![i
i
“book” — 2010/11/17 — 16:39 — page 3 —
i
i
i
i
i
i
1.2. LOCATION MODEL 3
1. If G is stochastically larger than F ((G(x) ≤ F(x)) for all x, then
T(G) ≥ T(F);
2. T(HaX+b) = aT(HX) + b, a > 0;
3. T(H−X) = −T(HX).
Then, we call θ = T(H) a location parameter of H.
Note that if X has location parameter θ it follows from the second item in
the above definition that the random variable e = X−θ has location parameter
0. Suppose X1, . . . , Xn is a random sample having the common distribution
function H(x) and θ = T(H) is a location parameter of interest. We express
this by saying that Xi follows the statistical location model,
Xi = θ + ei , i = 1, . . . , n , (1.2.1)
where e1, . . . , en are independent and identically distributed random variable
with distribution function F(x) and density function f(x) and location T(F) =
0. It follows that H(x) = F(x − θ) and that T(H) = θ. We next discuss three
examples of location parameters that we use throughout this chapter. Other
location parameters are discussed in Section 1.8. See Bickel and Lehmann
(1975) for additional discussion of location functionals.
Example 1.2.1 (The Median Location Functional). First define the inverse
of the cdf H(x) by H−1
(u) = inf{x : H(x) ≥ u}. Generally we suppose that
H(x) is strictly increasing on its support and this eliminates ambiguities on
the selection of the parameter. Now define θ1 = T1(H) = H−1
(1/2). This is the
median functional. Note that if G(x) ≤ F(x) for all x, then G−1
(u) ≥ F−1
(u)
for all u; and, in particular, G−1
(1/2) ≥ F−1
(1/2). Hence, T1(H) satisfies the
first condition for a location functional. Next let H∗
(x) = P(aX + b ≤ x) =
H[a−1
(x − b)]. Then it follows at once that H∗−1
(u) = aH−1
(u) + b and the
second condition is satisfied. The third condition follows with an argument
similar to the one for the second condition.
Example 1.2.2 (The Mean Location Functional). For the mean functional
let θ2 = T2(H) =
R
xdH(x), when the mean exists. Note that
R
xdH(x) =
R
H−1
(u)du. Now if G(x) ≤ F(x) for all x, then x ≤ G−1
(F(x)). Let x =
F−1
(u) and we have F−1
(u) ≤ G−1
(F(F−1
(u)) ≤ G−1
(u). Hence, T2(G) =
R
G−1
(u)du ≥
R
F−1
(u)du = T2(F) and the first condition is satisfied. The
other two conditions follow easily from the definition of the integral.
Example 1.2.3 (The Pseudo-Median Location Functional). Assume that X1
and X2 are independent and identically distributed, (iid), with distribution](https://image.slidesharecdn.com/16790-250623123356-0a045493/85/Robust-nonparametric-statistical-methods-2nd-ed-Edition-Thomas-P-Hettmansperger-24-320.jpg)
Robust nonparametric statistical methods 2nd ed Edition Thomas P Hettmansperger
- 1. Robust nonparametric statistical methods 2nd ed Edition Thomas P Hettmansperger download pdf https://ebookfinal.com/download/robust-nonparametric-statistical- methods-2nd-ed-edition-thomas-p-hettmansperger/ Visit ebookfinal.com today to download the complete set of ebook or textbook!
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Robust Statistical Methods with R 1st Edition Jana Jureckova https://ebookfinal.com/download/robust-statistical-methods-with-r-1st- edition-jana-jureckova/ Nonparametric Statistical Methods For Complete and Censored Data 1st Edition M.M. Desu (Author) https://ebookfinal.com/download/nonparametric-statistical-methods-for- complete-and-censored-data-1st-edition-m-m-desu-author/ Statistical methods 2nd ed Edition Rudolf J. Freund https://ebookfinal.com/download/statistical-methods-2nd-ed-edition- rudolf-j-freund/ Statistical Methods for Quality Improvement Wiley Series in Probability and Statistics 3rd Edition Thomas P. Ryan https://ebookfinal.com/download/statistical-methods-for-quality- improvement-wiley-series-in-probability-and-statistics-3rd-edition- thomas-p-ryan/
- 3. Clostridium difficile Methods and Protocols 2nd ed. 2016 Edition Adam P. Roberts https://ebookfinal.com/download/clostridium-difficile-methods-and- protocols-2nd-ed-2016-edition-adam-p-roberts/ Bayesian statistical modelling 2nd ed Edition Peter Congdon https://ebookfinal.com/download/bayesian-statistical-modelling-2nd-ed- edition-peter-congdon/ Discovering Statistics Using SPSS Introducing Statistical Methods S 2nd Edition Andy Field https://ebookfinal.com/download/discovering-statistics-using-spss- introducing-statistical-methods-s-2nd-edition-andy-field/ Psychological Testing A Practical Introduction 2nd Edition Thomas P. Hogan https://ebookfinal.com/download/psychological-testing-a-practical- introduction-2nd-edition-thomas-p-hogan/ Implementing six sigma smarter solutions using statistical methods 2nd Edition Forrest W. Breyfogle https://ebookfinal.com/download/implementing-six-sigma-smarter- solutions-using-statistical-methods-2nd-edition-forrest-w-breyfogle/
- 5. Robust nonparametric statistical methods 2nd ed Edition Thomas P Hettmansperger Digital Instant Download Author(s): Thomas P Hettmansperger; Joseph W McKean ISBN(s): 9781439809082, 1439809089 Edition: 2nd ed File Details: PDF, 5.09 MB Year: 2011 Language: english
- 6. Robust Nonparametric Statistical Methods Second Edition K10449_FM.indd 1 11/19/10 1:27 PM
- 7. MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors F. Bunea, V. Isham, N. Keiding, T. Louis, R. L. Smith, and H. Tong 1 Stochastic Population Models in Ecology and Epidemiology M.S. Barlett (1960) 2 Queues D.R. Cox and W.L. Smith (1961) 3 Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964) 4 The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis (1966) 5 Population Genetics W.J. Ewens (1969) 6 Probability, Statistics and Time M.S. Barlett (1975) 7 Statistical Inference S.D. Silvey (1975) 8 The Analysis of Contingency Tables B.S. Everitt (1977) 9 Multivariate Analysis in Behavioural Research A.E. Maxwell (1977) 10 Stochastic Abundance Models S. Engen (1978) 11 Some Basic Theory for Statistical Inference E.J.G. Pitman (1979) 12 Point Processes D.R. Cox and V. Isham (1980) 13 Identification of Outliers D.M. Hawkins (1980) 14 Optimal Design S.D. Silvey (1980) 15 Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981) 16 Classification A.D. Gordon (1981) 17 Distribution-Free Statistical Methods, 2nd edition J.S. Maritz (1995) 18 Residuals and Influence in Regression R.D. Cook and S. Weisberg (1982) 19 Applications of Queueing Theory, 2nd edition G.F. Newell (1982) 20 Risk Theory, 3rd edition R.E. Beard, T. Pentikäinen and E. Pesonen (1984) 21 Analysis of Survival Data D.R. Cox and D. Oakes (1984) 22 An Introduction to Latent Variable Models B.S. Everitt (1984) 23 Bandit Problems D.A. Berry and B. Fristedt (1985) 24 Stochastic Modelling and Control M.H.A. Davis and R. Vinter (1985) 25 The Statistical Analysis of Composition Data J. Aitchison (1986) 26 Density Estimation for Statistics and Data Analysis B.W. Silverman (1986) 27 Regression Analysis with Applications G.B. Wetherill (1986) 28 Sequential Methods in Statistics, 3rd edition G.B. Wetherill and K.D. Glazebrook (1986) 29 Tensor Methods in Statistics P. McCullagh (1987) 30 Transformation and Weighting in Regression R.J. Carroll and D. Ruppert (1988) 31 Asymptotic Techniques for Use in Statistics O.E. Bandorff-Nielsen and D.R. Cox (1989) 32 Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989) 33 Analysis of Infectious Disease Data N.G. Becker (1989) 34 Design and Analysis of Cross-Over Trials B. Jones and M.G. Kenward (1989) 35 Empirical Bayes Methods, 2nd edition J.S. Maritz and T. Lwin (1989) 36 Symmetric Multivariate and Related Distributions K.T. Fang, S. Kotz and K.W. Ng (1990) 37 Generalized Linear Models, 2nd edition P. McCullagh and J.A. Nelder (1989) 38 Cyclic and Computer Generated Designs, 2nd edition J.A. John and E.R. Williams (1995) 39 Analog Estimation Methods in Econometrics C.F. Manski (1988) 40 Subset Selection in Regression A.J. Miller (1990) 41 Analysis of Repeated Measures M.J. Crowder and D.J. Hand (1990) 42 Statistical Reasoning with Imprecise Probabilities P. Walley (1991) 43 Generalized Additive Models T.J. Hastie and R.J. Tibshirani (1990) 44 Inspection Errors for Attributes in Quality Control N.L. Johnson, S. Kotz and X. Wu (1991) K10449_FM.indd 2 11/19/10 1:27 PM
