Bayesian Bounds for Parameter Estimation and Nonlinear ...

Bayesian Bounds forParameter Estimation and

Nonlinear Filtering/Tracking

IEEE Press445 Hoes Lane

Piscataway, NJ 08854

IEEE Press Editorial BoardMohamed E. El-Hawary, Editor in Chief

R. Abari T. Chen R. J. Herrick S. Basu T. G. Croda S. V. KartalopoulosA. Chatterjee S. Farshchi M. S. Newman

B. M. Hammerli

Kenneth Moore, Director of IEEE Book and Information Services (BIS)Catherine Faduska, Senior Acquisitions Editor

Jeanne Audino, Project Editor

Bayesian Bounds forParameter Estimation and

Nonlinear Filtering/Tracking

Edited by

Harry L. Van TreesKristine L. Bell

WILEY-INTERSCIENCE

A JOHN WILEY & SONS, INC., PUBLICATION

IEEE PRESS

Copyright © 2007 by the Institute of Electrical and Electronics Engineers, Inc. All rights reserved.

Published by John Wiley & Sons, Inc. Published simultaneously in Canada.

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ISBN 978-0-470-12095-8

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

To Diane

and Stephen, Mark, Kathleen, PatriciaEileen, Harry, and Julia

and the next generation-Brittany, Erin, Thomas, Elizabeth, Emily,Dillon, Bryan, Julia, Robert, Margaret,Peter, Emma, Sarah, Harry, Rebecca, Molly, Jackson, Alexander, and Luke

HLV

To Jamie

and Julie and Lisa

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Contents

Preface xiii

Introduction 1Harry L. Van Trees and Kristine L. Bell

1 Bayesian Estimation: Static Parameters 21.1 Maximum Likelihood and Maximum a Posteriori Estimation 2

1.1.1 Nonrandom Parameters 21.1.2 Random Parameters 41.1.3 Hybrid Parameters 121.1.4 Examples 13

1.2 Covariance Inequality Bounds 331.2.1 Covariance Inequality 331.2.2 Bayesian Bounds 341.2.3 Scalar Parameters 35

1.2.3.1 Bayesian Cramér-Rao Bound 351.2.3.2 Weighted Bayesian Cramér-Rao Bound 351.2.3.3 Bayesian Bhattacharyya Bound 371.2.3.4 Bobrovsky-Zakai Bound 381.2.3.5 Weiss-Weinstein Bound 40

1.2.4 Vector Parameters 421.2.4.1 Bayesian Cramér-Rao Bound 441.2.4.2 Weighted Bayesian CRB 441.2.4.3 Bayesian Bhattacharyya Bound 441.2.4.4 Bobrovsky-Zakai Bound 441.2.4.5 Weiss-Weinstein Bound 45

1.2.5 Combined Bayesian Bounds 461.2.6 Nuisance Parameters 47

1.2.6.1 Nonrandom Unwanted Parameters 471.2.6.2 Random Parameters 48

1.2.7 Hybrid Parameters 501.2.8 Functions of the Parameter Vector 50

1.2.8.1 Scalar Parameters 501.2.8.2 Vector Parameters 52

1.2.9 Summary: Covariance Inequality Bounds 521.3 Ziv–Zakai Bounds 53

1.3.1 Scalar Parameters 531.3.2 Equally Likely Hypotheses 551.3.3 Vector Parameters 56

1.4 Method of Interval Estimation 581.5 Summary 62

2 Bayesian Estimation: Random Processes 622.1 Continuous-Time Processes and Continuous-Time Observations 62

2.1.1 Nonlinear Models 622.1.1.1 Linear AWGN Process and Observations 642.1.1.2 Linear AWGN Process, Nonlinear AWGN Observations 652.1.1.3 Nonlinear AWGN Process and Observations 672.1.1.4 Nonlinear Process and Observations 68

2.1.2 Bayesian Cramér-Rao Bounds: Continuous-Time 682.2 Continuous-Time Processes and Discrete-Time Observations 70

vii

2.2.1 Extended Kalman Filter 702.2.2 Bayesian Cramér-Rao Bound 712.2.3 Discretizing the Continuous-Time State Equation 71

2.3 Discrete-Time Processes and Discrete-Time Observations 722.3.1 Linear AWGN Process and Observations 732.3.2 General Nonlinear Model 74

