Journal of the Iranian Statistical Society

Quantile Approach of Generalized Cumulative Residual Information Measure of Order $(alpha,beta)$

Authors

1 Maharishi Dayanand University Rohtak-124001

2 UIET, M. D. University Rohtak-124001

3 DCRUST Murthal Sonipat

10.52547/jirss.19.2.67

Abstract

In this paper, we introduce the concept of quantile-based generalized cumulative residual entropy of order $(alpha,beta)$ for residual and past lifetimes and study their properties. Further we study the proposed information measure for series and parallel system when random variable are untruncated or truncated in nature and some characterization results are presented. At the end, we study generalized weighted dynamic cumulative residual entropy in terms of quantile function.

Keywords

  1. Abbasnejad, M., Arghami, N. R., 2011. $acute{Renyi}$ entropy properties of order statistics. Communications in StatisticsTheory and Methods 40, 40-52. Ajith, K. K., Abdul Sathar, E. I., 2020. Some Results on Dynamic Weighted Varma’s Entropy and its Applications. American Journal of Mathematical and Management Sciences 39(1), 90-98. Arnold, B. C, Balakrishnan, N. and Nagaraja, H. N., 1992. A First Course in Order Statistics. John Wiley and Sons, New York. Asadi, M., Zohrevand, Y., 2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931-1941. Baratpour, S., Khammar, A., H., 2018. A quantile-based generalized dynamic cumulative measure of entropy. Communications in Statistics- Theory and Methods 47(13), 3104-3117. Di Crescenzo, A., Longobardi, M., 2009. On cumulative entropies. Journal of Statistical Planning and Inference 139 (12), 4072-4087. David, H. A., Nagaraja, H.N., 2003. Order Statistics. John Wiley and Sons, New York. Gilchrist, W., 2000. Statistical modelling with quantile functions. Chapman and Hall/CRC, Boca Raton, FL. Govindarajulu, Z., 1977. A class of distributions useful in lifetesting and reliability with applications to nonparametric testing. In: C.P. Tsokos, I.N. Shimi (eds.) Theory and Applications of Reliability, vol. 1, 109-130. Acad. Press, New York. Guiasu, S., Picard, C.F., 1971. Borne infericutre de la Longuerur utile de certain codes, C. R. Acad. Sci. Paris Ser. A 273, 248-251. Hankin, R. K. S., Lee, A., 2006. A new family of nonnegative distributions. Austral. N. Z. J. Statist. 48, 67-78. Kayal, S., 2016. Characterization based on generalized entropy of order statistics. Communications in Statistics-Theory and Methods 45(15), 4628–4636. Kayal, S., Moharana, R. 2017. On weighted cumulative residual entropy. Journal of Statistics and Management Systems 20, 153-173. Kumar, V., Singh, N., 2018. Characterization of Lifetime Distribution Based on Generalized Interval Entropy. Statistics, Optimization and Information Computing 6, 547-559. Kumar, V., Singh, N., 2019. Quantile-based generalized entropy of order $(alpha,beta)$ for order statistics. Statistica 78(4), 299-318 . Kumar, V., Taneja, H. C., 2011. A Generalized entropy-based residual lifetime distributions. International Journal of Biomathematics 4(2), 171-184. Minimol, S., 2017. On generalized dynamic cumulative past entropy measure. Communications in Statistics-Theory and Methods 46(6), 2816–2822. Mirali, M., Baratpour, S., Fakoor, V., 2017. On weighted cumulative residual entropy. Communications in Statistics-Theory and Methods 46, 2857–2869. Misagh, F., Panahi, Y., Yari, G.H., Shahi, R. 2011. Weighted cumulative entropy and its estimation. In: 2011 IEEE International Conference on Quality and Reliability (ICQR). doi:10.1109/ICQR.2011.6031765 Nair, N. U., P. G. Sankaran, and N. Balakrishnan. 