- 8. 45 The Analysis of Contingency Tables, 2nd edition B.S. Everitt (1992) 46 The Analysis of Quantal Response Data B.J.T. Morgan (1992) 47 Longitudinal Data with Serial Correlation—A State-Space Approach R.H. Jones (1993) 48 Differential Geometry and Statistics M.K. Murray and J.W. Rice (1993) 49 Markov Models and Optimization M.H.A. Davis (1993) 50 Networks and Chaos—Statistical and Probabilistic Aspects O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall (1993) 51 Number-Theoretic Methods in Statistics K.-T. Fang and Y. Wang (1994) 52 Inference and Asymptotics O.E. Barndorff-Nielsen and D.R. Cox (1994) 53 Practical Risk Theory for Actuaries C.D. Daykin, T. Pentikäinen and M. Pesonen (1994) 54 Biplots J.C. Gower and D.J. Hand (1996) 55 Predictive Inference—An Introduction S. Geisser (1993) 56 Model-Free Curve Estimation M.E. Tarter and M.D. Lock (1993) 57 An Introduction to the Bootstrap B. Efron and R.J. Tibshirani (1993) 58 Nonparametric Regression and Generalized Linear Models P.J. Green and B.W. Silverman (1994) 59 Multidimensional Scaling T.F. Cox and M.A.A. Cox (1994) 60 Kernel Smoothing M.P. Wand and M.C. Jones (1995) 61 Statistics for Long Memory Processes J. Beran (1995) 62 Nonlinear Models for Repeated Measurement Data M. Davidian and D.M. Giltinan (1995) 63 Measurement Error in Nonlinear Models R.J. Carroll, D. Rupert and L.A. Stefanski (1995) 64 Analyzing and Modeling Rank Data J.J. Marden (1995) 65 Time Series Models—In Econometrics, Finance and Other Fields D.R. Cox, D.V. Hinkley and O.E. Barndorff-Nielsen (1996) 66 Local Polynomial Modeling and its Applications J. Fan and I. Gijbels (1996) 67 Multivariate Dependencies—Models, Analysis and Interpretation D.R. Cox and N. Wermuth (1996) 68 Statistical Inference—Based on the Likelihood A. Azzalini (1996) 69 Bayes and Empirical Bayes Methods for Data Analysis B.P. Carlin and T.A Louis (1996) 70 Hidden Markov and Other Models for Discrete-Valued Time Series I.L. MacDonald and W. Zucchini (1997) 71 Statistical Evidence—A Likelihood Paradigm R. Royall (1997) 72 Analysis of Incomplete Multivariate Data J.L. Schafer (1997) 73 Multivariate Models and Dependence Concepts H. Joe (1997) 74 Theory of Sample Surveys M.E. Thompson (1997) 75 Retrial Queues G. Falin and J.G.C. Templeton (1997) 76 Theory of Dispersion Models B. Jørgensen (1997) 77 Mixed Poisson Processes J. Grandell (1997) 78 Variance Components Estimation—Mixed Models, Methodologies and Applications P.S.R.S. Rao (1997) 79 Bayesian Methods for Finite Population Sampling G. Meeden and M. Ghosh (1997) 80 Stochastic Geometry—Likelihood and computation O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (1998) 81 Computer-Assisted Analysis of Mixtures and Applications— Meta-analysis, Disease Mapping and Others D. Böhning (1999) 82 Classification, 2nd edition A.D. Gordon (1999) 83 Semimartingales and their Statistical Inference B.L.S. Prakasa Rao (1999) 84 Statistical Aspects of BSE and vCJD—Models for Epidemics C.A. Donnelly and N.M. Ferguson (1999) 85 Set-Indexed Martingales G. Ivanoff and E. Merzbach (2000) K10449_FM.indd 3 11/19/10 1:27 PM
- 9. 86 The Theory of the Design of Experiments D.R. Cox and N. Reid (2000) 87 Complex Stochastic Systems O.E. Barndorff-Nielsen, D.R. Cox and C. Klüppelberg (2001) 88 Multidimensional Scaling, 2nd edition T.F. Cox and M.A.A. Cox (2001) 89 Algebraic Statistics—Computational Commutative Algebra in Statistics G. Pistone, E. Riccomagno and H.P. Wynn (2001) 90 Analysis of Time Series Structure—SSA and Related Techniques N. Golyandina, V. Nekrutkin and A.A. Zhigljavsky (2001) 91 Subjective Probability Models for Lifetimes Fabio Spizzichino (2001) 92 Empirical Likelihood Art B. Owen (2001) 93 Statistics in the 21st Century Adrian E. Raftery, Martin A. Tanner, and Martin T. Wells (2001) 94 Accelerated Life Models: Modeling and Statistical Analysis Vilijandas Bagdonavicius and Mikhail Nikulin (2001) 95 Subset Selection in Regression, Second Edition Alan Miller (2002) 96 Topics in Modelling of Clustered Data Marc Aerts, Helena Geys, Geert Molenberghs, and Louise M. Ryan (2002) 97 Components of Variance D.R. Cox and P.J. Solomon (2002) 98 Design and Analysis of Cross-Over Trials, 2nd Edition Byron Jones and Michael G. Kenward (2003) 99 Extreme Values in Finance, Telecommunications, and the Environment Bärbel Finkenstädt and Holger Rootzén (2003) 100 Statistical Inference and Simulation for Spatial Point Processes Jesper Møller and Rasmus Plenge Waagepetersen (2004) 101 Hierarchical Modeling and Analysis for Spatial Data Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand (2004) 102 Diagnostic Checks in Time Series Wai Keung Li (2004) 103 Stereology for Statisticians Adrian Baddeley and Eva B. Vedel Jensen (2004) 104 Gaussian Markov Random Fields: Theory and Applications Håvard Rue and Leonhard Held (2005) 105 Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu (2006) 106 Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood Youngjo Lee, John A. Nelder, and Yudi Pawitan (2006) 107 Statistical Methods for Spatio-Temporal Systems Bärbel Finkenstädt, Leonhard Held, and Valerie Isham (2007) 108 Nonlinear Time Series: Semiparametric and Nonparametric Methods Jiti Gao (2007) 109 Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis Michael J. Daniels and Joseph W. Hogan (2008) 110 Hidden Markov Models for Time Series: An Introduction Using R Walter Zucchini and Iain L. MacDonald (2009) 111 ROC Curves for Continuous Data Wojtek J. Krzanowski and David J. Hand (2009) 112 Antedependence Models for Longitudinal Data Dale L. Zimmerman and Vicente A. Núñez-Antón (2009) 113 Mixed Effects Models for Complex Data Lang Wu (2010) 114 Intoduction to Time Series Modeling Genshiro Kitagawa (2010) 115 Expansions and Asymptotics for Statistics Christopher G. Small (2010) 116 Statistical Inference: An Integrated Bayesian/Likelihood Approach Murray Aitkin (2010) 117 Circular and Linear Regression: Fitting Circles and Lines by Least Squares Nikolai Chernov (2010) 118 Simultaneous Inference in Regression Wei Liu (2010) 119 Robust Nonparametric Statistical Methods, Second Edition Thomas P. Hettmansperger and Joseph W. McKean (2011) K10449_FM.indd 4 11/19/10 1:27 PM
- 10. Thomas P. Hettmansperger Penn State University University Park, Pennsylvania, USA Joseph W. McKean Western Michigan University Kalamazoo, Michigan, USA Monographs on Statistics and Applied Probability 119 Robust Nonparametric Statistical Methods Second Edition K10449_FM.indd 5 11/19/10 1:27 PM
- 11. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4398-0908-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material repro- duced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copy- right.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifica- tion and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Hettmansperger, Thomas P., 1939- Robust nonparametric statistical methods / Thomas P. Hettmansperger, Joseph W. McKean. -- 2nd ed. p. cm. -- (Monographs on statistics and applied probability ; 119) Summary: “Often referred to as distribution-free methods, nonparametric methods do not rely on assumptions that the data are drawn from a given probability distribution. With an emphasis on Wilcoxon rank methods that enable a unified approach to data analysis, this book presents a unique overview of robust nonparametric statistical methods. Drawing on examples from various disciplines, the relevant R code for these examples, as well as numerous exercises for self-study, the text covers location models, regression models, designed experiments, and multivariate methods. This edition features a new chapter on cluster correlated data”-- Provided by publisher. Includes bibliographical references and index. ISBN 978-1-4398-0908-2 (hardback) 1. Nonparametric statistics. 2. Robust statistics. I. McKean, Joseph W., 1944- II. Title. III. Series. QA278.8.H47 2010 519.5--dc22 2010044858 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com K10449_FM.indd 6 11/19/10 1:27 PM
- 12. i i “book” — 2010/11/17 — 16:39 — page vii — i i i i i i vii Dedication: To Ann and to Marge
- 13. i i “book” — 2010/11/17 — 16:39 — page ix — i i i i i i Contents Preface xv 1 One-Sample Problems 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Location Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Geometry and Inference in the Location Model . . . . . . . . . 5 1.3.1 Computation . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Properties of Norm-Based Inference . . . . . . . . . . . . . . . 19 1.5.1 Basic Properties of the Power Function γS(θ) . . . . . 20 1.5.2 Asymptotic Linearity and Pitman Regularity . . . . . . 22 1.5.3 Asymptotic Theory and Efficiency Results for b θ . . . . 26 1.5.4 Asymptotic Power and Efficiency Results for the Test Based on S(θ) . . . . . . . . . . . . . . . . . . . . . . . 27 1.5.5 Efficiency Results for Confidence Intervals Based on S(θ) 29 1.6 Robustness Properties of Norm-Based Inference . . . . . . . . 32 1.6.1 Robustness Properties of b θ . . . . . . . . . . . . . . . . 33 1.6.2 Breakdown Properties of Tests . . . . . . . . . . . . . . 35 1.7 Inference and the Wilcoxon Signed-Rank Norm . . . . . . . . 38 1.7.1 Null Distribution Theory of T(0) . . . . . . . . . . . . 39 1.7.2 Statistical Properties . . . . . . . . . . . . . . . . . . . 40 1.7.3 Robustness Properties . . . . . . . . . . . . . . . . . . 46 1.8 Inference Based on General Signed-Rank Norms . . . . . . . . 48 1.8.1 Null Properties of the Test . . . . . . . . . . . . . . . . 50 1.8.2 Efficiency and Robustness Properties . . . . . . . . . . 51 1.9 Ranked Set Sampling . . . . . . . . . . . . . . . . . . . . . . . 57 1.10 L1 Interpolated Confidence Intervals . . . . . . . . . . . . . . 61 1.11 Two-Sample Analysis . . . . . . . . . . . . . . . . . . . . . . . 65 1.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ix