2.3.2.1 MMSE and MAP Estimation 742.3.2.2 Extended Kalman Filter 76

2.3.3 Recursive Bayesian Cramér–Rao Bounds 772.4 Global Recursive Bayesian Bounds 852.5 Summary 85

3 Outline of the Book 86

Part I Bayesian Cramér–Rao Bounds 871.1 H. L. Van Trees, Excerpts from Part I of Detection, Estimation, and Modulation Theory, pp. 66–86, 89

Wiley, New York, 1968 (reprinted Wiley 2001).1.2 M. P. Shutzenberger,” A generalization of the Fréchet-Cramér inequality in the case of Bayes estimation,” 110

Bulletin of the American Mathematical Society, vol. 63, no. 142, 1957.

Part II Global Bayesian Bounds 1112.1 B. Z. Bobrovsky, E. Mayer-Wolf, and M. Zakai, “Some classes of global Cramér–Rao bounds,” 113

Ann. Stat., vol. 15, pp. 1421–1438, 1987.2.2 H. L. Van Trees, Excerpts from Part I of Detection, Estimation, and Modulation Theory, pp. 273–286, 131

Wiley, New York, 1968 (reprinted 2001).2.3 D. Rife and R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” 144

IEEE Trans. Inform. Theory, vol. IT-20, no. 5, pp. 591–598, September 1974.2.4 R. J. McAulay and E. M. Hostetter, “Barankin bounds on parameter estimation,” IEEE Trans. Info. 152

Theory, vol. IT-17, no. 6, pp. 669–676, November 1971.2.5 R. Miller and C. Chang, “A modified Cramér–Rao bound and its applications, IEEE Trans. Info. 160

Theory, vol. 24, no. 3, pp. 398–400, May 1978.2.6 A. Weiss and E. Weinstein, “A lower bound on the mean-square error in random parameter estimation,” 163

IEEE. Trans. Info. Theory, vol. 31, no. 5, pp. 680–682, September 1985.2.7 E. Weinstein and A. J. Weiss, “Lower bounds on the mean square estimation error,” Proceedings of 166

the IEEE, vol. 73, no. 9, pp. 1433–1434, September 1985.2.8 E. Weinstein and A. J. Weiss, “A general class of lower bounds in parameter estimation,” IEEE Trans. 167

Info. Theory, vol. 34, no. 2, pp. 338–342, March 1988.2.9 J. S. Abel, “A bound on mean-square-estimate error,” IEEE. Trans. Info. Theory, vol. 39, no. 5, pp. 171

1675–1680, September 1993.2.10 A. Renaux, P. Forster, P. Larzabal, and C. Richmond, “The Bayesian Abel bound on the mean square 176

error,” ICASSP 2006, vol. 3, pp. III-9–12, Toulouse, France.2.11 J. Ziv and M. Zakai, “Some lower bounds on signal parameter estimation,” IEEE. Trans. Info. Theory, 180

vol. IT-15, no. 3, pp. 386–391, May 1969.2.12 L. P. Seidman, “Performance limitations and error calculations for parameter estimation,” Proc. IEEE, 186

vol. 58, no. 5, pp. 644–652, May 1970.2.13 D. Chazan, M. Zakai, and J. Ziv, “Improved lower bounds on signal parameter estimation,” IEEE Trans. 195

Info. Theory, vol. IT-21, no. 1, pp. 90–93, Jan. 1975.2.14 S. Bellini and G. Tartara, “Bounds on error in signal parameter estimation,” IEEE. Trans. Commun., 199

vol. COM-22, pp. 340–342, March 1974.2.14a S. Bellini and G. Tartara, “Corrections to ‘Bounds on error in signal parameter estimation,’ ” IEEE Trans. 201

Commun., vol. 23, no. 4, p. 486, April 1975.2.15 M. Wax and J. Ziv, “Improved bounds on the local mean-square error and the bias of parameter estimators,” 202

IEEE. Trans. Info. Theory, vol. 23, no. 4, pp. 529–530, July 1977.2.16 E. Weinstein, “Relations between Belini–Tartara, Chazan–Zakai–Ziv, and Wax–Ziv lower bounds,” 204

