Distribution-Free Statistical Methods

1 Basic concepts in distribution-free methods -- 1.1 Introduction -- 1.2 Randomization and exact tests -- 1.3 Test statistics and estimating equations -- 1.4 Consistency in the one parameter case -- 1.5 Confidence limits -- 1.6 Efficiency considerations in the one parameter case -- 1.6.1 Estimation -- 1.6.2 Hypothesis testing -- 1.7 Estimation of standard errors -- 1.8 Multiple samples and parameters -- 1.8.1 Introduction -- 1.8.2 Point estimation -- 1.8.3 Hypothesis testing -- 1.8.4 Confidence regions -- 1.9 Normal approximations -- 1.9.1 The need for normal and related approximations -- 1.9.2 The central limit theorem -- 1.9.3 Sampling from finite populations -- 1.9.4 Linear rank statistics -- 2 One-sample location problems -- 2.1 Introduction -- 2.1.1 The mean -- 2.1.2 The median -- 2.1.3 Other measures of location -- 2.2 The median -- 2.2.1 The sign statistic -- 2.2.2 The null distribution of the sign statistic -- 2.2.3 Hypothesis testing -- 2.2.4 Confidence limits for ¸ -- 2.2.5 Point estimation of ¸ -- 2.2.6 Estimating the standard error of the sample median -- 2.2.7 Efficiency considerations -- 2.2.8 Computational notes -- 2.3 Symmetric distributions -- 2.4 The mean statistic -- 2.4.1 Hypothesis testing -- 2.4.2 Confidence limits -- 2.4.3 Normal approximations -- 2.4.4 Point estimation and efficiency considerations -- 2.4.5 Computational note -- 2.5 The Wilcoxon signed rank statistic -- 2.5.1 The null distribution of W -- 2.5.2 Hypothesis testing -- 2.5.3 Confidence limits -- 2.5.4 Point estimation based on W -- 2.5.5 Efficiency -- 2.5.6 Estimating the variance of the Hodges-Lehmann estimate -- 2.5.7 Computational notes -- 2.6 Other rank based transformations -- 2.6.1 Scores based directly on ranks -- 2.6.2 The null distribution of w, -- 2.6.3 Hypothesis testing -- 2.6.4 Confidence limits -- 2.6.5 Point estimation -- 2.6.6 Efficiency -- 2.6.7 Optimum rank statistics -- 2.7 Robust transformations -- 2.8 M-estimates -- 2.8.1 Hypothesis testing and confidence limits -- 2.8.2 Point estimation and efficiency -- 2.9 M-estimation and scaling -- 2.9.1 Hypothesis testing -- 2.9.2 Point estimation and confidence limits -- 2.9.3 Estimating the variance of an M -estimate -- 2.10 L-estimates -- 2.11 Ties -- 2.12 Asymmetric distributions: M-estimates -- 3 Miscellaneous one-sample problems -- 3.1 Introduction -- 3.2 Dispersion: the interquartile range -- 3.2.1 Symmetric F, known location -- 3.2.2 General F -- 3.3 The sample distribution function -- 3.3.1 One-sided confidence bands for F -- 3.4 Estimation of densities -- 3.4.1 Estimation of F when some observations are censored -- 3.4.2 The actuarial method of estimating F -- 3.4.3 The product-limit estimate ofF -- 3.5 Paired comparisons -- 3.5.1 Signed rank tests -- 3.5.2 Sign tests -- 4 Two-sample problems -- 4.1 Types of two-sample problems -- 4.2 The basic randomization argument -- 4.3 Inference about location difference -- 4.3.1 Introduction -- 4.3.2 The two-sample mean statistic -- 4.3.3 The two-sample sign statistic -- 4.3.4 The two-sample rank sum statistic -- 4.3.5 1vo-sample transformed rank statistics -- 4.3.6 Robust transformations in the two-sample case -- 4.4 Multiplicative models -- 4.5 Proportional hazards (Lehmann alternative) -- 4.5.1 The Wilcoxon statistic and inference about a -- 4.5.2 The 'log-rank' test and inference about a -- 4.5.3 Conditional likelihood and the log-rank test -- 4.5.4 The log-rank test and censored observations -- 4.6 Dispersion alternatives -- 4.6.1 