- 14. i i “book” — 2010/11/17 — 16:39 — page x — i i i i i i x CONTENTS 2 Two-Sample Problems 77 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.2 Geometric Motivation . . . . . . . . . . . . . . . . . . . . . . 78 2.2.1 Least Squares (LS) Analysis . . . . . . . . . . . . . . . 81 2.2.2 Mann-Whitney-Wilcoxon (MWW) Analysis . . . . . . 82 2.2.3 Computation . . . . . . . . . . . . . . . . . . . . . . . 84 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.4 Inference Based on the Mann-Whitney-Wilcoxon . . . . . . . . 87 2.4.1 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.4.2 Confidence Intervals . . . . . . . . . . . . . . . . . . . 97 2.4.3 Statistical Properties of the Inference Based on the MWW 97 2.4.4 Estimation of ∆ . . . . . . . . . . . . . . . . . . . . . . 102 2.4.5 Efficiency Results Based on Confidence Intervals . . . . 103 2.5 General Rank Scores . . . . . . . . . . . . . . . . . . . . . . . 105 2.5.1 Statistical Methods . . . . . . . . . . . . . . . . . . . . 109 2.5.2 Efficiency Results . . . . . . . . . . . . . . . . . . . . . 110 2.5.3 Connection between One- and Two-Sample Scores . . . 113 2.6 L1 Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.6.1 Analysis Based on the L1 Pseudo-Norm . . . . . . . . . 115 2.6.2 Analysis Based on the L1 Norm . . . . . . . . . . . . . 119 2.7 Robustness Properties . . . . . . . . . . . . . . . . . . . . . . 122 2.7.1 Breakdown Properties . . . . . . . . . . . . . . . . . . 122 2.7.2 Influence Functions . . . . . . . . . . . . . . . . . . . . 123 2.8 Proportional Hazards . . . . . . . . . . . . . . . . . . . . . . . 125 2.8.1 The Log Exponential and the Savage Statistic . . . . . 126 2.8.2 Efficiency Properties . . . . . . . . . . . . . . . . . . . 129 2.9 Two-Sample Rank Set Sampling (RSS) . . . . . . . . . . . . . 131 2.10 Two-Sample Scale Problem . . . . . . . . . . . . . . . . . . . 133 2.10.1 Appropriate Score Functions . . . . . . . . . . . . . . . 133 2.10.2 Efficacy of the Traditional F-Test . . . . . . . . . . . . 142 2.11 Behrens-Fisher Problem . . . . . . . . . . . . . . . . . . . . . 144 2.11.1 Behavior of the Usual MWW Test . . . . . . . . . . . . 144 2.11.2 General Rank Tests . . . . . . . . . . . . . . . . . . . . 146 2.11.3 Modified Mathisen’s Test . . . . . . . . . . . . . . . . . 147 2.11.4 Modified MWW Test . . . . . . . . . . . . . . . . . . . 149 2.11.5 Efficiencies and Discussion . . . . . . . . . . . . . . . . 150 2.12 Paired Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 152 2.12.1 Behavior under Alternatives . . . . . . . . . . . . . . . 156 2.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
- 15. i i “book” — 2010/11/17 — 16:39 — page xi — i i i i i i CONTENTS xi 3 Linear Models 165 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.2 Geometry of Estimation and Tests . . . . . . . . . . . . . . . . 166 3.2.1 The Geometry of Estimation . . . . . . . . . . . . . . . 166 3.2.2 The Geometry of Testing . . . . . . . . . . . . . . . . . 169 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.4 Assumptions for Asymptotic Theory . . . . . . . . . . . . . . 177 3.5 Theory of Rank-Based Estimates . . . . . . . . . . . . . . . . 180 3.5.1 R Estimators of the Regression Coefficients . . . . . . . 180 3.5.2 R Estimates of the Intercept . . . . . . . . . . . . . . . 185 3.6 Theory of Rank-Based Tests . . . . . . . . . . . . . . . . . . . 191 3.6.1 Null Theory of Rank-Based Tests . . . . . . . . . . . . 191 3.6.2 Theory of Rank-Based Tests under Alternatives . . . . 197 3.6.3 Further Remarks on the Dispersion Function . . . . . . 201 3.7 Implementation of the R Analysis . . . . . . . . . . . . . . . . 203 3.7.1 Estimates of the Scale Parameter τϕ . . . . . . . . . . 204 3.7.2 Algorithms for Computing the R Analysis . . . . . . . 207 3.7.3 An Algorithm for a Linear Search . . . . . . . . . . . . 210 3.8 L1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 3.9 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3.9.1 Properties of R Residuals and Model Misspecification . 214 3.9.2 Standardization of R Residuals . . . . . . . . . . . . . 220 3.9.3 Measures of Influential Cases . . . . . . . . . . . . . . 227 3.10 Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 231 3.11 Correlation Model . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.11.1 Huber’s Condition for the Correlation Model . . . . . . 240 3.11.2 Traditional Measure of Association and Its Estimate . 242 3.11.3 Robust Measure of Association and Its Estimate . . . . 243 3.11.4 Properties of R Coefficients of Multiple Determination 245 3.11.5 Coefficients of Determination for Regression . . . . . . 250 3.12 High Breakdown (HBR) Estimates . . . . . . . . . . . . . . . 252 3.12.1 Geometry of the HBR Estimates . . . . . . . . . . . . 252 3.12.2 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.12.3 Asymptotic Normality of b βHBR . . . . . . . . . . . . . 256 3.12.4 Robustness Properties of the HBR Estimates . . . . . . 260 3.12.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 263 3.12.6 Implementation and Examples . . . . . . . . . . . . . . 264 3.12.7 Studentized Residuals . . . . . . . . . . . . . . . . . . 265 3.12.8 Example on Curvature Detection . . . . . . . . . . . . 267 3.13 Diagnostics for Differentiating between Fits . . . . . . . . . . 268 3.14 Rank-Based Procedures for Nonlinear Models . . . . . . . . . 276 3.14.1 Implementation . . . . . . . . . . . . . . . . . . . . . . 279
- 16. i i “book” — 2010/11/17 — 16:39 — page xii — i i i i i i xii CONTENTS 3.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 4 Experimental Designs: Fixed Effects 291 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 4.2 One-way Design . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.2.1 R Fit of the One-way Design . . . . . . . . . . . . . . . 294 4.2.2 Rank-Based Tests of H0 : µ1 = · · · = µk . . . . . . . . 296 4.2.3 Tests of General Contrasts . . . . . . . . . . . . . . . . 299 4.2.4 More on Estimation of Contrasts and Location . . . . . 300 4.2.5 Pseudo-observations . . . . . . . . . . . . . . . . . . . 302 4.3 Multiple Comparison Procedures . . . . . . . . . . . . . . . . 304 4.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 311 4.4 Two-way Crossed Factorial . . . . . . . . . . . . . . . . . . . . 313 4.5 Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . 317 4.6 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . 321 4.7 Rank Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 325 4.7.1 Monte Carlo Study . . . . . . . . . . . . . . . . . . . . 327 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 5 Models with Dependent Error Structure 337 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 5.2 General Mixed Models . . . . . . . . . . . . . . . . . . . . . . 337 5.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . 342 5.3 Simple Mixed Models . . . . . . . . . . . . . . . . . . . . . . . 342 5.3.1 Variance Component Estimators . . . . . . . . . . . . . 343 5.3.2 Studentized Residuals . . . . . . . . . . . . . . . . . . 344 5.3.3 Example and Simulation Studies . . . . . . . . . . . . 346 5.3.4 Simulation Studies of Validity . . . . . . . . . . . . . . 347 5.3.5 Simulation Study of Other Score Functions . . . . . . . 349 5.4 Arnold Transformations . . . . . . . . . . . . . . . . . . . . . 350 5.4.1 R Fit Based on Arnold Transformed Data . . . . . . . 351 5.5 General Estimating Equations (GEE) . . . . . . . . . . . . . . 356 5.5.1 Asymptotic Theory . . . . . . . . . . . . . . . . . . . . 359 5.5.2 Implementation and a Monte Carlo Study . . . . . . . 360 5.5.3 Example: Inflammatory Markers . . . . . . . . . . . . . 362 5.6 Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 5.6.1 Asymptotic Theory . . . . . . . . . . . . . . . . . . . . 368 5.6.2 Wald-Type Inference . . . . . . . . . . . . . . . . . . . 370 5.6.3 Linear Models with Autoregressive Errors . . . . . . . 372 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