IEEE. Trans. Info. Theory, vol. 34, no. 2, pp. 342–343, March, 1988.2.17 K. L. Bell, Y. Steinberg, Y. Ephraim, and H. L. Van Trees, “Extended Ziv–Zakai lower bound for vector 206

parameter estimation,” IEEE. Trans. Info. Theory, vol. 43, no. 2, pp. 624–637, March 1997.

viii Contents

2.18 K. L. Bell, Y. Ephraim, and H. L. Van Trees, “Explicit Ziv–Zakai lower bound for bearing 220estimation,” IEEE. Trans. Signal Process., vol. 44, no. 11, pp. 2810–2814, November 1996.

2.19 S. Basu and Y. Bresler, “A global lower bound on parameter estimation error with periodic distortion 235functions,” IEEE. Trans. Info. Theory, vol. 46, no. 3, pp. 1145–1150, May 2000.

2.20 H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part III, Chapter 10, pp. 275–308, Wiley, 241New York, 1971 (reprinted Wiley 2001).

2.21 F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” 274IEEE Trans. on Signal Process., vol. 53, no. 4, pp. 1359–1373, April 2005.

2.22 C. D. Richmond, “Capon algorithm mean-squared error threshold SNR prediction and probability 289of resolution,” IEEE Trans. on Signal Process., vol. 53, no. 8, pp. 2748–2764, August 2005.

2.23 C. Richmond, “Mean-squared error and threshold SNR prediction of maximum-likelihood signal 306parameter estimation with estimated colored noise covariances,” IEEE. Trans. Info. Theory, vol. 52,no. 5, pp. 2146–2164, May 2006.

2.24 L. Najjar-Atallah, P. Larzabal, and P. Forster, “Threshold region determination of ML estimation in known 325phase data-aided frequency synchronization,” IEEE Signal Process. Letters, vol. 12, no. 9, pp. 605–608, September 2005.

2.25 L. D. Brown and R. C. Liu, “Bounds on the Bayes and minimax risk for signal parameter estimation,” 329IEEE. Trans. Info. Theory, vol. 39, no. 4, pp. 1386–1394, July 1993.

2.26 J. K. Thomas, L. L. Scharf, and D. W. Tufts, “The probability of a subspace swap in the SVD,” IEEE 338Trans. Signal Process., vol. 43, no. 3, pp. 730–736, March 1995.

Part III Hybrid Bayesian Bounds 3453.1 Y. Rockah and P. M. Schultheiss, “Array shape calibration using sources in unknown locations—Part I: 347

far-field sources,” IEEE Trans. Acoust., Speech Signal Process., vol. 35, no. 3, pp. 286–299, March 1987.

3.2 I. Reuven and H. Messer, “A Barankin-type lower bound on the estimation error of a hybrid parameter 361vector,” IEEE. Trans. Info. Theory, vol. 43, no. 3, pp. 1084–1093, May 1997.

3.3 J. Tabrikian and J. Krolik, “Efficient computation of the Bayesian Cramér–Rao bound on estimating 371parameters of Markov models,” IEEE Conf. Acoustics, Speech, and Sig. Process., ICASSP’99,pp. 1761–1764, 1999.

3.4 S. Buzzi, M. Lops, and S. Sardellitti, “Further results on Cramér–Rao bounds for parameter estimation 375in long-code DS/CDMA systems,” IEEE Trans. Sig. Process., vol. 53, no. 3, pp. 1216–1221, March 2005.

3.5 P. Tichavsky and K. Wong, “Quasi-fluid-mechanics-based quasi-Bayesian Cramér–Rao bounds for 381deformed towed-array direction finding,” IEEE Trans. Signal Process., vol. 52, no. 1, pp. 36–47,Jan. 2004.

