# Order Statistics: Theory & Methods

**By**

- Gerard Meurant

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. This volume focuses on theory and methods, and volume 17 deals primarily with applications. Each volume has been divided into parts, each part specializing in one aspect of order statistics. The articles in this volume have been classified into nine parts. 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 in this volume.

These two volumes form a useful and valuable reference work for theoretical researchers, applied scientists and engineers, and graduate students involved in the area of order statistics.

### Book information

- Published: July 1998
- Imprint: ELSEVIER
- ISBN: 978-0-444-82091-4

### Table of Contents

Preface. Contributors.**Part I. Introduction and Basic Properties. Order Statistics: An Introduction**(N. Balakrishnan, C.R. Rao). Introduction. Marginal distributions of order statistics. Joint distributions of order statistics. Properties. Moments and product moments. Recurrence relations and identities. Bounds. Approximations. Characterizations. Asymptotics. Best linear unbiased estimation and prediction. Inference under censoring. Results for some specific distributions. Outliers and robust inference. Goodness-of-fit tests. Related statistics. Generalizations. References.

**Order Statistics: A Historical Perspective**(H.L. Harter, N. Balakrishnan). Introduction. Distribution theory and properties. Measures of central tendency and dispersion. Regression coefficients. Treatment of outliers and robust estimation. Maximum likelihood estimators. Best linear unbiased estimators. Recurrence relations and identities. Bounds and approximations. Distribution-free tolerance procedures. Prediction. Statistical quality control and range. Multiple comparisons and studentized range. Ranking and selection procedures. Extreme values. Plotting positions on probability paper. Simulation methods. Ordered characteristic roots. Goodness-of-fit tests. Characterizations. Moving order statistics and applications. Order statistics under non-standard conditions. Multivariate order statistics and concomitants. Records. References.

**Computer Simulation of Order Statistics**(P.R. Tadikamalla, N. Balakrishnan). Introduction. Direct generation of order statistics. Generation of uniform (0,1) ordered samples. Generation of progressive Type-II censored ordered statistics. Miscellaneous topics. References.

**Part II. Orderings and Bounds. Lorenz Ordering of Order Statistics and Record Values**(B.C. Arnold, J.A. Villasenor). Introduction. The Lorenz order. Order statistics and record values. Lorenz ordering of order statistics. Lorenz ordering of record values. Remarks. References.

**Stochastic Ordering of Order Statistics**(P.J. Boland

*et al.,).*Introduction. Stochastic orderings. Stochastic order for order statistics from one sample. Stochastic order for order statistics from two samples. Acknowledegment. References.

**Bounds for Expectations of**(T. Rychlik). Introduction. Distribution bounds. Moment and support bounds. Moment bounds for restricted families. Quantile bounds for restricted families. References.

*L*-Estimates**Part III. Relations and Identities. Recurrence Relations and Identities for Moments of Order Statistics**(N. Balakrishnan, K.S. Sultan). Introduction. Notations. Recurrence relations for single moments. Recurrence relations for product moments. Relations between moments of order statistics from two related populations. Normal and half normal distributions. Cauchy distribution. Logistic and related distributions. Gamma and related distributions. Exponential and related distributions. Power funtion and related distributions. Pareto and related distributions. Rayleigh distribution. Linear-exponential distribution. Lomax distribution. Log-logistic and related distributions'. Burr and truncated Burr distributions. Doubly truncated parabolic and skewed distributions. Mixture of two exponential distributions. Doubly truncated Laplace distribution. A class of probability distributions. Acknowledgement. References.

**Part IV. Characterizations. Recent Approaches to Characterizations Based on Order Statistics and Record Values**(C.R. Rao, D.N. Shanbhag). Introduction. Some basic tools. Characterizations based on order statistics. Characterizations involving record values and monotonic stochastic processes. Acknowledgement. References.

**Characterizations of Distributions via Identically Distributed Functions of Order Statistics**(U. Gather

*et al.,*). Introduction. Characterizations of exponential distributions based on normalized spacings. Related characterizations of other continuous distributions. Characterizations of uniform distributions. Characterizations of specific continuous distributions. Characterizations of geometric and other discrete distributions. References.

**Characterizations of Distributions by Recurrence Relations and Identities for Moments of Order Statistics**(U. Kamps). Introduction. Characterizations by sequences of moments and complete function sequences. Characterizations of exponential distributions. Related characterizations in classes of distributions. Characterizations based on a single identity. Characterizations of normal and other distributions by product moments. References.

**Part V. Extremes and Asymptotics. Univariate Extreme Value Theory and Applications.**(J. Galambos). Introduction. The classical models. Applications and statistical inference. Deviations from the classical models. Acknowledgements. References.