- 17. i i “book” — 2010/11/17 — 16:39 — page xiii — i i i i i i CONTENTS xiii 6 Multivariate 377 6.1 Multivariate Location Model . . . . . . . . . . . . . . . . . . . 377 6.2 Componentwise Methods . . . . . . . . . . . . . . . . . . . . . 382 6.2.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 385 6.2.2 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 6.2.3 Componentwise Rank Methods . . . . . . . . . . . . . 390 6.3 Spatial Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 392 6.3.1 Spatial Sign Methods . . . . . . . . . . . . . . . . . . . 392 6.3.2 Spatial Rank Methods . . . . . . . . . . . . . . . . . . 399 6.4 Affine Equivariant and Invariant Methods . . . . . . . . . . . 403 6.4.1 Blumen’s Bivariate Sign Test . . . . . . . . . . . . . . 403 6.4.2 Affine Invariant Sign Tests . . . . . . . . . . . . . . . . 405 6.4.3 The Oja Criterion Function . . . . . . . . . . . . . . . 413 6.4.4 Additional Remarks . . . . . . . . . . . . . . . . . . . 418 6.5 Robustness of Estimates of Location . . . . . . . . . . . . . . 419 6.5.1 Location and Scale Invariance: Componentwise Methods 419 6.5.2 Rotation Invariance: Spatial Methods . . . . . . . . . . 420 6.5.3 The Spatial Hodges-Lehmann Estimate . . . . . . . . . 421 6.5.4 Affine Equivariant Spatial Median . . . . . . . . . . . . 421 6.5.5 Affine Equivariant Oja Median . . . . . . . . . . . . . 422 6.6 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 6.6.1 Test for Regression Effect . . . . . . . . . . . . . . . . 425 6.6.2 The Estimate of the Regression Effect . . . . . . . . . 431 6.6.3 Tests of General Hypotheses . . . . . . . . . . . . . . . 432 6.7 Experimental Designs . . . . . . . . . . . . . . . . . . . . . . . 439 6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 A Asymptotic Results 447 A.1 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . 447 A.2 Simple Linear Rank Statistics . . . . . . . . . . . . . . . . . . 448 A.2.1 Null Asymptotic Distribution Theory . . . . . . . . . . 449 A.2.2 Local Asymptotic Distribution Theory . . . . . . . . . 450 A.2.3 Signed-Rank Statistics . . . . . . . . . . . . . . . . . . 457 A.3 Rank-Based Analysis of Linear Models . . . . . . . . . . . . . 460 A.3.1 Convex Functions . . . . . . . . . . . . . . . . . . . . . 463 A.3.2 Asymptotic Linearity and Quadraticity . . . . . . . . . 464 A.3.3 Asymptotic Distance between b β and e β . . . . . . . . . 467 A.3.4 Consistency of the Test Statistic Fϕ . . . . . . . . . . . 468 A.3.5 Proof of Lemma 3.5.1 . . . . . . . . . . . . . . . . . . . 469 A.4 Asymptotic Linearity for the L1 Analysis . . . . . . . . . . . . 470 A.5 Influence Functions . . . . . . . . . . . . . . . . . . . . . . . . 473
- 18. i i “book” — 2010/11/17 — 16:39 — page xiv — i i i i i i xiv CONTENTS A.5.1 Influence Function for Estimates Based on Signed-Rank Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 474 A.5.2 Influence Functions for Chapter 3 . . . . . . . . . . . . 476 A.5.3 Influence Function of b βHBR of Section 3.12.4 . . . . . . 482 A.6 Asymptotic Theory for Section 3.12.3 . . . . . . . . . . . . . . 484 A.7 Asymptotic Theory for Section 3.12.7 . . . . . . . . . . . . . . 491 A.8 Asymptotic Theory for Section 3.13 . . . . . . . . . . . . . . . 492 References 495 Author Index 521 Index 527
- 19. i i “book” — 2010/11/17 — 16:39 — page xv — i i i i Preface Basically, I’m not interested in doing research and I never have been. I’m interested in understanding, which is quite a different thing. And often to understand something you have to work it out yourself because no one else has done it. David Blackwell describing himself as a “dilettante” in a 1983 interview for Mathematical People, a collection of profiles and interviews. I don’t believe I can really do without teaching. The reason is, I have to have something so that when I don’t have any ideas and I’m not getting anywhere I can say to myself, “At least I’m living; at least I’m doing something; I’m making some contribution”-it’s just psychological. Richard Feynman Nonparametric inference methods, especially those derived from ranks, have a long and successful history extending back to early work by Frank Wilcoxon in 1945. In the first edition of this monograph we developed rank- based methods from the unifying theme of geometry and continue this ap- proach in the second edition. The least squares norm is replaced by a weighted L1 norm, and the resulting statistical interpretations are similar to those of least squares. This results in rank-based methods or L1 methods depending on the choice of weights. The rank-based methods proceed much like the tra- ditional analysis. Using the norm, models are easily fitted. Diagnostics pro- cedures can then be used to check the quality of fit (model criticism) and to locate outlying points and points of high influence. Upon satisfaction with the fit, rank-based inferential procedures can then be used to conduct the statisti- cal analysis. The advantages of rank-based methods include better power and efficiency at heavy-tailed distributions and robustness against various model violations and pathological data. In the first edition we extended rank methods from univariate location models to linear models and multivariate models, providing a much more ex- tensive set of tools and methods for data analysis. The second edition provides xv
- 20. i i “book” — 2010/11/17 — 16:39 — page xvi — i i i i i i xvi PREFACE additional models (including models with dependent error structure and non- linear models) and methods and extends significantly the possible analyses based on ranks. In the second edition we have retained the material on one- and two-sample problems (Chapters 1 and 2) along with the basic development of rank meth- ods in the linear model (Chapter 3) and fixed effects experimental designs (Chapter 4). Chapter 5, from the first edition, on high breakdown R esti- mates has been condensed and moved to Chapter 3. In addition, Chapter 3 now contains a new section on rank procedures for nonlinear models. Se- lected topics from the first four chapters provide a basic graduate course in rank-based methods. The methods are fully illustrated and the theory fully developed. The prerequisites are a basic course in mathematical statistics and some background in applied statistics. For a one semester course, we suggest the first seven sections of Chapter 1, the first four sections of Chapter 2, the first seven sections plus section 9 in Chapter 3, and the first four sections of Chapter 4, and then choice of topics depending on interest. The new Chapter 5 deals with models with dependent error structure. New material on rank methods for mixed models is included along with material on general estimating equations, GEE. Finally, a section on time series has been added. As in the first edition, this new material is illustrated on data sets and R software is made available to the reader. Chapter 6 in both editions deals with multivariate models. In the sec- ond edition we have added new material on the development of affine in- variant/equivariant sign methods based on transform-retransform techniques. The new methods are computationally efficient as opposed to the earlier affine invariant/equivariant methods. The methods developed in the book can be computed using R li- braries and functions. These libraries are discussed and illustrated in the relevant sections. Information on several of these packages and func- tions (including Robnp, ww, and Rfit) can be obtained at the web site http://www.stat.wmich.edu/mckean/index.html. Hence, we have again ex- panded significantly the available set of tools and inference methods based on ranks. We have included the data sets for many of our examples in the book. For others, the reader can obtain the data at the Chapman and Hall web site. See also the site http://www.stat.wmich.edu/mckean/index.html for information on the data sets used in this book. We are indebted to many of our students and colleagues for valuable dis- cussions, stimulation, and motivation. In particular, the first author would like to express his sincere thanks for many stimulating hours of discussion with Steve Arnold, Bruce Brown, and Hannu Oja while the second author wants to express his sincere thanks for discussions over the years with Ash Abebe,
- 21. i i “book” — 2010/11/17 — 16:39 — page xvii — i i i i i i xvii Kim Crimin, Brad Huitema, John Kapenga, John Kloke, Joshua Naranjo, M. Rashid, Jerry Sievers, Jeff Terpstra, and Tom Vidmar. We both would like to express our debt to Simon Sheather, our friend, colleague, and co-author on many papers. We express our thanks to Rob Calver, Sarah Morris, and Michele Dimont of Chapman & Hall/CRC for their assistance in the preparation of this book. Thomas P. Hettmansperger Joseph W. McKean