Part IV Constrained Cramér–Rao Bounds 3934.1 J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE. Trans. 395

Info. Theory, vol. 36, no. 6, pp. 1285–1301, November 1990.4.2 T. L. Marzetta, “A simple derivation of the constrained multiple parameter Cramér–Rao bound,” 412

IEEE. Trans. Signal Process., vol. 41, no. 6, pp. 2247–2249, June 1993.4.3 P. Stoica and B. C. Ng, “On the Cramér–Rao bound under parametric constraints,” IEEE Signal Process 415

Letters, vol. 5, no. 7, pp. 177–179, July 1998.4.4 T. J. Moore, B. M. Sadler, and R. J. Kozick, “Regularity and strict identifiability in MIMO systems,” 418

IEEE. Trans. Signal Process., vol. 50, no. 8, pp. 1831–1842, August 2002.4.5 S. T. Smith, “Covariance, subspace, and intrinsic Cramér–Rao bounds,” IEEE. Trans. Signal Process., 430

vol. 53, no. 5, pp. 1610–1630, May 2005.4.6 A. O. Hero, J. A. Fessler, and M. Usman, “Exploring estimator bias-variance tradeoffs using the uniform 451

CR bound,” IEEE. Trans. Signal Process., vol. 44, no. 8, pp. 2026–2041, August 1996.4.7 A. Nehorai and M. Hawkes, “Performance bounds for estimating vector systems,” IEEE. Trans. Signal 467

Process., vol. 48, no. 6, pp. 1737–1749, June 2000.4.8 L. T. McWhorter and L. L. Scharf, “Properties of quadratic covariance bounds,” Proc. 27th Annual 480

Asilomar Conf. on Signals, Systems, and Computers, Asilomar, CA, vol. 2, pp. 1176–1180,November 1993.

Contents ix

Part V Applications: Static Parameters 4855.1 A. Weiss and E. Weinstein, “Fundamental limitations in passive time delay estimation—Part I: 487

Narrow-band systems,” IEEE Trans. Acoustics, Speech, Sig. Process., vol. ASSP-31, no. 2,pp. 472–486, April 1983.

5.2 A. Bartow and H. Messer, “Lower bound on the achievable DSP performance for localizing step-like 502continuous signals in noise,” IEEE. Trans. Signal Process., vol. 46, no. 8, pp. 2195–2201, August1998.

5.3 B. Sadler and R. Kozick, “A survey of time delay estimation performance bounds,” Fourth IEEE 509Workshop on Sensor Array and Multichannel Process., pp. 282–288, 12–14 July 2006.

5.4 W. Xu, A. Baggeroer, and C. Richmond, “Bayesian bounds for matched-field parameter estimation,” 516IEEE. Trans. Signal Process., vol. 52, no. 12, pp. 3293–3305, December, 2004.

5.5 J. Tabrikian and J. Krolik, “Barankin bounds for source localization in an uncertain ocean environment,” 529IEEE. Trans. Signal Process., vol. 47, no. 11, pp. 2917–2927, November 1999.

5.6 Ü. Oktel and R. Moses, “A Bayesian approach to array geometry design,” IEEE. Trans. Signal Process., 540vol. 53, no. 5, pp. 1919–1923, May 2005.

5.7 F. Athley, “Optimization of element positions for direction finding with sparse arrays,” 11th IEEE SPW 545on Stat. Proc., pp. 516–519, August 2001.

5.8 F. Athley and C. Engdahl, “Direction-of-arrival estimation using separated subarrays,” 34th IEEE 549Asilomar Conf. on SSC, vol. 1, pp. 585–589, October 2001.

5.9 H. Nguyen and and H. L. Van Trees, “Comparison of performance bounds for DOA estimation,” 554IEEE Seventh SP Workshop on Statistical Signal and Array Processing, pp. 313–316, June 1994.

5.10 Y. Qi and H. Kobayashi, “On geolocation accuracy with prior information in non-line-of-sight 558environment,” IEEE 56th Vehicular Tech. Conf. Proc., vol. 1, pp. 285–288, Sept. 2002.

5.11 S. C. White and N. C. Beaulieu, “On the application of the Cramér–Rao and detection theory bounds 562to mean square error of symbol timing recovery,” IEEE Trans. Comm., vol. 40, no. 10, pp. 1635–1643, October 1992.

5.12 A. Pinkus and J. Tabrikian, “Barankin bound for range and Doppler estimation using orthogonal signal 571transmission,” IEEE Conf. on Radar, pp. 94–99, April 2006.

5.13 J. Tabrikian, “Barankin bounds for target localization by MIMO radars,” Fourth IEEE Workshop on 577Sensor Array and Multichannel Process., pp. 278–281, July 2006.

