Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Predicting depth value of the future depth-based multivariate record

  • Samaneh Tata (Department of Statistics, Faculty of Mathematical Sciences, Shahid Beheshti University) ;
  • Mohammad Reza Faridrohani (Department of Statistics, Faculty of Mathematical Sciences, Shahid Beheshti University)
  • Received : 2023.02.19
  • Accepted : 2023.02.25
  • Published : 2023.09.30
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Abstract

The prediction problem of univariate records, though not addressed in multivariate records, has been discussed by many authors based on records values. There are various definitions for multivariate records among which depth-based records have been selected for the aim of this paper. In this paper, by means of the maximum likelihood and conditional median methods, point and interval predictions of depth values which are related to the future depth-based multivariate records are considered on the basis of the observed ones. The observations derived from some elements of the elliptical distributions are the main reason of studying this problem. Finally, the satisfactory performance of the prediction methods is illustrated via some simulation studies and a real dataset about Kermanshah city drought.

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References

  1. Ahmadi J, Jafari Jozani M, Marchand E, and Parsian A (2009). Prediction of k-records from a general class of distributions under balanced type loss functions, Metrika, 70, 19-33. https://doi.org/10.1007/s00184-008-0176-5
  2. Ahsanullah M (1980). Linear prediction of record values for the two parameter exponential distribution, Annals of the Institute of Statistical Mathematics, 32, 363-368. https://doi.org/10.1007/BF02480340
  3. Ahsanullah M (2009). Records and concomitants, Bulletin of the Malaysian Mathematical Sciences Society. Second Series, 32, 101-117.
  4. Ahsanullah M and Nevzorov VB (2015). Records via Probability Theory, Atlantis Press, Paris.
  5. Arnold BC, Balakrishnan N, and Nagaraja HN (2011). Records, John Wiley & Sons, New York.
  6. Berred AM (1998). Prediction of record values, Communications in Statistics-Theory and Methods, 27, 2221-2240. https://doi.org/10.1080/03610929808832224
  7. DeGroot MH (2005). Optimal Statistical Decisions, John Wiley & Sons, New York.
  8. Gnedin A (2007). The chain records, Electronic Journal of Probability, 12, 767-786. https://doi.org/10.1214/EJP.v12-410
  9. Henningsen A and Toomet O (2011). maxLik: A package for maximum likelihood estimation in R, Computational Statistics, 26, 443-458. https://doi.org/10.1007/s00180-010-0217-1
  10. Hwang HK and Tsai TH (2010). Multivariate records based on dominance, Electronic Journal of Probability, 15, 1863-1892. https://doi.org/10.1214/EJP.v15-825
  11. Liu RY (1992). Data depth and multivariate rank tests. In Proceedings of 2nd International Conference on Stat. Data Analysis Based on the L-1 Norm and Related Methods Y Dodge Ed, North-Holland, 279-294.
  12. Liu RY, Parelius JM, and Singh K (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion and a rejoinder by liu and singh), The Annals of Statistics, 27, 783-858. https://doi.org/10.1214/aos/1018031259
  13. Liu RY and Singh K (1993). A quality index based on data depth and multivariate rank tests, Journal of the American Statistical Association, 88, 252-260. https://doi.org/10.1080/01621459.1993.10594317
  14. Nevzorov VB (2001). Records: Mathematical Theory, American Mathematical Society, Providence, RI.
  15. Raqab MZ, Ahmadi J, and Doostparast M (2007). Statistical inference based on record data from Pareto model, Statistics, 41, 105-118. https://doi.org/10.1080/02331880601106579
  16. Resnick SI (1973). Record values and maxima, The Annals of Probability, 1, 650-662. https://doi.org/10.1214/aop/1176996892
  17. Serfling R (2006). Depth functions in nonparametric multivariate inference, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 72, 1-16. https://doi.org/10.1090/dimacs/072/01
  18. Smith RL (1988). Forecasting records by maximum likelihood, Journal of the American Statistical Association, 83, 331-338. https://doi.org/10.1080/01621459.1988.10478602
  19. Tat S and Faridrohani MR (2021). A new type of multivariate records: Depth-based records, Statistics, 55, 296-320. https://doi.org/10.1080/02331888.2021.1925280
  20. Tukey J (1975). Mathematics and picturing data, Proceedings of International Congress of Mathematicians, 2, 523-531.
  21. Zuo Y (2003). Projection-based depth functions and associated medians, The Annals of Statistics, 31, 1460-1490. https://doi.org/10.1214/aos/1065705115
  22. Zuo Y and Serfling R (2000a). General notions of statistical depth function, The Annals of Statistics, 28, 461-482. https://doi.org/10.1214/aos/1016218226
  23. Zuo Y and Serfling R (2000b). Structural properties and convergence results for contours of sample statistical depth functions, The Annals of Statistics, 28, 483-499. https://doi.org/10.1214/aos/1016218227