- 22. i i “book” — 2010/11/17 — 16:39 — page 1 — i i i i i i Chapter 1 One-Sample Problems 1.1 Introduction Traditional statistical procedures are widely used because they offer the user a unified methodology with which to attack a multitude of problems, from simple location problems to highly complex experimental designs. These procedures are based on least squares fitting. Once the problem has been cast into a model then least squares offers the user: 1. a way of fitting the model by minimizing the Euclidean normed distance between the responses and the conjectured model; 2. diagnostic techniques that check the adequacy of the fit of the model, explore the quality of fit, and detect outlying and/or influential cases; 3. inferential procedures, including confidence procedures, tests of hypothe- ses, and multiple comparison procedures; 4. computational feasibility. Procedures based on least squares, though, are easily impaired by outlying observations. Indeed one outlying observation is enough to spoil the least squares fit, its associated diagnostics and inference procedures. Even though traditional inference procedures are exact when the errors in the model follow a normal distribution, they can be quite inefficient when the distribution of the errors has longer tails than the normal distribution. For simple location problems, nonparametric methods were proposed by Wilcoxon (1945). These methods consist of test statistics based on the ranks of the data and associated estimates and confidence intervals for location pa- rameters. The test statistics are distribution free in the sense that their null distributions do not depend on the distribution of the errors. It was soon 1
- 23. i i “book” — 2010/11/17 — 16:39 — page 2 — i i i i i i 2 CHAPTER 1. ONE-SAMPLE PROBLEMS realized that these procedures are almost as efficient as the traditional meth- ods when the errors follow a normal distribution and, furthermore, are often much more efficient relative to the traditional methods when the error distri- butions deviate from normality; see Hodges and Lehmann (1956). These pro- cedures possess both robustness of validity and power. In recent years these nonparametric methods have been extended to linear and nonlinear models. In addition, from the perspective of modern robustness theory, contrary to least squares estimates, these rank-based procedures have bounded influence functions and positive breakdown points. Often these nonparametric procedures are thought of as disjoint methods that differ from one problem to another. In this text, we intend to show that this is not the case. Instead, these procedures present a unified methodology analogous to the traditional methods. The four items cited above for the tra- ditional analysis hold for these procedures too. Indeed the only operational difference is that the Euclidean norm is replaced by another norm. There are computational procedures available for the rank-based pro- cedures discussed in this book. We offer the reader a collection of com- putational functions written in the software language R; see the site http://www.stat.wmich.edu/mckean/. We refer to these computational algo- rithms as robust nonparametric R algorithms or Robnp. For the chapters on linear models we make use of the set of algorithms ww written by Terpstra and McKean (2005) and the R package Rfit developed by Kloke and McKean (2010). We discuss these functions throughout the text and use them in many of the examples, simulation studies, and exercises. The programming language R (see Ihaka and Gentleman, 1996) is freeware and can run on all (PC, Mac, Linux) platforms. To download the R software and accompanying informa- tion, visit the site http://www.r-project.org/. The language R has intrinsic functions for computation of some of the procedures discussed in this and the next chapter. 1.2 Location Model In this chapter we consider the one-sample location problem. This allows us to explore some useful concepts such as distribution freeness and robustness in a simple setting. We extend many of these concepts to more complicated situations in later chapters. We need to first define a location parameter. For a random variable X we often subscript its distribution function by X to avoid confusion. Definition 1.2.1. Let T(H) be a function defined on the set of distribution functions. We say T(H) is a location functional if
- 24. i i “book” — 2010/11/17 — 16:39 — page 3 — i i i i i i 1.2. LOCATION MODEL 3 1. If G is stochastically larger than F ((G(x) ≤ F(x)) for all x, then T(G) ≥ T(F); 2. T(HaX+b) = aT(HX) + b, a > 0; 3. T(H−X) = −T(HX). Then, we call θ = T(H) a location parameter of H. Note that if X has location parameter θ it follows from the second item in the above definition that the random variable e = X−θ has location parameter 0. Suppose X1, . . . , Xn is a random sample having the common distribution function H(x) and θ = T(H) is a location parameter of interest. We express this by saying that Xi follows the statistical location model, Xi = θ + ei , i = 1, . . . , n , (1.2.1) where e1, . . . , en are independent and identically distributed random variable with distribution function F(x) and density function f(x) and location T(F) = 0. It follows that H(x) = F(x − θ) and that T(H) = θ. We next discuss three examples of location parameters that we use throughout this chapter. Other location parameters are discussed in Section 1.8. See Bickel and Lehmann (1975) for additional discussion of location functionals. Example 1.2.1 (The Median Location Functional). First define the inverse of the cdf H(x) by H−1 (u) = inf{x : H(x) ≥ u}. Generally we suppose that H(x) is strictly increasing on its support and this eliminates ambiguities on the selection of the parameter. Now define θ1 = T1(H) = H−1 (1/2). This is the median functional. Note that if G(x) ≤ F(x) for all x, then G−1 (u) ≥ F−1 (u) for all u; and, in particular, G−1 (1/2) ≥ F−1 (1/2). Hence, T1(H) satisfies the first condition for a location functional. Next let H∗ (x) = P(aX + b ≤ x) = H[a−1 (x − b)]. Then it follows at once that H∗−1 (u) = aH−1 (u) + b and the second condition is satisfied. The third condition follows with an argument similar to the one for the second condition. Example 1.2.2 (The Mean Location Functional). For the mean functional let θ2 = T2(H) = R xdH(x), when the mean exists. Note that R xdH(x) = R H−1 (u)du. Now if G(x) ≤ F(x) for all x, then x ≤ G−1 (F(x)). Let x = F−1 (u) and we have F−1 (u) ≤ G−1 (F(F−1 (u)) ≤ G−1 (u). Hence, T2(G) = R G−1 (u)du ≥ R F−1 (u)du = T2(F) and the first condition is satisfied. The other two conditions follow easily from the definition of the integral. Example 1.2.3 (The Pseudo-Median Location Functional). Assume that X1 and X2 are independent and identically distributed, (iid), with distribution
- 25. i i “book” — 2010/11/17 — 16:39 — page 4 — i i i i i i 4 CHAPTER 1. ONE-SAMPLE PROBLEMS function H(x). Let Y = (X1 + X2)/2. Then Y has distribution function H∗ (y) = P(Y ≤ y) = R H(2y − x)h(x)dx. Let θ3 = T3(H) = H∗−1 (1/2). To show that T3 is a location functional, suppose G(x) ≤ F(x) for all x. Then G∗ (y) = Z G(2y − x)g(x) dx = Z Z 2y−x −∞ g(t) dt g(x) dx ≤ Z Z 2y−x −∞ f(t) dt g(x) dx = Z Z 2y−t −∞ g(x) dt f(t) dx ≤ Z Z 2y−t −∞ f(x) dt f(t) dx = F∗ (y) ; hence, as in Example 1.2.1, it follows that G∗−1 (u) ≥ F∗−1 (u) and, hence, that T3(G) ≥ T3(F). For the second property, let W = aX + b where X has distribution function H and a 0. Then W has distribution function FW (t) = H((t − b)/a). Then by the change of variable z = (x − b)/a, we have F∗ W (y) = Z H 2y − x − b a 1 a h x − b a dx = Z H 2 y − b a − z h(z) dz . Thus the defining equation for T3(FW ) is 1 2 = Z H 2 T3(FW ) − b a − z h(z) dz , which is satisfied for T3(FW ) = aT3(H) + b. For the third property, let V = −X where X has distribution function H. Then V has distribution function FV (t) = 1 − H(−t). Hence, by the change in variable z = −x, F∗ V (y) = Z (1 − H(−2y + x))h(−x) dx = 1 − Z H(−2y − z))h(z) dz . Because the defining equation of T3(FV ) can be written as 1 2 = Z H(2(−T3(FV )) − z)h(z) dz , it follows that T3(FV ) = −T3(H). Therefore, T3 is a location functional. It has been called the pseudo-median by Hoyland (1965) and is more appropriate for symmetric distributions. The next theorem characterizes all the location functionals for a symmetric distribution.