2013. Quantile-based reliability analysis. New York: Springer Nair, N. U., Sankaran, P. G., 2009. Quantile-based reliability analysis. Communications in Statistics Theory and Methods 38, 222-232. Nanda, A. K., Sankaran, P. G. and Sunoj, S. M. (2014). Residual Renyi entropy: A Quantile approach. Statistics and Probability Letters, 85, 114-121. Navarro, J., del Aguila, Y., Majid, A., 2010. Journal of Statistical Planning and Inference, 140(1), 310-322. Rao, M. Chen, Y. Vemuri, B. C. and Wang, F., 2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220-1228. Rao, M., 2005. More on a new concept of entropy and information. Journal of Theoretical Probability 18, 967-981. Sankaran, P.G., Sunoj, S.M., 2017. Quantile-based cumulative entropies. Communications in Statistics Theory and Methods 46 (2), 805-814. Shannon, C. E., 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379–423. Sheraz, M., Dedub, S., Predaa, V., 2015. Entropy Measures for Assessing Volatile Markets. Procedia Economics and Finance 22, 655-662. Sreelakshmi, N., Kattumannil, S.K., Asha, G., 2018. Quantile-based tests for expontiality against dmrq and nbue alternatives. Journal of Korean Statistics Society, 47(2), 185-200. Sunoj, S. M., Krishnan, A. S., Sankaran, P. G., 2017. Quantile-based entropy of order statistics. Journal of the Indian Society for Probability and Statistics 18, 1-17. Sunoj, S. M., Sankaran, P. G., 2012. Quantile-based entropy function. Statistics and Probability Letters 82, 1049-1053. Sunoj, S. M., Sankaran, P. G., Nanda, A. K., 2013. Quantile-based entropy function in past lifetime. Statistics and Probability Letters 83, 366-372. Thapliyal, R., Taneja, H. C., 2012. Generalized entropy of order statistics. Applied Mathematics 3, 1977-1982. Thapliyal, R., Taneja, H. C., Kumar, V., 2015. Characterization results based on non-additive entropy of order statistics. Physica A: Statistical Mechanics and its Applications, 417, 297-303. Taneja, H.C., Hooda, D.S. and Tuteja, R.K. (1985). Coding theorems on a generalized ‘useful’ information. Soochow Journal of Mathematics, 11, 123-131. Toomaj, A., Di Crescenzo, A., 2020. Connections between weighted generalized cumulative residual entropy and variance. Mathematics 8(7), 1072. Wang, F., Vemuri, B. C., 2007. Non-Rigid multimodal image registration using cross-cumulative residual entropy. International Journal of Computer Vision 74(2), 201-215. Van Staden, P.J., Loots, M. R., 2009. L-moment estimation for the generalized lambda distribution. Third Annual ASEARC Conference, New Castle, Australia.
  2. Abbasnejad, M., Arghami, N. R., 2011. $acute{Renyi}$ entropy properties of order statistics. Communications in StatisticsTheory and Methods 40, 40-52. [DOI:10.1080/03610920903353683]
  3. Abbasnejad, M., Arghami, N. R., 2011. $acute{Renyi}$ entropy properties of order statistics. Communications in StatisticsTheory and Methods 40, 40-52. [DOI:10.1080/03610920903353683]
  4. Ajith, K. K., Abdul Sathar, E. I., 2020. Some Results on Dynamic Weighted Varma's Entropy and its Applications. American Journal of Mathematical and Management Sciences 39(1), 90-98. [DOI:10.1080/01966324.2019.1642817]
  5. Ajith, K. K., Abdul Sathar, E. I., 2020. Some Results on Dynamic Weighted Varma's Entropy and its Applications. American Journal of Mathematical and Management Sciences 39(1), 90-98. [DOI:10.1080/01966324.2019.1642817]
  6. Arnold, B. C, Balakrishnan, N. and Nagaraja, H. N., 1992. A First Course in Order Statistics. John Wiley and Sons, New York.
  7. Arnold, B. C, Balakrishnan, N. and Nagaraja, H. N., 1992. A First Course in Order Statistics. John Wiley and Sons, New York.
  8. Asadi, M., Zohrevand, Y., 2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931-1941. [DOI:10.1016/j.jspi.2006.06.035]
  9. Asadi, M., Zohrevand, Y., 2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931-1941. [DOI:10.1016/j.jspi.2006.06.035]
  10. Baratpour, S., Khammar, A., H., 2018. A quantile-based generalized dynamic cumulative measure of entropy. Communications in Statistics- Theory and Methods 47(13), 3104-3117. [DOI:10.1080/03610926.2017.1348520]
  11. Baratpour, S., Khammar, A., H., 2018. A quantile-based generalized dynamic cumulative measure of entropy. Communications in Statistics- Theory and Methods 47(13), 3104-3117. [DOI:10.1080/03610926.2017.1348520]
