# Handbook of Statistics 17: Order Statistics: Applications

**Edited by**

- N. Balakrishnan, Department of Mathematics and Statistics, Mcmaster University, Hamilton, Ont. Canada, L8S 4K1.
- C.R. Rao, Department of Statistics, The Pennysylvania State University, University Park, PA 16802-2111, USA

Volume 17 of the Handbook of Statistics is the concluding volume covering Order Statistics. Dealing primarily with Applications, it is divided into six parts as follows: Results for Specific Distributions, Linear Estimation, Inferential Methods, Prediction, Goodness-of-fit Tests and Applications. Major theoretical advances were made in this area of research, and in the course of these developments order statistics has also found important applications in many diverse areas. These include life-testing and reliability, robustness studies, statistical quality control, filtering theory, signal processing, image processing, and radar target detection.

Theoretical researchers working on theoretical and methodological advancements on order statistics and applied statisticians and engineers developing new and innovative applications of order statistics have been successfully brought together to create this handbook. For the convenience of readers, the subject matter has been divided into two volumes. Volume 16 focuses on theory and methods. Each volume has been divided into parts, each part specializing in one aspect of order statistics. An elaborate Author Index as well as a Subject Index is presented in both volumes in order to facilitate easy access to all the material included.

Theoretical researchers, applied scientists and engineers, and graduate students involved in the area of order statistics, will find these two volumes dealing with order statistics and applications an invaluable source of information.

### Book information

- Published: July 1998
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-82922-1

### Reviews

...Part of a two-volume set (Handbooks 16 and 17) covering most of the important theoretical and applied aspects of order statistics..

Journal of Economic Literature, 1999

.....this is an extremely helpful contribution.....

Ultramicroscopy, Vol. 80, 2000

### Table of Contents

Preface. Contributors.**Part I. Results for Specific Distributions. Order Statistics in Exponential Distribution**(A.P. Basu, B. Singh). Introduction. Order statistics and its properties. Censored data. Inference concerning several exponential populations. Order restricted inference. Bayesian inference. Acknowlegement. References.

**Higher Order Moments of Order Statistics from Exponential and Right-truncated Exponential Distributions and Applications to Life-testing Problems**(N. Balakrishnan, S.S. Gupta). Introduction. Relations for single moments. Relations for double moments. Relations for triple moments. Relations for quadruple moments. Applications to inference for the one-parameter exponential distribution. Generalized results for the right-truncated exponential distribution. Illustrative examples. Acknowledgements. References.

**Log-gamma Order Statistics and Linear Estimation of Parameters**(N. Balakrishnan, P.S. Chan). Introduction. Single moments of order statistics. Product moments of order statistics. Best linear unbiased estimators. Illustrative example. Acknowledgements. References.

**Recurrence Relations for Single and Product Moments or Order Statistics from a Generalized Logistic Distribution with Applications to Inference and Generalizations to Double Truncation**(N. Balakrishnan, R. Aggarwala). Generalized logistic distribution. Introduction. Recurrence relations for single moments. Recurrence relations for product moments. Recursive computational algorithm. Best linear unbiased estimators. Maximum likelihood estimation. Numerical example. Doubly truncated generalized logistic distribution. Introduction. Recurrence relations for single moments. Recurrence relations for product moments. Recursive algorithm. Acknowledgements. References.

**Order Statistics from the Type III Generalized Logistic Distribution and Applications**(N. Balakrishnan, S.K. Lee). Introduction. Type III generalized logistic distribution. Order statistics and moments. BLUEs of location and scale parameters. MLEs of location and scale parameters. Comparison of the BLUEs with the MLEs. Illustrative examples. References.

**Part II. Linear Estimation. Estimation of Scale Parameter Based on Fixed Set of Order Statistics**(S.K. Sarkar, W. Wang). Introduction. Linear estimators. The positivity of the best unbiased

*L*-estimator. Nonlinear estimators. Extension of the positivity results to censored scale regression model. Concluding remarks. References.

**Optimal Linear Inference Using Selected Order Statistics in Location-Scale Models**(M.M. Ali, D. Umbach). Introduction. Preliminaries. Optimality criteria for estimation. Specific distributions. Tests of significance. Testing goodness-of-fit. References.

**(J.R.M. Hosking). Introduction. Introductory examples. Single-sample problems. More complicated problems. References.**

*L*-Estimation**On Some**(S. Alimoradi, A.K. Md. E. Saleh). Introduction. Regression quantiles and their properties.