- 26. Random documents with unrelated content Scribd suggests to you:
- 27. which a sliding element containing a fulminate strikes. The sliding block carries a small charge of black powder which is set off by the fulminate, thus igniting the train which leads to the high explosive charge detonator. Were this sliding block left free to slide back and forth at all times it would be unsafe to transport the fuze, as it might be set off by accident. There must be therefore some means of holding it safely away from the anvil until it is desired to detonate the charge. There are thus two conflicting conditions to be met: safety during transportation and sensitiveness at the point of departure. It may not be understood at first why sensitiveness at the point of departure should be a condition to be met. Suffice it to say that all fuzes are designed to arm at discharge or soon after leaving the bore for they must be ready to act at any time after leaving the muzzle. Were they to be safe during flight they might be so safe that the remaining velocity would not be sufficient to set them off. All fuzes are designed to arm as we say either during travel through the bore or immediately after. Methods of Arming. Spring method.—Let us suppose that after our projectile has started on its way the sliding block is free to move within a cavity at the forward end of which is the anvil. If the projectile comes to a sudden drop or even sudden reduction of velocity the block if unrestrained will, according to the principle of inertia, keep on going till something stops it. The something in this case is the anvil and the fulminate cap is set off. But it is not so simple. For while the projectile is in flight it is acted upon by the air resistance and slows down but the block in the cavity of the head is not subjected to this resistance. It therefore gains on the projectile or creeps forward in the cavity unless restrained as it is by a spring. Now one more point and this type of fuze is complete. We supposed that our block was free to slide. For safety’s sake it is pinned to the cavity. Again we call upon inertia to bread the pin so as to leave the block free to slide. The strength of the pin is calculated so that the force of inertia of the mass of the block is greater than the resistance of the safety pin and
- 28. when the projectile starts the pin breaks and the spring forces the block to the rear of the cavity until the sudden stop of the projectile permits the block to slide forward as explained. Such a fuze requires a comparatively high initial velocity and is not adapted to howitzers using low muzzle velocities. There are three other methods in use to arm the fuze. They are inertia of a sleeve; centrifugal force and powder pellet system, that is, combustion of a grain of powder holding the sliding block from the anvil by means of an arm resting against the unburned powder grain. These are more sensitive than the type described. In the first system, a sleeve fitting around the plunger carrying the cap slides to the rear by inertia when the projectile starts and two clips engage in notches on the plunger body making the sleeve and plunger thereafter move as one body, they are thus held together by a plunger spring which before arming held the plunger away from the anvil. The safety spring held the sleeve and plunger away from the anvil and after arming prevents forward creeping by the plunger and sleeve now locked together. Upon striking, the plunger and sleeve move forward as one body and the cap strikes the anvil. In centrifugal systems the primer plunger is kept safely away from the anvil by a lock which is kept in place by springs. When the rotational velocity reaches a certain point the force of the springs is overcome by the centrifugal force and the locks are thrown aside or opened and the plunger is free to move forward on impact. In the powder pellet system (the one largely used by the Germans) there is a well or channel filled with compressed powder, this is set off by a fulminate cap which is fired by inertia, a small plunger-anvil striking the cap. When the powder is consumed it leaves a channel into which an arm attached to the sliding block carrying the igniting fulminate for the charge may slide, thus permitting the block to slide forward to the anvil fixed in the forward part of the cavity. It is held from creeping forward after the compressed powder is burned by a safety spring, thus insuring sufficiently hard an impact to set off the cap.
- 29. Heretofore in our service the fulminating cap has been fixed and the plunger carried the anvil or as we call it the firing pin. Such is now the system in our base detonating fuzes, and in our combination fuze. The new point detonating fuzes are patterned after the French and are practically French fuzes. Fuzes Classification. Fuses are classified as: (a) Percussion if it acts on impact, producing a low order of explosion. (b) Time when it acts in the air at a certain point of the trajectory. (c) Combination if it is able to act in the air or upon impact. (d) Detonating when it contains a fulminate which will bring about detonation upon impact. The detonator may be separate or incorporated in the fuse. For the 75-mm gun and the 155-mm howitzer it forms a part of the fuze. Many fuzes are armed on set-back. An exception to this is the long detonating fuse, MK 111, which is armed by the unrolling of a brass spiral holding together two half rings made of steel so fitted as to prevent the anvil and the head of the fuse from getting close together. The spiral unrolls when the rotational velocity of the projectile reaches a certain speed, thus drawing away the two steel rings and arming the fuse.
- 30. DETONATING FUZE—MARK-III. DETONATING FUZE—MARK-V. It is of great importance that the spiral spring be not unrolled during transportation or storage. This is prevented by winding a tape of tarred canvas around the spirals, the head being covered by a thin band of tinfoil. Just before loading the projectile the head and tape are removed by pulling the free end of the tape.
- 31. The following precautions concerning fuses must be rigidly observed to prevent grave accidents: 1. All detonators and detonating point fuses must be fitted with a felt washer underneath, thus insuring proper seating in the central tube. 2. Never disassemble a fuse by unscrewing. 3. Any fuse, the parts of which have become accidentally unscrewed, must be destroyed at once. If fired it may cause a premature burst; if handled a grave accident may result. 4. Any fuse or projectile which has been fired is dangerous, because it may then be able to detonate by a very slight shock. It is forbidden to touch it. 5. Never remove the tin hood from the long fuse before having screwed the fuse in the central tube. 6. After having removed the tin hood, be sure that the spiral is in its proper position. Never use a long fuse without the spiral. 7. Be sure the men understand that this spiral must not be removed. It has happened that men have removed this spiral, thinking that it was a device similar to the safety ring in trench mortar fuzes, MK VII E. 8. See that the ring of the long fuze which connects the powder train to the fuze body cannot be unscrewed. If it can be unscrewed the fuze should be sent back to the depot. 9. If it is necessary to remove a shell with a long fuze by means of the rammer, be sure to have a special rammer cup in the shape of a hollow cylinder of wood which will fit between the shell and the rammer. 10. Time and combination fuzes cannot be made absolutely water- tight; the cover must therefore not be removed until the projectile is about to be loaded. Fuse Tables.
