Old Bands, New Tracks—Revisiting the Band Model for Robust Hypothesis Testing

The density band model proposed by Kassam [1] for robust hypothesis testing is revisited. First, a novel criterion for the general characterization of least favorable distributions is proposed that unifies existing results. This criterion is then used to derive an implicit definition of the least favorable distributions under band uncertainties. In contrast to the existing solution, it only requires two scalar values to be determined and eliminates the need for case-by-case statements. Based on this definition, a generic fixed-point algorithm is proposed that iteratively calculates the least favorable distributions for arbitrary band specifications. Finally, three different types of robust tests that emerge from band models are discussed and a numerical example is presented to illustrate their potential use in practice.

[1]  Joseph Mitola,et al.  Cognitive radio: making software radios more personal , 1999, IEEE Wirel. Commun..

[2]  V. Yohai,et al.  Robust Statistics: Theory and Methods , 2006 .

[3]  Neri Merhav,et al.  When is the generalized likelihood ratio test optimal? , 1992, IEEE Trans. Inf. Theory.

[4]  R. Cooke.  Real and Complex Analysis , 2011 .

[5]  Bernard C. Levy,et al.  Principles of Signal Detection and Parameter Estimation , 2008 .

[6]  P. J. Huber,et al.  Minimax Tests and the Neyman-Pearson Lemma for Capacities , 1973 .

[7]  Abdelhak M. Zoubir,et al.  Robust hypothesis testing for modeling errors , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[8]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[9]  H. Vincent Poor,et al.  Minimax robust decentralized detection , 1994, IEEE Trans. Inf. Theory.

[10]  Shuguang Cui,et al.  Optimal Linear Cooperation for Spectrum Sensing in Cognitive Radio Networks , 2008, IEEE Journal of Selected Topics in Signal Processing.

[11]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[12]  Stephen P. Boyd,et al.  A minimax theorem with applications to machine learning, signal processing, and finance , 2007, 2007 46th IEEE Conference on Decision and Control.

[13]  Hüseyin Arslan,et al.  A survey of spectrum sensing algorithms for cognitive radio applications , 2009, IEEE Communications Surveys & Tutorials.

[14]  H. Vincent Poor,et al.  Robust matched filters , 1983, IEEE Trans. Inf. Theory.

[15]  Frederick R. Forst,et al.  On robust estimation of the location parameter , 1980 .

[16]  F. Österreicher,et al.  On the construction of least favourable pairs of distributions , 1978 .

[17]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[18]  Saleem A. Kassam,et al.  Robust hypothesis testing for bounded classes of probability densities , 1981, IEEE Trans. Inf. Theory.

[19]  P. Pardalos,et al.  Minimax and applications , 1995 .

[20]  P. J. Huber.  A Robust Version of the Probability Ratio Test , 1965 .

[21]  Zhai Xuping,et al.  Energy-detection based spectrum sensing for cognitive radio , 2007 .

[22]  K. B. Letaief,et al.  Optimization of cooperative spectrum sensing with energy detection in cognitive radio networks , 2009, IEEE Transactions on Wireless Communications.

[23]  Ali H. Sayed,et al.  An optimal error nonlinearity for robust adaptation against impulsive noise , 2013, 2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC).

[24]  Ferdinand Österreicher,et al.  Statistical information and discrimination , 1993, IEEE Trans. Inf. Theory.

[25]  Bernard C. Levy,et al.  Robust Hypothesis Testing With a Relative Entropy Tolerance , 2007, IEEE Transactions on Information Theory.

[26]  S.A. Kassam,et al.  Robust techniques for signal processing: A survey , 1985, Proceedings of the IEEE.

[27]  Gutti Jogesh Babu.  Bootstrap Techniques for Signal Processing , 2005, Technometrics.

[28]  Lei Chen,et al.  Efficient energy detector for spectrum sensing in complex Gaussian noise , 2010, WCSP.

[29]  Zheng Zhou,et al.  Energy Detection Based Spectrum Sensing for Cognitive Radios in Noise of Uncertain Power , 2008, 2008 International Symposium on Communications and Information Technologies.

[30]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[31]  Adityanand Guntuboyina,et al.  Sharp Inequalities for $f$ -Divergences , 2014, IEEE Trans. Inf. Theory.

[32]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .

[33]  Song-xi Chen,et al.  Probability Density Function Estimation Using Gamma Kernels , 2000 .

[34]  H. Komiya.  Elementary proof for Sion's minimax theorem , 1988 .

[35]  M. Sion.  On general minimax theorems , 1958 .