*L*-estimation in Linear Regression Models*L*-estimation of the parameters of a linear model based on a few selected regression quantiles with known error distributions. Trimmed least-squared estimation of regression parameters and its asymptotic distribution. Trimmed estimation of regression parameters under uncertain prior information. Acknowledgement. References.

**Part III. Inferential Methods. The Role of Order Statistics in Estimating Threshold Parameters**(A.C. Cohen). Introduction. The exponential distribution. The Weibull distribution. The lognormal distribution. The gamma distribution. The Inverse Gaussian distribution. Errors of estimates. Illustrative examples. Acknowledgement. References.

**Parameter Estimation under Multiply Type-II Censoring**(F. Kong). Introduction. Best linear estimation. Maximum likelihood estimation. Approximate maximum likelihood estimation. Interval estimation for exponential distribution. References.

**On Some Aspects of Ranked Set Sampling in Parametric Estimation**(N.N. Chuiv, B.K. Sinha). Introduction. Estimation of a normal mean and a normal variance. Estimation of an exponential mean. Estimation of parameters in a two parameter exponential distribution. Estimation of the location parameter of a Cauchy distribution. Estimation of location and scale parameters of a logistic distribution. Estimation of parameters in Weibull and extreme-value distributions. References.

**Some Uses of Order Statistics in Bayesian Analysis**(S. Geisser). Introduction. Discordancy testing. Suspicious circumstances. Examples. Ransacked data. Conditional predictive discordancy (CPD) tests. Combinations of largest and smallest. Ordering future values. Multivariate problems. Acknowledgement. References.

**Inverse Sampling Procedures to Test for Homogeneity in a Multinominal Distribution**(S. Panchapakesan

*et al.*). Introduction. The proposed inverse sampling procedures. Critical values, power and expected sample size. Comparison with the standard

*X*

^{2}-test. The combined procedure. Conclusions. Acknowledgements. References.

**Part IV. Prediction. Prediction of Order Statistics**(K.S. Kaminsky, P.I. Nelson). Introduction. Prediction preliminaries. Assumptions and notation. Point prediction. Interval prediction. Concluding remarks. References.

**Part V. Goodness-Of-Fit Tests. The Probability Plot: Tests of Fit Based on the Correlation Coefficient**(R.A. Lockhart, M.A. Stephens). Introduction. Distribution theory for the correlation coefficient. Tests for the normal distribution. Tests for the uniform distribution. Power of correlation tests. Appendix. References.

**Distribution Assessment**(S. Shapiro). Introduction. Probability plotting. Regression type tests. Use of spacings of the order statistics. References.

**Part VI. Applications. Application of Order Statistics to Sampling Plans for Inspection by Variables**(H. Schneider, F. Barbera). Introduction. Sampling plans for inspection by variables. Robustness of variable sampling plans for normal distributed characteristics. Failure censored sampling plans. Reduction of test times for life-test sampling plans. Conclusion. References.

**Linear Combinations of Ordered Symmetric Observations with Applications to Visual Acuity**(M. Viana). Introduction. Models and basic results. Correlations and linear regressions. Maximum likelihood and large-sample estimates. An exact test for &ggr; = 0. Numerical examples. Acknowledgement. References.

**Order-Statistic Filtering and Smoothing of Time-Series: Part I**(G.R. Arce

*et al.*). Introduction. The estimators. &agr;-trimmed

*L*filters. Optimization. Filter lattice structures. Piecewise linear structure of

^{j}l*L*filters. Applications. Conclusion. References.

^{j}l**Order-Statistic Filtering and Smoothing of Time-Series: Part II**(K.E. Barner, G.R. Arce). Introduction. The median filter. Weighted median filters. Time-Rank coupling extensions: PWOS filters. Optimization techniques. Applications to image restoration. Conclusion. References.

**Order Statistics in Image Processing**(S.T. Acton, A.C. Bovik). Introduction. Order statistic filters. Spatial/temporal extensions. Morphological filters. Related OS applications. Conclusions. References.

**Order Statistics Application to CFAR Radar Target Detection**(R. Viswanathan). Introduction. Order statistics based CFAR tests for Rayleigh clutter. Order statistics based tests for non-Rayleigh clutter. Conclusion. Acknowledgement. References.

**Author Index. Subject Index. Contents of Previous Volumes.**