022, 10, 3371. https://doi.org/10.3390/math10183371
Adamska J, Bielak Ł, Janczura J, Wyłomańska A. From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case. Mathematics. 2022; 10(18):3371. https://doi.org/10.3390/math10183371
Chicago/Turabian StyleAdamska, Julia, Łukasz Bielak, Joanna Janczura, and Agnieszka Wyłomańska. 2022. "From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case" Mathematics 10, no. 18: 3371. https://doi.org/10.3390/math10183371
APA StyleAdamska, J., Bielak, Ł., Janczura, J., & Wyłomańska, A. (2022). From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case. Mathematics, 10(18), 3371. https://doi.org/10.3390/math10183371