Order Statistics: Applications - GBV

[Pages:9]Order Statistics: Applications

Edited by

N. Balakrishnan

Department of Mathematics and Statistics McMaster University

Hamilton, Ontario, Canada

C. R. Rao

Center for Multivariate Analysis Department of Statistics, The Pennsylvania State University

University Park, PA, USA

1998 ELSEVIER AMSTERDAM ? LAUSANNE ? NEW YORK ? OXFORD ? SHANNON ? SINGAPORE TOKYO

Table of contents

Preface v

Contributors xvii

PART I. RESULTS FOR SPECIFIC DISTRIBUTIONS

Ch. 1. Order Statistics in Exponential Distribution 3 Asit P. Basu and Bahadur Singh

1. Introduction 3 2. Order statistics and its properties 3 3. Censored data 5 4. Inference concerning several exponential populations 11 5. Order restricted inference 14 6. Bayesian inference 20

Acknowledgement 22 References 22

Ch. 2. Higher Order Moments of Order Statistics from Exponential and Right-truncated Exponential Distributions and Applications to Life-testing Problems 25 N. Balakrishnan and Shanti S. Gupta

1. Introduction 25 2. Relations for Single moments 26 3. Relations for double moments 28 4. Relations for triple moments 31 5. Relations for quadruple moments 36 6. Applications to inference for the one-parameter exponential distribution 43 7. Generalized results for the right-truncated exponential distribution 45 8. Illustrative examples 55

Acknowledgements 58 References 58

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Ch. 3. Log-gamma Order Statistics and Linear Estimation of Parameters 61

N. Balakrishnan and P. S. Chan

1. Introduction 61 2. Single moments of order statistics 63 3. Product moments of order statistics 72 4. Best linear unbiased estimators 76 5. Illustrative example 79

Acknowledgements 82 References 82

Ch. 4. Recurrence Relations for Single and Product Moments of Order Statistics from a Generalized Logistic Distribution with Applications to Inference and Generalizations to Double Truncation 85

N. Balakrishnan and Rita Aggarwala

I. Generalized logistic distribution 85 1. Introduction 85 2. Recurrence relations for Single moments 88 3. Recurrence relations for product moments 91 4. Recursive computational algorithm 103 5. Best linear unbiased estimators 104 6. Maximum likelihood estimation 105 7. Numerical example 108 II. Doubly truncated generalized logistic distribution 116 8. Introduction 116 9. Recurrence relations for Single moments 117 10. Recurrence relations for product moments 119 11. Recursive algorithm 125

Acknowledgements 125 References 125

Ch. 5. Order Statistics from the Type III Generalized Logistic Distribution and Applications 127

N. Balakrishnan and S. K. Lee

1. Introduction 127 2. Type III generalized logistic distribution 128 3. Order statistics and moments 132 4. BLUEs of location and scale parameters 143 5. MLEs of location and scale parameters 147 6. Comparison of the BLUEs with the MLEs 149 7. Illustrative examples 150

References 154

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PART II. LINEAR ESTIMATION

Ch. 6. Estimation of Scale Parameter Based on a Fixed Set of Order Statistics 159 Sanat K. Sarkar and Wenjin Wang

1. Introduction 159 2. Linear estimators 161 3. The positivity of the best unbiased L-estimator 165 4. Nonlinear estimators 166 5. Extension of the positivity results to censored scale regression model 177 6. Concluding remarks 179

References 180

Ch. 7. Optimal Linear Inference Using Selected Order Statistics in Location-Scale Models 183 M. Masoom Ali and Dale Umbach

1. Introduction 183 2. Preliminaries 184 3. Optimality criteria for estimation 189 4. Specific distributions 193 5. Tests of significance 203 6. Testing goodness-of-fit 205

References 207

Ch. 8. L-Estimation 215 J. R. M. Hosking

1. Introduction 215 2. Introductory examples 216 3. Single-sample problems 220 4. More complicated problems 230

References 233

Ch. 9. On Some L-estimation in Linear Regression Models 237 Soroush Alimoradi and A. K. Md. Ehsanes Saleh

1. Introduction 237 2. Regression quantiles and their properties 238 3. L-estimation of the parameters of a linear model based

on a few selected regression quantiles with known error distributions 241

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Trimmed least-squared estimation of regression parameters and its asymptotic distribution 251 Trimmed estimation of regression parameters under uncertain prior Information 271 Acknowledgement 278 References 279

PART III. INFERENTIAL METHODS

Ch. 10. The Role of Order Statistics in Estimating Threshold Parameters

A. Clifford Cohen

1. Introduction 283 2. The exponential distribution 284 3. The Weibull distribution 286 4. The lognormal distribution 288 5. The gamma distribution 292 6. The Inverse Gaussian distribution 297 7. Errors of estimates 300 8. Illustrative examples 308