  12. Di Crescenzo, A., Longobardi, M., 2009. On cumulative entropies. Journal of Statistical Planning and Inference 139 (12), 4072-4087. [DOI:10.1016/j.jspi.2009.05.038]
  13. Di Crescenzo, A., Longobardi, M., 2009. On cumulative entropies. Journal of Statistical Planning and Inference 139 (12), 4072-4087. [DOI:10.1016/j.jspi.2009.05.038]
  14. David, H. A., Nagaraja, H.N., 2003. Order Statistics. John Wiley and Sons, New York. [DOI:10.1002/0471722162]
  15. David, H. A., Nagaraja, H.N., 2003. Order Statistics. John Wiley and Sons, New York. [DOI:10.1002/0471722162]
  16. Gilchrist, W., 2000. Statistical modelling with quantile functions. Chapman and Hall/CRC, Boca Raton, FL. [DOI:10.1201/9781420035919]
  17. Gilchrist, W., 2000. Statistical modelling with quantile functions. Chapman and Hall/CRC, Boca Raton, FL. [DOI:10.1201/9781420035919]
  18. Govindarajulu, Z., 1977. A class of distributions useful in lifetesting and reliability with applications to nonparametric testing. In: C.P. Tsokos, I.N. Shimi (eds.) Theory and Applications of Reliability, vol. 1, 109-130. Acad. Press, New York.
  19. Govindarajulu, Z., 1977. A class of distributions useful in lifetesting and reliability with applications to nonparametric testing. In: C.P. Tsokos, I.N. Shimi (eds.) Theory and Applications of Reliability, vol. 1, 109-130. Acad. Press, New York.
  20. Guiasu, S., Picard, C.F., 1971. Borne infericutre de la Longuerur utile de certain codes, C. R. Acad. Sci. Paris Ser. A 273, 248-251.
  21. Guiasu, S., Picard, C.F., 1971. Borne infericutre de la Longuerur utile de certain codes, C. R. Acad. Sci. Paris Ser. A 273, 248-251.
  22. Hankin, R. K. S., Lee, A., 2006. A new family of nonnegative distributions. Austral. N. Z. J. Statist. 48, 67-78. [DOI:10.1111/j.1467-842X.2006.00426.x]
  23. Hankin, R. K. S., Lee, A., 2006. A new family of nonnegative distributions. Austral. N. Z. J. Statist. 48, 67-78. [DOI:10.1111/j.1467-842X.2006.00426.x]
  24. Kayal, S., 2016. Characterization based on generalized entropy of order statistics. Communications in Statistics-Theory and Methods 45(15), 4628-4636. [DOI:10.1080/03610926.2014.927491]
  25. Kayal, S., 2016. Characterization based on generalized entropy of order statistics. Communications in Statistics-Theory and Methods 45(15), 4628-4636. [DOI:10.1080/03610926.2014.927491]
  26. Kayal, S., Moharana, R. 2017. On weighted cumulative residual entropy. Journal of Statistics and Management Systems 20, 153-173. [DOI:10.1080/09720510.2016.1224471]
  27. Kayal, S., Moharana, R. 2017. On weighted cumulative residual entropy. Journal of Statistics and Management Systems 20, 153-173. [DOI:10.1080/09720510.2016.1224471]
  28. Kumar, V., Singh, N., 2018. Characterization of Lifetime Distribution Based on Generalized Interval Entropy. Statistics, Optimization and Information Computing 6, 547-559. [DOI:10.19139/soic.v6i4.328]
  29. Kumar, V., Singh, N., 2018. Characterization of Lifetime Distribution Based on Generalized Interval Entropy. Statistics, Optimization and Information Computing 6, 547-559. [DOI:10.19139/soic.v6i4.328]
  30. Kumar, V., Singh, N., 2019. Quantile-based generalized entropy of order $(alpha,beta)$ for order statistics. Statistica 78(4), 299-318 .
  31. Kumar, V., Singh, N., 2019. Quantile-based generalized entropy of order $(alpha,beta)$ for order statistics. Statistica 78(4), 299-318 .
  32. Kumar, V., Taneja, H. C., 2011. A Generalized entropy-based residual lifetime distributions. International Journal of Biomathematics 4(2), 171-184. [DOI:10.1142/S1793524511001416]
  33. Kumar, V., Taneja, H. C., 2011. A Generalized entropy-based residual lifetime distributions. International Journal of Biomathematics 4(2), 171-184. [DOI:10.1142/S1793524511001416]
  34. Minimol, S., 2017. On generalized dynamic cumulative past entropy measure. Communications in Statistics-Theory and Methods 46(6), 2816-2822. [DOI:10.1080/03610926.2015.1053928]