- 32. Tables showing American and French fuses to be used by our Field Artillery, with information concerning markings, color, time of delay, size of fuse, etc. DETONATING FUSES. Time of delay. Color. Size of Fuse. Corresponding to. Cannon. MK I 2- 100 White head. Short. Russian 3GT. 3” gun for target practice only. M II (non delay) 2- 100 8”, 9.2”, MK II (non delay) 2- 100 White top. Short. 204-m/m. MK II (short delay) 5- 100 Black top. Short. Modified. Gun and Howitzer. M II (long delay) 15- 100 Black head. Short. Russian. MK III (Supersensitive) zero No color. Long. French IAL. 75 G; 3.8”G and H; 4.7 in. G and H; 6”H; 155H; all gas shells. MK IV (non-delay) 2- 100 White top. Short. French 24/31 SR (99- 15). Howitzer only. MK IV (short delay) 5- 100 Black top. Short. French 24/31 AR (99- 15). Howitzer only. MK IV (long delay) 15- 100 Black top violet detonator socket. Short. French 24/31 SR (99- 15). Howitzer only. MK V (non-delay) 2- 100 White top. Short. French 24/31 SR (99- 08). All guns, but not Howitzers. MK V (short delay) 5- Black top. Short. French 24/31 AR (99- All guns, but not Howitzers.
- 33. 100 08). Mark—VII (non delay) 2- 100 White. Short 6” T. M. Mark VII (long delay) 20- 100 Black top with violet detonator socket Short 6” T. M. Letter “E” after mark VII indicates safety device. Note:—All American point detonating fuses are stamped on head cap in letters and figures, .125 in high, with name of use, amount of delay, initials of loader, lot and number; thus: PDF. MIV, xx Delay, FA, Lot No. xx. 45-SECOND COMBINATION FUZE MARK I.
- 34. 21 SECOND COMBINATION FUZE MODEL OF 1907 M. COMBINATION FUSES. Fuse. Total time burning Sec. Corresponding French Type. On what projectile used. By what cannon fired. Wt. of fuse. 21 s/comb. F. A., 1907 M. 21 22/31M 1897, 24 sec. Com. Shrapnel. MKi. All 3” and 75-mm guns 1¼ lbs. 21 s/comb. F.A., 1915 21 22/31M 1916, 24 sec. AA. Com. Shrapnel. MKi. All 3” and 75-mm guns 1¼ lbs. 31 s/comb. F. A. 1915 31 30/55M 1889, 40 sec. Com. Shrapnel. 4.7” gun. 2 lbs. 45 s/comb. F. A. 1907 M. 45 Same as above. 30/55M 1889, 40 sec. Com. Shrapnel, MKi. 155 How.
- 35. 30/55M 1913, 40 sec. AA. C. S. Shell AA MKiii AA. Shrapnel. 4.7” gun Anti- aircraft. ACTION OF AMERICAN AND FRENCH DETONATING FUSES. Time zero 1/100 2/100 5/100 15/100 Color No color. Red. White. Black. Black with violet socket. American MKii None being made. MK i MK ii (SD) MK ii (LD) Detonating Fuse is considered MK ii (ND) MK ii (SD) MK iv (LD) unsafe MK iV (ND) MK V (SD) safe MK iV (ND) MK V (SD) Fuses Will be abandoned by French MK V (ND) French detonating fuses iAL. 1 SR. AR. LR. Notes on Ammunition Marking. Marks on H. E. Shell. These are of two kinds. (a) Stamped marks made with a steel punch on the body of the projectile just above the rotating band. These refer to the manufacture of the projectile. (b) Painted marks or bands which are clearly visible. They refer to the loading, to the weight of the projectile and to the special purposes for which the projectile is to be used. Painted marks referring to loading are found on the ogive. H. E. shells are usually painted red.
- 36. Marks referring to weight are painted in black just above the rotating band, as follows: L.— very light. +— light. ++— normal. +++— heavy. ++++— very heavy. A white cross above these marks means that a plate has been welded on the base. These marks are also painted on the boxes. Shells fitted with cartridge cases (fixed ammunition) are not painted below the rotating bands. Special Shell. Incendiary shells.—These incendiary shells are filled with some flame-producing liquid, alumino thermic charge or incendiary cylinder composed of slow burning linstock and string strongly impregnated with saltpeter. Markings.—Green with red ogive. All shells containing black powder are more or less incendiary. Percussion shrapnel is incendiary.
- 38. Star Shells.—For 155 howitzer. Upon bursting, they liberate eight white stars fitted with silken parachutes. The stars are projected backward through the base of the projectile at the point of burst. The parachutes open, the stars descending very slowly, illuminating the surrounding objects for about 45 sec. The best height of burst is about 300 m.; the burst interval should not be over 300. These shells are also incendiary. Markings: a blue star and an “E.” Gas shells are either toxic or tear-producing. (a) Toxic shells are numbered either 4 or 5. The liquids 4 and 5 volatize, immediately upon contact with the air. The gases are quickly diffused and easily carried by the wind. Effect.—Liquid 4 acts immediately and is felt instantly. Liquid 5, on the contrary, works more slowly and its effects are apparent only after several hours. Markings: Green with white bands, and numbers 4 or 5 on the ogive. (b) Tear shells.—These shells are numbered 11, 12 and 13. They are filled with two liquids, either mixed or separated, one liquid being tear producing, the other smoke producing. When the shell bursts, a greater part of the liquid is volatilized, the remainder being projected to the ground in small drops which volatize with variable speed. Markings: Green with numbers 11, 12 or 13 on the ogive. Tracer shell.—This shell is fitted with a time fuse which ignites the inside charge, the flames of which pass through the holes in the ogive thus tracing the trajectory. Tracer shells are used in fire for adjustment on aircraft. They are also incendiary. Markings: White with blue ogive. Letter “T” painted on body. PRECAUTIONS IN SEPARATE LOADING PROJECTILES. All projectiles must be seated accurately and carefully in loading, otherwise not only inaccurate fire will result but also premature detonations may occur.
- 39. Rotating bands should be smoothed and lightly greased just before loading. In transport and in storage the bands should be protected by rope bands, straw tithes, etc., to prevent deformation.