Acknowledgement 312 References 312

Ch. 11. Parameter Estimation under Multiply Type-II Censoring 315

Fanhui Kong

1. Introduction 315 2. Best linear estimation 315 3. Maximum likelihood estimation 319 4. Approximate maximum likelihood estimation 323 5. Interval estimation for exponential distribution 326

References 333

Ch. 12. On Some Aspects of Ranked Set Sampling in Parametric Estimation 337

Nora Ni Chuiv and Bimal K. Sinha

1. Introduction 337 2. Estimation of a normal mean and a normal variance 341 3. Estimation of an exponential mean 347 4. Estimation of parameters in a two parameter exponential distribution 351 5. Estimation of the location parameter of a Cauchy distribution 358 6. Estimation of location and scale parameters of a logistic distribution 364 7. Estimation of parameters in Weibull and extreme-value distributions 370

References 375

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Ch. 13. Some Uses of Order Statistics in Bayesian Analysis 379 Seymour Geisser

1. Introduction 379 2. Discordancy testing 379 3. Suspicious circumstances 380 4. Examples 381 5. Ransacked data 386 6. Conditional predictive discordancy (CPD) tests 389 7. Combinations of largest and smallest 392 8. Ordering future values 395 9. Multivariate problems 398

Acknowledgement 399 References 399

Ch. 14. Inverse Sampling Procedures to Test for Homogeneity in a Multinomial Distribution 401 S. Panchapakesan, Aaron Childs, B. H. Humphrey and N. Balakrishnan

1. Introduction 401 2. The proposed inverse sampling procedures 402 3. Critical values, power and expected sample size 403 4. Comparison with the Standard x2-test 409 5. The combined procedure 412 6. Conclusions 426

Acknowledgements 426 References 426

PART IV. PREDICTION

Ch. 15. Prediction of Order Statistics 431 Kenneth S. Kaminsky and Paul I. Nelson

1. Introduction 431 2. Prediction preliminaries 432 3. Assumptions and notation 433 4. Point prediction 434 5. Interval prediction 440 6. Concluding remarks 448

References 448

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PART V. GOODNESS-OF-FIT TESTS

Ch. 16. The Probability Plot: Tests of Fit Based on the Correlation Coefficient 453 R. A. Lockhart and M. A. Stephens

1. Introduction 453 2. Distribution theory for the correlation coefficient 457 3. Tests for the normal distribution 460 4. Tests for the uniform distribution 464 5. Power of correlation tests 467 A. Appendix 468

References 472

Ch. 17. Distribution Assessment 475 Samuel Shapiro

1. Introduction 475 2. Probability plotting 476 3. Regression type tests 480 4. Use of spacings of the order statistics 488

References 492

PART VI. APPLICATIONS

Ch. 18. Application of Order Statistics to Sampling Plans for Inspection by Variables 497 Helmut Schneider and Frances Barbera

1. Introduction 497 2. Sampling plans for inspection by variables 498 3. Robustness of variable sampling plans for normal distributed characteristics 499 4. Failure censored sampling plans 500 5. Reduction of test times for life-test sampling plans 506 6. Conclusion 509

References 509

Ch. 19. Linear Combinations of Ordered Symmetrie Observations with Applications to Visual Acuity 513 Marios Viana

1. Introduction 513 2. Models and basic results 514

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3. Correlations and linear regressions 515 4. Maximum likelihood and large-sample estimates 518 5. An exact test for 7 = 0 519 6. Numerical examples 520

Acknowledgement 524 References 524

Ch. 20. Order-Statistic Filtering and Smoothing of Time-Series: Part I 525

Gonzalo R. Arce, Yeong-Taeg Kim and Kenne th E. Barner

1. Introduction 525 2. The estimators 527 3. a-trimmed Vi Alters 535 4. Optimization 536 5. Filter lattice structures 540 6. Piecewise linear structure oiL'l Alters 542 7. Applications 544 8. Conclusion 553

References 553

Ch. 21. Order-Statistic Filtering and Smoothing of Time-Series: Part II 555

Kenne th E. Barner and Gonzalo R. Arce

1. Introduction 555 2. The median filter 559 3. Weighted median Alters 570 4. Time-Rank coupling extensions: PWOS Alters 580 5. Optimization techniques 591 6. Applications to image restoration 598 7. Conclusion 601

References 602

Ch. 22. Order Statistics in Image Processing 603

Scott T. Acton and Alan C. Bovik

1. Introduction 603 2. Order statistic Alters 605 3. Spatial/temporal extensions 626 4. Morphological Alters 628 5. Related OS applications 635 6. Conclusions 638

References 638

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