5.14 A. Renaux, “Weiss–Weinstein bound for data-aided carrier estimation,” IEEE Sig. Proc. Letters, 581vol. 14, no. 4, pp. 283–286, April 2007.

Part VI Nonlinear Stochastic Dynamic Systems 5856.1 H. J. Kushner, “On the differential equations satisfied by conditional probability densities of Markov 587

processes, with applications,” J. SIAM on Control, vol. 2, pp. 106–119, 1964.6.2 Y. C. Ho and R. C. K. Lee, “A Bayesian approach to problems in stochastic estimation and control,” 601

IEEE Trans. Auto. Control, vol. 9, no. 4, pp. 333–339, October 1964.6.3 H. Cox, “On the estimation of state variables and parameters for noisy dynamic systems,” IEEE Trans. 608

Auto. Control, vol. 9, no. 1, pp. 5–12, January 1964.6.4 H. L. Van Trees, “Bounds on the accuracy attainable in the estimation of continuous random processes,” 616

IEEE Trans. Info. Theory, vol. 12, no. 3, pp. 298–305, July 1966.6.5 D. L. Snyder and I. B. Rhodes, “Filtering and control performance bounds with implication on asymptotic 624

Separation,” Automatica, vol. 8, pp. 747–753, Nov. 1972.6.6 B. Z. Bobrovsky, and M. Zakai, “A lower bound on the estimation error for Markov processes,” IEEE Trans. 631

Auto. Control, vol. 20, no. 6, 785–788, December 1975.6.7 B. Z. Bobrovsky and M. Zakai, “A lower bound on the estimation error for certain diffusion processes,” 635

IEEE Trans. Info. Theory, vol. IT-22, no. 1, pp. 45–52, January 1976.6.8 B. Bobrovsky, M. Zakai, and O. Zeitouni, “Error ounds for the nonlinear filtering of signals with small 643

diffusion coefficients,” IEEE Trans. Info. Theory, vol. 34, no. 4, pp. 710–721, July 1988.6.9 J. H. Taylor, “The Cramér–Rao estimation error lower bound computation for deterministic nonlinear 655

systems,” IEEE Trans. Auto. Control, vol. 24, pp. 343–344, April 1979.6.10 C. B. Chang, “Two lower bounds on the covariance for nonlinear estimation problems,” IEEE Trans. Auto. 657

Control, vol. AC-26, no. 6, pp. 1294–1297, Dec. 1981.6.11 H. L. Van Trees, Excerpts from Part II of Detection, Estimation, and Modulation Theory, pp. 134–153, 660

Wiley, New York, 1971 (reprinted Wiley 2003).

x Contents

6.12 M. Zakai and J. Ziv, “Lower and upper bounds on the optimal filtering error of certain diffusion 679processes,” IEEE Trans. Info. Theory, vol. IT-18, no. 3, pp. 325–331, May 1972.

6.13 P. Tichavský, C. Muravchik, and A. Nehorai, “Posterior Cramér–Rao bounds for discrete-time nonlinear 686filtering,” IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1386–1396, May 1998.

6.14 M. Šimandl, J. Královec, and P. Tichavský, “Filtering, predictive and smoothing Cramér–Rao bounds for 697discrete-time nonlinear dynamic systems,” Automatica, vol. 37, pp. 1703–1716, 2001.

6.15 I. Rapoport and Y. Oshman, “Recursive Weiss–Weinstein lower bounds for discrete-time nonlinear 711filtering,” 43rd IEEE Conf. on Decision and Control, vol. 3, pp. 2662–2667, Dec. 2004.

6.16 S. Reece and D. Nicholson, “Tighter alternatives to the Cramér–Rao lower bound for discrete-time 717filtering,” 7th Intl. Conf. on Info. Fusion, vol. 1, pp. 101–106, 25–28 July 2005.

6.17 M. S. Arulampalam, S. Maskell, N. Gordon and T. Clapp, “A tutorial on particle filters for online 723nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Sig. Process, vol. 50, no. 2, pp. 174–188, February 2002.

