Distribution-Dependent Distance of First Two Moments

Closeness measures between distributions, between vectors, and between matrices abound. Many practical problems, however, call for measures of closeness between the first two moments of two unspecified distributions given only a sample of one of them or of a third distribution without other information. We present several metrics for such problems that demand special, distribution-dependent solutions, and show their good qualities. We demonstrate their rich applications in various areas, such as estimation performance analysis, efficiency of distributed fusion, metrized Kullback-Leibler divergence, decision performance evaluation, credibility of estimators, filter initialization, and empirical distribution function problems.

[1]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[2]  A.R. Runnalls,et al.  Kullback-Leibler Approach to Gaussian Mixture Reduction , 2007, IEEE Transactions on Aerospace and Electronic Systems.

[3]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[4]  X. Rong Li,et al.  Iterative Mid-Range with Application to Estimation Performance Evaluation , 2015, IEEE Signal Processing Letters.

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[7]  X. Rong Li,et al.  Nonlinear Estimation by LMMSE-Based Estimation With Optimized Uncorrelated Augmentation , 2015, IEEE Transactions on Signal Processing.

[8]  Vesselin P. Jilkov,et al.  A survey of maneuvering target tracking: approximation techniques for nonlinear filtering , 2004, SPIE Defense + Commercial Sensing.

[9]  X. R. Li,et al.  Optimal Linear Estimation Fusion — Part IV : Optimality and Efficiency of Distributed Fusion , 2001 .

[10]  Elena Deza,et al.  Encyclopedia of Distances , 2014 .

[11]  S. Luo,et al.  Informational distance on quantum-state space , 2004 .

[12]  Zhansheng Duan,et al.  Comprehensive evaluation of decision performance , 2008, 2008 11th International Conference on Information Fusion.

[13]  Georg Lindgren,et al.  Recursive estimation in mixture models with Markov regime , 1991, IEEE Trans. Inf. Theory.

[14]  X. Rong Li,et al.  Measuring Estimator's Credibility: Noncredibility Index , 2006, 2006 9th International Conference on Information Fusion.