  35. Minimol, S., 2017. On generalized dynamic cumulative past entropy measure. Communications in Statistics-Theory and Methods 46(6), 2816-2822. [DOI:10.1080/03610926.2015.1053928]
  36. Mirali, M., Baratpour, S., Fakoor, V., 2017. On weighted cumulative residual entropy. Communications in Statistics-Theory and Methods 46, 2857-2869. [DOI:10.1080/03610926.2015.1053932]
  37. Mirali, M., Baratpour, S., Fakoor, V., 2017. On weighted cumulative residual entropy. Communications in Statistics-Theory and Methods 46, 2857-2869. [DOI:10.1080/03610926.2015.1053932]
  38. Misagh, F., Panahi, Y., Yari, G.H., Shahi, R. 2011. Weighted cumulative entropy and its estimation. In: 2011 IEEE International Conference on Quality and Reliability (ICQR). doi:10.1109/ICQR.2011.6031765 [DOI:10.1109/ICQR.2011.6031765]
  39. Misagh, F., Panahi, Y., Yari, G.H., Shahi, R. 2011. Weighted cumulative entropy and its estimation. In: 2011 IEEE International Conference on Quality and Reliability (ICQR). doi:10.1109/ICQR.2011.6031765 [DOI:10.1109/ICQR.2011.6031765]
  40. Nair, N. U., P. G. Sankaran, and N. Balakrishnan. 2013. Quantile-based reliability analysis. New York: Springer [DOI:10.1007/978-0-8176-8361-0]
  41. Nair, N. U., P. G. Sankaran, and N. Balakrishnan. 2013. Quantile-based reliability analysis. New York: Springer [DOI:10.1007/978-0-8176-8361-0]
  42. Nair, N. U., Sankaran, P. G., 2009. Quantile-based reliability analysis. Communications in Statistics Theory and Methods 38, 222-232. [DOI:10.1080/03610920802187430]
  43. Nair, N. U., Sankaran, P. G., 2009. Quantile-based reliability analysis. Communications in Statistics Theory and Methods 38, 222-232. [DOI:10.1080/03610920802187430]
  44. Nanda, A. K., Sankaran, P. G. and Sunoj, S. M. (2014). Residual Renyi entropy: A Quantile approach. Statistics and Probability Letters, 85, 114-121. [DOI:10.1016/j.spl.2013.11.016]
  45. Nanda, A. K., Sankaran, P. G. and Sunoj, S. M. (2014). Residual Renyi entropy: A Quantile approach. Statistics and Probability Letters, 85, 114-121. [DOI:10.1016/j.spl.2013.11.016]
  46. Navarro, J., del Aguila, Y., Majid, A., 2010. Journal of Statistical Planning and Inference, 140(1), 310-322. [DOI:10.1016/j.jspi.2009.07.015]
  47. Navarro, J., del Aguila, Y., Majid, A., 2010. Journal of Statistical Planning and Inference, 140(1), 310-322. [DOI:10.1016/j.jspi.2009.07.015]
  48. Rao, M. Chen, Y. Vemuri, B. C. and Wang, F., 2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220-1228. [DOI:10.1109/TIT.2004.828057]
  49. Rao, M. Chen, Y. Vemuri, B. C. and Wang, F., 2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220-1228. [DOI:10.1109/TIT.2004.828057]
  50. Rao, M., 2005. More on a new concept of entropy and information. Journal of Theoretical Probability 18, 967-981. [DOI:10.1007/s10959-005-7541-3]
  51. Rao, M., 2005. More on a new concept of entropy and information. Journal of Theoretical Probability 18, 967-981. [DOI:10.1007/s10959-005-7541-3]
  52. Sankaran, P.G., Sunoj, S.M., 2017. Quantile-based cumulative entropies. Communications in Statistics Theory and Methods 46 (2), 805-814. [DOI:10.1080/03610926.2015.1006779]
  53. Sankaran, P.G., Sunoj, S.M., 2017. Quantile-based cumulative entropies. Communications in Statistics Theory and Methods 46 (2), 805-814. [DOI:10.1080/03610926.2015.1006779]
  54. Shannon, C. E., 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379-423. [DOI:10.1002/j.1538-7305.1948.tb01338.x]
  55. Shannon, C. E., 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379-423. [DOI:10.1002/j.1538-7305.1948.tb01338.x]
  56. Sheraz, M., Dedub, S., Predaa, V., 2015. Entropy Measures for Assessing Volatile Markets. Procedia Economics and Finance 22, 655-662. [DOI:10.1016/S2212-5671(15)00279-8]
  57. Sheraz, M., Dedub, S., Predaa, V., 2015. Entropy Measures for Assessing Volatile Markets. Procedia Economics and Finance 22, 655-662. [DOI:10.1016/S2212-5671(15)00279-8]