Part VII Applications: Nonlinear Dynamic Systems 7397.1 V. Aidala and S. Hammel, “Utilization of modified polar coordinates for bearings-only tracking,” IEEE 741

Trans. Auto. Control, vol. AC-28, no. 3, pp. 283–294, March 1983.7.2 M. Hernandez, B. Ristic, and A. Farina, “A performance bound for manoeuvering target tracking using 753

best-fitting Gaussian distribution,” Proc. 7th Int. Conf. Information Fusion, FUSION 2005,” Philadelphia, PA, pp. 1–8, July 2005.

7.3 J. M. Passerieux and D. Van Cappel, “Optimal observer maneuver for bearings-only tracking,” IEEE 761Trans. Aero. and Elect. Syst., vol. 34, no. 3, pp. 777–788, July 1998.

7.4 K. L. Bell and H. L. Van Trees, “Posterior Cramér–Rao bound for tracking target bearing,” 13th Annual 773Workshop on Adaptive Sensor Array Process. (ASAP 2005), MIT Lincoln Lab, Lexington, MA, June 2005.

7.5 R. Niu, P. Willett and Y. Bar-Shalom, “Matrix CRLB scaling due to measurements of uncertain origin,” 779IEEE Trans. Signal Process., vol. 49, no. 7, pp. 1325–1335, July 2001.

7.6 M. Hernandez, A. Farina, and B. Ristic, “PCRLB for tracking in cluttered environments: Measurement 790sequence conditioning approach,” IEEE Trans. Aero. Elect. Syst., vol. 42, no. 2, pp. 680–704, April 2006.

7.7 X. Zhang, P. Willett, and Y. Bar-Shalom, “Dynamic Cramér–Rao bound for target tracking in clutter,” 814IEEE Trans. Aero. Elect. Syst., vol. 41, no. 4, pp. 1154–1167, October 2005.

7.8 B. Ristic, A. Farina, and M. Hernandez, “Cramér–Rao lower bound for tracking multiple targets,” IEE 828Proc. on Radar, Sonar and Navig., vol. 151, no. 3, pp. 129–134, June 2004.

7.9 C. Hue, J-P. Le Cadre, and P. Pérez, “Posterior Cramér–Rao bounds for multi-target tracking,” IEEE Trans. 834Aero. Elect. Syst., vol. 42, no. 1, pp. 37–49, January 2006.

7.10 F. E. Daum, “Bounds on performance for multiple target tracking,” IEEE Trans. Auto. Control, vol. 35, 847no. 4, pp. 443–446, April 1990.

7.11 N. Bergman, L. Ljung, and F. Gustafsson, “Point-mass filter and Cramér–Rao bound for terrain-aided 850navigation,” IEEE Proc. of the 36th IEEE Conf. on Decision & Control, vol. 1, pp. 565–570, December 1997.

7.12 H. L. Van Trees, K. L. Bell, and Y. Wang, “Bayesian Cramér–Rao bounds for multistatic radar,” IEEE 856International Waveform Diversity & Design Conference, January 2006.

7.13 I. Rapoport and Y. Oshman, “Weiss–Weinstein lower bounds for Markovian systems. Part 2: Applications 860to fault-tolerant filtering,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 2031–2042, May 2007.

7.14 K. L. Bell and H. L. Van Trees, “Combined Cramér–Rao/Weiss–Weinstein bound for tracking target bearing,” 8724th Annual IEEE Workshop on Sensor Array and Multi-Channel Processing (SAM 2006), Waltham, MA, pp. 273–277, July 2006.

Part VIII Statistical Literature 8778.1 R. D. Gill and B. Y. Levit, “Applications of the Van Trees inequality: A Bayesian Cramér–Rao bound,” 879

Bernoulli 1 (1/2), pp. 59–79, 1995.8.2 B. L. S. Prakasa Rao, “On Cramér–Rao type integral inequalities.” Calcutta Stat. Assoc. Bulletin, 900

(H. K. Nandi Mem. Spec. Vol.), vol. 40, nos. 157–60, pp. 183–205, 1991.

Contents xi

8.3 J. J. Gart, “An extension of the Cramér–Rao inequality.” Annals Math. Stat., vol. 30, no. 2, pp. 367–380, 9231959.

8.4 M. Ghosh, “Cramér–Rao bounds for posterior variances.” Stat. Prob. Letters, vol. 17, pp. 173–178, 9371993.

References 943

Author Index 951

xii Contents

xiii

xiv Preface

Preface xv

xvi Preface