- 40. CHAPTER XIII CARE AND PRESERVATION. OILS AND CLEANING MATERIAL, TOOLS AND ACCESSORIES FOR ARTILLERY MATERIEL WITH THEIR USE. In order that all parts of the materiel may function easily, it is necessary that all the working and bearing surfaces may be properly cleaned and lubricated with the appropriate lubricant. Where such surfaces are not directly accessible, oil holes are provided; these holes should be kept free from grit and dirt. Except during oiling, they should be kept fully closed by the means provided. For use in service, for the cleaning and preservation of this materiel, the ordnance department issues hydroline oil, lubricating oil, clock oil, vaseline, sperm oil, coal oil, neat’s-foot oil and light slushing oil. Each of these oils are suited for the particular purpose for which it is issued, as stated below, and care should be taken that it is not used for other purposes. Hydroline oil.—Used in the recoil cylinders of the carriage and for no other purpose. Never used as a lubricant. It is characterized by its low freezing point and its non-corrosive action on metals. Lubricating oil (Engine oil Number 1).—A light petroleum oil used exclusively in all oil holes of the materiel, and in lubricating such parts as wheels and axles, guns and cradle slides, cradle pintle and socket elevating and traversing mechanisms, exterior of cylinders, brake bearings, hinges, different surfaces of breechblocks, threads, breech recess, et cetera. Clock oil.—Used on the spindle and all gearings of the Battery Commander’s telescope, bearings of the panoramic sight, range
- 41. quadrants and fuze setters. In all cases clock oil should be used only when the instruments mentioned are disassembled for cleaning. It should be applied by dropping from the end of the dropper attached to the end of the cork. In case of emergency, use as a substitute either sperm oil or Engine oil No. 1, in the order mentioned. Vaseline (Petrolatum).—The heavy petroleum oil free from rosin. Used on the worm gears and the worm racks of the panoramic sight, the hand and bracket fuze setter, B. C. telescope, and on the micrometer screw and bushing of the quadrant. The spare parts of the breech mechanism should also be coated with vaseline and each piece then wrapped in paper to prevent the oil from being rubbed off. Sperm oil.—A lighter lubricant than the lubricating oils, and may be used on the gears of sights, fuze setters, ranges, quadrants, parts of revolvers, et cetera; lubricating oil may also be used on such parts. It is also used as a temporary rust preventive. Its low viscosity and light body make it unsuitable for this purpose for more than a few days. Coal oil.—Used for cleaning purposes. In the field it may be used for lanterns. Coal oil for general illuminating purposes is furnished by the quartermaster department. Neat’s-foot Oil.—An animal oil used for softening and preserving leather. Applied with a moistened cloth to the flesh side of moistened leather. Light slushing oil.—The heavy petroleum oil similar to cosmic. Used as a rust preventive. Essentially a mineral oil containing a large per cent of rosin. Prescribed for use in the protection and preservation of all bright or unpainted of steel or iron on all parts of the equipment when the materiel is to remain unused for an appreciable length of time. Its use as a lubricant for mobile artillery is forbidden. Before applying the slushing oil to any surface, the parts should be thoroughly cleaned so as to be free from rust, coal oil, lubricating oil, et cetera, as their presence will cause rusting under the slushing oil. The slushing oil should then be applied in a thin, uniform coat, since this is all that is necessary to give good protection. Except in very cold weather it can be applied by using a
- 42. paint brush as when painting, in cold weather it should be applied by stippling—that is, by holding the brush perpendicular to the surface to be coated and then tapping the surface with the point of the brush. It can be applied through the bore of the gun by a slush brush issued for that purpose. In cold weather it should be warmed before used in the bore of the gun. It may be readily removed by burlap or waste dipped coal oil. Borax.—Issued for use as a flux in welding. Lavaline.—A metal polish issued interchangeable with Gibson’s soap polish. Used on bits and collars. Lye, powdered.—When dissolved in water, one pound to six quarts with sufficient lime to give a consistence of paint, is used to remove old and blistered paint. Napthaline.—A moth preventive, effective only after eggs and grubs already present have been removed. Used in the storage of blankets, et cetera. Polish, Gibson’s Soap.—A metal polish issued interchangeably with lavaline. Used on bits and collars. Paint, rubberine.—Used in connection with loading ammunition in accordance with instructions regarding the same. Primer, brown enamel.—A hard, quick drying enamel used for painting parts of horse collars, draft springs, et cetera. Sal Soda, Bicarbonate of Soda.—A saturated solution of soda and water makes an alkaline solution that will not rust. The solution must be saturated, that is, at least 20% or one-fourth pound of soda (6 heaping spoonfuls to one cup of water). This solution is an effective solvent of powder fouling and should always be used after firing, whether metal fouling solution is to be used or not. It reduces the labor of cleaning with oil alone by more than half. Used also in a weaker solution (one-half pound to 8 quarts of water) in washing surfaces to be painted and to remove dirt and grease. Soap, H. H.—A neutral naphtha soap used in washing blankets, web and cloth equipment. Applied in the form of a solution (one cake
- 43. to 9 cups of hot water). If for any cause this soap is not obtainable, a good laundry soap (ivory or equal) may be used, but in no case should yellow soap containing a large percentage of alkali be used. Soap, castile.—An alkaline soap used in cleaning leather equipment. Applied on a moistened sponge. Soap, saddle (Hollingshead).—A soap used as a dressing for leather equipment. Applied with a thick lather on a moistened sponge. Swabbing solution, contains.—Ammonium persulphate, 60 grains or one half spoonful smoothed off. Ammonia 28%, 6 oz. or ⅜ of a pint or 12 spoonfuls. Water, 4 oz. or ¼ pint or 8 spoonfuls. Dissolve the ammonium persulphate in the water and add the ammonia. Keep in a tightly corked bottle. Pour out only what is necessary at a time and keep the bottle corked. TOOLS AND ACCESSORIES. In the repair of all equipment, it is literally true that “a stitch in time saves nine,” and that a timely repair will save the entire article. Tool Kits will be kept complete and serviceable; edges of cold chisels free from nicks; drifts and punches properly shaped immediately after using; and files kept clean. To prevent unscrewing, copper wire is used to lash nuts and other threaded parts which are not secured by split pins. Contents of Leather Pouch for Spare Parts (carried in Trail Boxes of 3-inch Guns):— For Breechblock— 50 Split pins 1 Block latch and spring 1 Firing pin and spring 1 Firing pin sleeve 2 Handy oilers, 5-16 inch Hinge pin catch
- 44. 1 1 Lever latch spring 1 Locking bolt, nut and pin 1 Locking bolt spring 2 Oil hole covers with screws 1 Pallet pin 1 Sear 2 Trigger shaft detent For Hand Fuze Setter— 2 Corrector scale screws 1 Guide plate lock screw 2 Index bar screws 1 Index plunger and spring 2 Oil hole screws 1 Range index 3 Range ring screws 1 Stop pin screw For Bracket Fuze Setter— 1 Corrector scale screw 3 Guide screws 4 Housing screws split washers 1 Knob washer 1 Range worm crank knob taper pin 1 Range worm crank handle 2 Range ring screws 3 Split pins (0.125) 1 Spring and spring cover with screw 2 Stop pins with rivets For Cylinder— 1 Drain-plug, cylinder 1 Elevating traversing lock spring 1 Filling plug (piston rod)
- 45. 5 Rings Garlock packing, ¾ in Special Wrenches, Spanners, other tools and accessories will be used only for the purposes for which they are intended. This purpose is usually stamped upon the tool. In assembling or disassembling parts of the materiel, no part will be struck directly with a hammer. If force is necessary, a piece of wood or copper should be interposed between the hammer and the part struck. All nuts are provided with split pins as keepers. A pair of wire cutting pliers is provided for use in pulling split pins, cutting wire lashings, etc. When a nut is assembled the split pin should always be inserted and properly opened. Axes, hatchets, picks, pick mattocks and shovels are carried on the carriage for use in the field and will not be put to other uses. The working edges will be kept bright and lightly oiled, the edges being sharpened if intended for cutting, or smooth if intended for digging. Deformed blades, edges or points should be straightened at the anvil and forge or in a vise. Shovel points are straightened with a hammer on a block of wood. The side edges of shovel blades should not be used as a mattock, as such treatment will deform the blade. In the field, split handles should be wrapped with a cord until they can be replaced by new handles. Canvas Buckets are used for watering animals, for washing carriages and equipment. Whenever possible, they should be dried before folding and replacing in the holders on the carriages. A rip or hole may be patched and made practically water-tight by a coat of shellac. Lanterns are used for illuminating purposes in the field only. Paulins are used to cover the harness and guns when in the field or in park. On the march they are carried on the carriages, being folded to serve as seat cushions. Holes and tears should always be properly sewed, stitched or darned as soon as practicable.
- 46. Picket Ropes are used in the field as drag ropes for the carriages or as picket lines for the animals. The ropes must be in a serviceable condition and free from knots. To keep them in a serviceable condition, splicing may often be necessary. CARE AND CLEANING OF THE DIFFERENT PARTS OF THE CARRIAGES. To disassemble and to clean the cylinder.—For cleaning, the cylinder is dismounted and emptied and the cylinder head, counter recoil buffer, and piston rod removed. The interior of the cylinder, the piston, the counter recoil buffer and the stuffing box should then be thoroughly cleaned by the use of cotton waste. The removal of the packing is not necessary in cleaning. The cylinder bore should be carefully inspected, and if any rust has formed it should be removed with coal oil, using if necessary, fine emery cloth. The latter must be used with great care to prevent any increase in the clearance between the piston and the cylinder. If rubbing, burring, or scoring of the parts is noted, the rough spots should be carefully smoothed down by a skilled workman with a dead smooth file or with emery cloth, and the cause of the roughness ascertained and removed. Where unusual rubbing or scoring has occurred, the facts will be reported to the Officer of the Ordnance Department charged with the duty of keeping the battery in repair, for his information and action. The exterior of the cylinder should be kept well oiled and free from rust and dirt, and an inspection made at least once a month to ascertain its condition. Where rust has formed it should be removed with coal oil, and, if necessary, emery cloth. For shipment or storage, or where the carriage is to stand without firing for extended periods, the cylinder should be coated with the light slushing oil used for the bore of guns. To fill the recoil cylinder.—If the cylinder is not completely filled, loss of stability will occur and higher stresses than normal will be developed in the carriage. For this reason the cylinder should be filled with the greatest care, a commissioned should, himself, verify that the cylinder is full and that no air is left in it, exception of the void
- 47. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookfinal.com