Minimum chi-square estimation

Abstract: A common task in signal processing is the estimation of the parameters of a probability distribution function Perhaps the most frequently encountered estimation problem is the estimation of the mean of a signal in noise In many parameter estimation problems the situation is more complicated because direct access to the data necessary to estimate the parameters is impossible, or some of the data are missing Such difficulties arise when an outcome is a result of an accumulation of simpler outcomes, or when outcomes are clumped together, for example, in a binning or histogram operation There may also be data dropouts or clustering in such a way that the number of underlying data points is unknown (censoring and/or truncation) The EM (expectation-maximization) algorithm is ideally suited to problems of this sort, in that it produces maximum-likelihood (ML) estimates of parameters when there is a many-to-one mapping from an underlying distribution to the distribution governing the observation The EM algorithm is presented at a level suitable for signal processing practitioners who have had some exposure to estimation theory

Abstract: An algorithm for quantum-state estimation based on the maximum-likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be overestimated since they do not guarantee the positive definiteness of the reconstructed density matrix.

Abstract: 1. Introduction, Coverage, Philosophy, and Computation. 2. The Linear Model. 3. Least-Squares Estimation: Batch Processing. 4. Least-Squares Estimation: Singular-Value Decomposition. 5. Least-Squares Estimation: Recursive Processing. 6. Small Sample Properties of Estimators. 7. Large Sample Properties of Estimators. 8. Properties of Least-Squares Estimators. 9. Best Linear Unbiased Estimation. 10. Likelihood. 11. Maximum-Likelihood Estimation. 12. Multivariate Gaussian Random Variables. 13. Mean-Squared Estimation of Random Parameters. 14. Maximum A Posteriori Estimation of Random Parameters. 15. Elements of Discrete-Time Gauss-Markov Random Sequences. 16. State Estimation: Prediction. 17. State Estimation: Filtering (The Kalman Filter). 18. State Estimation: Filtering Examples. 19. State Estimation: Steady-State Kalman Filter and Its Relationships to a Digital Wiener Filter. 20. State Estimation: Smoothing. 21. State Estimation: Smoothing (General Results). 22. State Estimation for the Not-So-Basic State-Variable Model. 23. Linearization and Discretization of Nonlinear Systems. 24. Iterated Least Squares and Extended Kalman Filtering. 25. Maximum-Likelihood State and Parameter Estimation. 26. Kalman-Bucy Filtering. A. Sufficient Statistics and Statistical Estimation of Parameters. B. Introduction to Higher-Order Statistics. C. Estimation and Applications of Higher-Order Statistics. D. Introduction to State-Variable Models and Methods. Appendix A: Glossary of Major Results. Appendix B: Estimation of Algorithm M-Files. References. Index.