A randomized exact test of dispersion -- 4.6.2 Comparing interquartile ranges -- 4.6.3 Rank test for dispersion -- 5 Straight line regression -- 5.1 The model and some preliminaries -- 5.2 Inference about f3 only -- 5.2.1 Inference based on untransformed residuals -- 5.2.2 Rank transformation of residuals -- 5.2.3 Sign transformation -- 5.2.4 Optimal weights for statistics of type T -- 5.2.5 Theil's statistic, Kendall's rank correlation -- 5.2.6 Robust transfonnations -- 5.2.7 Computational notes -- 5.3 Joint inference about a and fJ -- 5.3.1 Median regression -- 5.3.2 Symmetric untransfonned residuals -- 5.3.3 Symmetric residuals: signed rank method -- 5.3.4 Symmetric residuals: scores based on ranks -- 5.3.5 Symmetric residuals: robust transformations -- 6 Multiple regression and general linear models -- 6.1 Introduction -- 6.2 Plane regression: two independent variables -- 6.2.1 Inference about slopes: joint conditional distributiom -- ~ 6.3 Rank statistics for slopes -- 6.4 Sign statistics for slopes -- 6.4.1 Joint confidence regions -- 6.4.2 Point estimation -- 6.4.3 Consistency and efficiency -- 6.4.4 Estimating the covariance matrix -- 6.5 Inference about intercepts and slopes -- 6.5.1 Sign statistics -- 6.5.2 Symmetric residuals -- 6.5.3 Signed ranks -- 6.5.4 Robust transforms -- 6.6 General linear models -- 6.7 Inference about slopes only -- 6.7.1 Mean statistics -- 6.7.2 One-way analysis of variance -- 6.7.3 Randomized blocks: two-way analysis of variance -- 6.8 Exact inference using restricted randomization -- 6.8.1 Inference about individual regression coefficients -- 6.8.2 Rank transfonnations -- 6.8.3 Grouping and restricted randomization -- 7 Bivariate problems -- 7.1 Introduction -- 7.2 Tests of correlation -- 7.2.1 Conditional pennutation tests: the product moment correlation coefficient -- 7.2.2 Rank transfonnation: Spearman correlation -- 7.2.3 Sign transfonnation -- 7.2.4 Kendall's 1" and Theil's statistic -- 7.2.5 Mean square successive difference test -- 7.2.6 Contingency tables, correlation ratios -- 7.2.7 Computational notes -- 7.3 One-sample location -- 7.3.1 Medians -- 7.3.2 Hypothesis testing -- 7.3.3 Confidence regions -- 7.3.4 Point estimation -- 7.3.5 Symmetric distributions -- 7.3.6 Hypothesis testing -- 7.3.7 Confidence regions -- 7.3.8 Point estimation -- 7.3.9 Symmetric distributions: transfonnation of observations -- 7.3.10 Symmetric distributions: sign statistics -- 7.3.11 Symmetric distributions: rank statistics -- 7.3.12 Hypothesis testing and confidence limits -- 7.3.13 Point estimation -- 7.4 Two-sample location problems -- 7.4.1 Introduction: randomization -- 7.4.2 Medians and sign tests -- 7.4.3 Testing a specified (Bx. By) -- 7.4.4 Confidence regions and point estimation -- 7.4.5 Mean statistics -- 7.4.6 Hypothesis testing and confidence limits -- 7.4.7 Point estimation -- 7.4.8 Alternative calculation of QA -- 7.4.9 Rank statistics -- 7.4.10 Confidence regions, point estimation -- 7.4.11 Other transfonnations of (u;, v;) -- 7.5 Three-sample location problems -- 8 Miscellaneous complements -- 8.1 Linearization representation -- 8.2 Asymptotic relative efficiency -- 8.3 Estimating equations and the smoothing of statistics -- 8.4 Least squares smoothing -- 8.5 Kernel gradient estimates -- 8.5.1 Kernel density estimation -- 8.5.2 Estimating the mean density -- 8.6 Bootstrap estimation of standard errors -- 8.7 Conditional standard errors -- 8.7.1 Introduction and definitions -- 8.7.2 Large sample calculations -- References -- Index.