Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

On Complete Convergence for Weighted Sums of Pairwise Negatively Quadrant Dependent Sequences

  • Ko, Mi-Hwa (Division of Mathematics and Informational Statistics, WonKwang University)
  • Received : 2011.12.12
  • Accepted : 2012.01.31
  • Published : 2012.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we prove the complete convergence for weighted sums of pairwise negatively quadrant dependent random variables. Some results on identically distributed and negatively associated setting of Liang and Su (1999) are generalized and extended to the pairwise negative quadrant dependence case.

Keywords

References

  1. Bai, Z. and Su, C. (1985). The complete convergence for partial sums of i.i.d. random variables, Science China Mathematics, A28, 1261-1277
  2. Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing Probability Models, Holt, Rinehart and Winston, New York.
  3. Cheng, P. E. (1995). A note on strong convergence rates in nonparametric regression, Statistics and Probability Letters, 24, 357-364. https://doi.org/10.1016/0167-7152(94)00195-E
  4. Gut, A. (1992). Complete convergence for arrays, Periodica Mathematica Hungarica, 25, 51-75. https://doi.org/10.1007/BF02454383
  5. Gut, A. (1993). Complete convergence and Cesaro summation for i.i.d. random variables, Probability Theory and Related Fields, 97, 169-178. https://doi.org/10.1007/BF01199318
  6. Hsu, P. L. and Robbins, H. (1947). Complete convergence and the law of large numbers, Proceedings of the National Academy of Sciences of the United States of America, 33, 25-31.
  7. Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications, Annals of Statistics, 11, 286-295. https://doi.org/10.1214/aos/1176346079
  8. Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions, Wiley, New York.
  9. Kafles, D. and Bhaskara Rao, M. (1982). Weak consistency of least squares estimators in linear models, Journal of Multivariate Analysis, 12, 186-198. https://doi.org/10.1016/0047-259X(82)90014-8
  10. Kuczmaszewska, A. (2009). On complete convergence for arrays of rowwise negatively associated random variables, Statistics and Probability Letters, 79, 116-124. https://doi.org/10.1016/j.spl.2008.07.030
  11. Lehmann, E. L. (1966). Some concepts of dependence, Annals of Mathematical Statistics, 37, 1137-1153. https://doi.org/10.1214/aoms/1177699260
  12. Li, D. L., Rao, M. B., Jiang, T. F. and Wang, X. C. (1995). Complete convergence and almost sure convergence of weighted sums of random variables, Journal of Theoretical Probability, 8, 49-76. https://doi.org/10.1007/BF02213454
  13. Liang, H. Y. (2000). Complete convergence for weighted sums of negatively associated random variables, Statistics and Probability Letters, 48, 317-325. https://doi.org/10.1016/S0167-7152(00)00002-X
  14. Liang, H. Y. and Su, C. (1999). Complete convergence for weighted sums of NA sequences, Statistics and Probability Letters, 45, 85-95. https://doi.org/10.1016/S0167-7152(99)00046-2
  15. Rao, C. R. and Zhao, M. T. (1992). Linear representation of M-estimates in linear models, Canadian Journal of Statistics, 20, 359-368. https://doi.org/10.2307/3315607
  16. Tolley, H. D. and Norman, J. E. (1979). Time on trial estimates with bivariate risk, Biometrika, 66, 285-291. https://doi.org/10.1093/biomet/66.2.285
  17. Wu, Q. (2006). Probability Limit Theory for Mixing Sequences, Science Press, Beijing, China, 170-176, 206-211.