# An Introduction to Nonparametric Statistics

Probability BackgroundProbability Distributions for ObservationsGaussian DistributionUniform DistributionLaplace DistributionCauchy DistributionLogistic DistributionExponential DistributionLocation and Scale FamiliesSampling DistributionsBinomial Distribution˜2-distributionT-distributionF-distributionElementary Tasks in Frequentist InferenceHypothesis TestingOne-Sided Hypothesis TestsTwo-Sided Hypothesis TestsP-valuesConfidence IntervalsP-value InversionTest Inversion with Pivotal StatisticsA Problematic ExampleExercisesOne-Sample Nonparametric InferenceParametric Inference on MeansEstimation Using AveragesOne-Sample Testing for Gaussian ObservationsThe Need for Distribution-Free TestsOne-Sample Median MethodsEstimates of the Population MedianHypothesis Tests Concerning the Population MedianCon dence Intervals for the MedianInference for Other QuantilesComparing TestsPower, Sample Size, and Effect SizePowerSample and Effect SizesEffciency CalculationsExamples of Power CalculationsDistribution Function EstimationExercisesTwo-Sample TestingTwo-Sample Approximately Gaussian InferenceTwo-Sample Approximately Gaussian Inference on ExpectationsApproximately Gaussian Dispersion InferenceGeneral Two-Sample Rank TestsNull Distributions of General Rank StatisticsMoments of Rank StatisticsA First Distribution-Free TestThe Mann-Whitney-Wilcoxon TestExact and Approximate Mann-Whitney ProbabilitiesMoments and Approximate NormalityOther Scoring SchemesUsing Data as Scores: the Permutation TestEmpirical Levels and Powers of Two-Sample TestsAdaptation to the Presence of Tied ObservationsMann-Whitney-Wilcoxon Null HypothesesE ciency and Power of Two-Sample TestsEfficacy of the Gaussian-Theory TestEfficacy of the Mann-Whitney-Wilcoxon TestSummarizing Asymptotic Relative EfficiencyPower for Mann-Whitney-Wilcoxon TestingTesting Equality of DispersionTwo-Sample Estimation and Con dence IntervalsInversion of the Mann-Whitney-Wilcoxon TestTests for Broad AlternativesExercisesMethods for Three or More GroupsGaussian-Theory MethodsContrastsMultiple ComparisonsGeneral Rank TestsMoments of General Rank SumsConstruction of a Chi-Square-Distributed StatisticThe Kruskal-Wallis TestKruskal-Wallis Approximate Critical ValuesOther Scores for Multi-Sample Rank Based TestsMultiple ComparisonsOrdered AlternativesPowers of TestsPower of Tests for Ordered AlternativesPower of Tests for Unordered AlternativesE ciency CalculationsOrdered AlternativesUnordered AlternativesExercisesGroup Differences with BlockingGaussian Theory ApproachesPaired ComparisonsMultiple Group ComparisonsNonparametric Paired ComparisonsEstimating the Population Median Di erenceCon dence IntervalsSigned-Rank Statistic Alternative DistributionTwo-Way Non-Parametric Analysis of VarianceDistribution of Rank SumsA Generalization of the Test of FriedmanThe Balanced CaseThe Unbalanced CaseMultiple Comparisons and ScoringTests for a Putative Ordering in Two-Way LayoutsExercisesBivariate MethodsParametric ApproachPermutation InferenceNonparametric CorrelationRank CorrelationAlternative Expectation of the Spearman CorrelationKendall'sBivariate Semi-Parametric Estimation via CorrelationInversion of the Test of Zero CorrelationInversion of the Pearson CorrelationInversion of Kendall's TInversion of the Spearman CorrelationExercisesMultivariate AnalysisStandard Parametric ApproachesMultivariate EstimationOne-Sample TestingTwo-Sample TestingNonparametric Multivariate EstimationEquivariance PropertiesNonparametric One-Sample Testing ApproachesMore General Permutation SolutionsCon dence Regions for a Vector Shift ParameterTwo-Sample MethodsHypothesis TestingPermutation TestingPermutation Distribution ApproximationsExercisesDensity EstimationHistogramsKernel Density EstimatesExercisesRegression Function EstimatesStandard Regression InferenceKernel and Local Regression SmoothingIsotonic RegressionSplinesQuantile RegressionFitting the Quantile Regression ModelResistant RegressionExercisesResampling TechniquesThe Bootstrap IdeaThe Bootstrap Sampling SchemeUnivariate Bootstrap TechniquesThe Normal MethodBasic IntervalThe Percentile MethodBCa MethodSummary So Far, and More ExamplesBootstrapping Multivariate Data SetsRegression Models and the Studentized Bootstrap MethodFixed X BootstrapThe JackknifeExamples of Biases of the Proper OrderBias CorrectionCorrecting the Bias in Mean EstimatorsCorrecting the Bias in Quantile EstimatorsExercisesAppendix A: Analysis Using the SAS SystemAppendix B: Construction of Heuristic Tables and Figures Using RBibliography