  58. Sreelakshmi, N., Kattumannil, S.K., Asha, G., 2018. Quantile-based tests for expontiality against dmrq and nbue alternatives. Journal of Korean Statistics Society, 47(2), 185-200. [DOI:10.1016/j.jkss.2017.12.003]
  59. Sreelakshmi, N., Kattumannil, S.K., Asha, G., 2018. Quantile-based tests for expontiality against dmrq and nbue alternatives. Journal of Korean Statistics Society, 47(2), 185-200. [DOI:10.1016/j.jkss.2017.12.003]
  60. Sunoj, S. M., Krishnan, A. S., Sankaran, P. G., 2017. Quantile-based entropy of order statistics. Journal of the Indian Society for Probability and Statistics 18, 1-17. [DOI:10.1007/s41096-016-0014-4]
  61. Sunoj, S. M., Krishnan, A. S., Sankaran, P. G., 2017. Quantile-based entropy of order statistics. Journal of the Indian Society for Probability and Statistics 18, 1-17. [DOI:10.1007/s41096-016-0014-4]
  62. Sunoj, S. M., Sankaran, P. G., 2012. Quantile-based entropy function. Statistics and Probability Letters 82, 1049-1053. [DOI:10.1016/j.spl.2012.02.005]
  63. Sunoj, S. M., Sankaran, P. G., 2012. Quantile-based entropy function. Statistics and Probability Letters 82, 1049-1053. [DOI:10.1016/j.spl.2012.02.005]
  64. Sunoj, S. M., Sankaran, P. G., Nanda, A. K., 2013. Quantile-based entropy function in past lifetime. Statistics and Probability Letters 83, 366-372. [DOI:10.1016/j.spl.2012.09.016]
  65. Sunoj, S. M., Sankaran, P. G., Nanda, A. K., 2013. Quantile-based entropy function in past lifetime. Statistics and Probability Letters 83, 366-372. [DOI:10.1016/j.spl.2012.09.016]
  66. Thapliyal, R., Taneja, H. C., 2012. Generalized entropy of order statistics. Applied Mathematics 3, 1977-1982. [DOI:10.4236/am.2012.312272]
  67. Thapliyal, R., Taneja, H. C., 2012. Generalized entropy of order statistics. Applied Mathematics 3, 1977-1982. [DOI:10.4236/am.2012.312272]
  68. Thapliyal, R., Taneja, H. C., Kumar, V., 2015. Characterization results based on non-additive entropy of order statistics. Physica A: Statistical Mechanics and its Applications, 417, 297-303. [DOI:10.1016/j.physa.2014.09.047]
  69. Thapliyal, R., Taneja, H. C., Kumar, V., 2015. Characterization results based on non-additive entropy of order statistics. Physica A: Statistical Mechanics and its Applications, 417, 297-303. [DOI:10.1016/j.physa.2014.09.047]
  70. Taneja, H.C., Hooda, D.S. and Tuteja, R.K. (1985). Coding theorems on a generalized 'useful' information. Soochow Journal of Mathematics, 11, 123-131.
  71. Taneja, H.C., Hooda, D.S. and Tuteja, R.K. (1985). Coding theorems on a generalized 'useful' information. Soochow Journal of Mathematics, 11, 123-131.
  72. Toomaj, A., Di Crescenzo, A., 2020. Connections between weighted generalized cumulative residual entropy and variance. Mathematics 8(7), 1072. [DOI:10.3390/math8071072]
  73. Toomaj, A., Di Crescenzo, A., 2020. Connections between weighted generalized cumulative residual entropy and variance. Mathematics 8(7), 1072. [DOI:10.3390/math8071072]
  74. Wang, F., Vemuri, B. C., 2007. Non-Rigid multimodal image registration using cross-cumulative residual entropy. International Journal of Computer Vision 74(2), 201-215. [DOI:10.1007/s11263-006-0011-2]
  75. Wang, F., Vemuri, B. C., 2007. Non-Rigid multimodal image registration using cross-cumulative residual entropy. International Journal of Computer Vision 74(2), 201-215. [DOI:10.1007/s11263-006-0011-2]
  76. Van Staden, P.J., Loots, M. R., 2009. L-moment estimation for the generalized lambda distribution. Third Annual ASEARC Conference, New Castle, Australia.
  77. Van Staden, P.J., Loots, M. R., 2009. L-moment estimation for the generalized lambda distribution. Third Annual ASEARC Conference, New Castle, Australia.
Journal of the Iranian Statistical Society
Volume 19, Issue 2
December 2020
Pages 67-99
  • Receive Date: 23 July 2022
  • Revise Date: 10 May 2024
  • Accept Date: 23 July 2022