Asymptotic Value of the Probability That the First Order Statistic Is from Null Hypothesis


When every element of a random vector X =(X1,X2,...,Xn) assumes the cumulative distribution function F0 and F1 with probability p and 1 - p, respectively, we have shown that the probability S0 that the first order statistic of X is originally under F0 can be expressed as . We have also shown that , where  and   with  the support of (x) . Applications and implications of the results are discussed in the performance of wideband spectrum sensing schemes.

Share and Cite:

Song, I. , Lee, S. , Park, S. and Yoon, S. (2013) Asymptotic Value of the Probability That the First Order Statistic Is from Null Hypothesis. Applied Mathematics, 4, 1702-1705. doi: 10.4236/am.2013.412231.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] T. Yücek and H. Arslan, “A Survey of Spectrum Sensing Algorithms for Cognitive Radio Applications,” IEEE Communications Surveys and Tutorials, Vol. 11, No. 1, 2009, pp. 116-130.
[2] Z. Quan, S. Cui, A. H. Sayed and H. V. Poor, “Optimal Multiband Joint Detection for Spectrum Sensing in Cognitive Radio Networks,” IEEE Transactions on Signal Processing, Vol. 57, No. 3, 2009, pp. 1128-1140.
[3] A. Taherpour, M. Nasiri-Kenari and S. Gazor, “Multiple Antenna Spectrum Sensing in Cognitive Radios,” IEEE Transactions on Wireless Communications, Vol. 9, No. 2, 2010, pp. 814-823.
[4] P. Paysarvi-Hoseini and N. C. Beaulieu, “Optimal Wideband Spectrum Sensing Framework for Cognitive Radio Systems,” IEEE Transactions on Signal Processing, Vol. 59, No. 3, 2011, pp. 1170-1182.
[5] T. An, H.-K. Min, S. Lee and I. Song, “Likelihood Ratio Test for Wideband Spectrum Sensing,” Proceedings of IEEE Pacific Rim Conference on Communications, Computers and Signal Processing, Victoria, 27-29 August 2013.
[6] H. A. David and H. N. Nagaraja, “Order Statistics,” 3rd edition, John Wiley and Sons, New York, 2003.
[7] I. Song, K. S. Kim, S. R. Park and C. H. Park, “Principles of Random Processes,” Kyobo, Seoul, 2009.
[8] V. K. Rohatgi and A. K. Md. E. Saleh, “An Introduction to Probability and Statistics,” 2nd edition, John Wiley and Sons, New York, 2001.
[9] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products,” Academic, New York, 1980.
[10] J. Hajek, Z. Sidak and P. K. Sen, “Theory of Rank Tests,” 2nd edition, Academic, New York, 1999.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.