**Order Statistics: Asymptotics in Applications**(P. Kumar Sen). Introduction. Some basic results in order statistics. Some basic asymptotics in order statistics. Robust estimation and order statistics: asymptotics in applications. Trimmed LSE and regression quantiles. Asymptotics for concomitants of order statistics. Concomitant

*-functionals and nonparametric regression. Applications of order statistics in some reliability problems. TTT asymptotics and tests for aging properties. Concluding remarks. References.*

**L****Zero-One Laws for Large Order Statistics**(R.J. Tomkins, H. Wang). Introduction. Zero-One laws for the upper-case probability. Zero-One laws for the lower-case probability. Zero-One laws for the lower-case probability when ranks vary. Acknowledgements. References.

**Part VI. Robust Methods. Some Exact Properties of Cook's**(D.R. Jensen, D.E. Ramirez). Introduction. Preliminaries. The structure of Cook's

*D*_{1}**. Normal-Theory properties. Modified versions of**

*D*_{1}**. Summary. References.**

*D*_{1}**Generalized Recurrence Relations for Moments of Order Statistics from Non-Identical Pareto and Truncated Pareto Random Variables with Applications to Robustness**(A. Childs, N. Balakrishnan). Introduction. Relations for single moments. Relations for product moments. Results for the multiple-outlier model (with a slippage of

*observations). Generalization to the truncated Pareto distribution. Robustness of the MLE and BLUE. Robustness of the censored BLUE. Conclusions. Acknowledgements. Appendix A. Appendix B. References.*

**p****Part VII. Resampling Methods. A Semiparametric Bootstrap for Simulating Extreme Order Statistics**(R.L. Strawderman, D. Zelterman). Introduction. A semiparametric bootstrap approximation to

*. A saddlepoint approximation to the bootstrap distribution. Numerical implementation. Simulation results. Example: The British coal mining data. Acknowledgements. References.*

**X**_{j}**Approximations to Distributions of Sample Quantiles**(C. Ma, J. Robinson). Introduction and definitions. Smirnov's lemma. Normal approximation. Saddlepoint approximation. Bootstrap approximation. References.

**Part VIII. Related Statistics. Concomitants of Order Statistics**(H.A. David, H.N. Nagaraja). Introduction and summary. Finite-sample distribution theory and moments. Asymptotic theory. Estimation and test of hypotheses. The rank of

**Y**_{[r:n]}. Selection through an associated variable. Functions of concomitants. References.

**A Record of Records**(V.B. Nevzorov, N. Balakrishnan). Introduction. Classical records. Definitions. Representations of record times and record values using sums of independent terms. Distributions and probability structure of record times. Moments of record times and numbers of records. Limit theorems for record times. Inter-Record times. Distributions and probability structure of record values in sequences of continuous random variable. Limit theorems for record values from continuous distributions. Record values from discrete distributions. Weak records. Bounds and approximations for moments of record values. Recurrence relations for moments of record values. Joint distributions of record times and record values. Generalizations of the classical record model.

*k*

^{th}record times.

*k*

^{th}inter-record times.

*k*

^{th}record values for the continuous case.

*k*

^{th}record values for the discrete case.Weak

*k*

^{th}record values.

*k*-records. Records in sequences of dependent random variables. Random record models. Nonstationary record models. Multivariate records. Relations between records and other probabilistic and statistical problems. Nonclassical characterizations based on records. Processes associated with records. Diverse results. Acknowledgement. References.

_{n}**Part IX. Related Processes. Weighted Sequential Empirical Type Processes with Applications to Change-Point Problems**(B. Szyszkowicz). Introduction. Weighted empirical processes based on observations. "Bridge-type" two-time parameter empirical processes. Weighted empirical processes based on ranks. Weighted empirical processes based on sequential ranks. "Bridge-type" empirical processes of sequential ranks. Contiguous alternatives. Weighted multi-time parameter empirical processes. Acknowledgement. References.

**Sequential Quantile and Bahadur-Kiefer Processes**(M. Csörgö, B. Szyszkowic). Introduction: Basic notions, definitions and some preliminary results. Deviations between the general and uniform quantile processes and their sequential versions. Weighted sequential quantile processes in supremum and

*L*-metrics. A summary of the classical Bahadur-Kiefer process theory via strong invariance principles. An extension of the classical Bahadur-Kiefer process theory via strong invariance principles. An outline of a sequential version of the extended Bahadur-Kiefer process theory via strong invariance principles. Acknowledgement. References.

_{p}**Author Index. Subject Index. Contents of Previous Volumes.**