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<title>Smooth Tests of Goodness
of Fit
Using R
second edition
by J.C.W. Rayner, O. Thas and D.J. Best
University of Newcastle, Australia
Ghent University, Belgium
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<p style="padding-top: 0pt; " class="paragraph_style">Smooth Tests of Goodness <br /></p>
<p style="padding-bottom: 0pt; " class="paragraph_style">of Fit<br />Using R<br /><span style="line-height: 22px; " class="style_1">second edition<br /><br /><br /></span><span style="line-height: 22px; " class="style_2">by J.C.W. Rayner, O. Thas and D.J. Best</span><span style="line-height: 22px; " class="style_1"><br />University of Newcastle, Australia<br />Ghent University, Belgium<br /></span></p>
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<p style="padding-top: 0pt; " class="paragraph_style_1">Preface<span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1">1 Introduction <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>1.1 The Problem Defined<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>1.2 A Brief History of Smooth Tests <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>1.3 Monograph Outline <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>1.4 Examples <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">2 Pearson’s<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>Test <span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.1 Introduction <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.2 Foundations <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.3 The Pearson<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>Test - an Update<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.3.1 Notation, Definition of the Test, and Class Construction<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.3.2 Power related properties <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.3.3 The Sample Space Partition Approach<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.4<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>Tests of Composite Hypotheses <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>2.5 Examples <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">3 Asymptotically Optimal Tests <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>3.1 Introduction<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>3.2 The Likelihood Ratio, Wald, and Score Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>3.3 The Likelihood Ratio, Wald and Score Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>3.4 Generalised Score Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">4 Neyman Smooth Tests For Simple Null Hypotheses <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>4.1 Neyman’s<span style="line-height: 14px; " class="style_4"> </span>Ψ<span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>test <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>4.2 Neyman Smooth Tests for Uncategorised Simple Null Hypotheses<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>4.3 The Choice of Order<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>4.4 Examples<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>4.5 EDF Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">5 Neyman Smooth Tests for Categorised Simple Null Hypotheses <span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.1 Smooth tests for completely specified multinomials<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.2<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>Effective Order<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.3 Components of<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span><br /></p>
<p class="paragraph_style_1"> 5.3.1 Construction of the Components <span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.3.2 Power Study<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.3.3 Diagnostic Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.3.4 Cressie and Read Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.4 Examples<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.5 Class Construction<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.5.1 The Alternatives<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.5.2 Results of the Simulation Study<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.5.3 Discussion <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.6 A More Comprehensive Class of Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>5.7 Overlapping Cells Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">6 Smooth Tests for Uncategorised Composite Null Hypotheses<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.1 Smooth Tests for Uncategorised Composite Null Hypotheses<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.2 Smooth Tests for the Univariate Normal Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.2.1 The Construction of the Smooth Test<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.2.2 Simulation Study <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.2.3 Examples<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.2.4 Relationship with a Test of Thomas and Pierce <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.3 Smooth Tests for the Exponential Distribution <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.4 Smooth Tests for Multivariate Normal Distribution <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.5 Smooth Tests for the Bivariate Poisson Distribution <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.5.1 Definitions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.5.2 Score Tests for the Bivariate Poisson Model <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.5.3 A Smooth Covariance Test<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.5.4 Variance Tests <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.5.5 A Competitor for the Index of Dispersion Test<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.5.6 Revised Index of Dispersion and Crockett Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>6.6 Components of the Rao-Robson<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>Statistic<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">7 Smooth Tests for Categorised Composite Null Hypotheses<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>7.1 Neyman Smooth Tests for Composite Multinomials<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>7.2 Components of the Pearson-Fisher Statistic<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>7.3 Composite Overlapping Cells and Cell Focusing<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>7.4 Comparison of Pearson-Fisher and Rao-Robson<span style="line-height: 14px; " class="style_4"> </span>X<span style="line-height: 14px; " class="style_4"> </span><span style="line-height: 8px; " class="style_5">2</span><span style="line-height: 14px; " class="style_4"> </span>Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">8 Smooth Tests for Certain Discrete Distributions <span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.1 Smooth Tests for Discrete Composite Null Hypotheses<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.2 Smooth and EDF Tests for the Univariate Poisson Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.2.1 Definitions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.2.2 Size and Power Study<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.2.3 Examples <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.3 Smooth and EDF Tests for the Binomial Distribution <span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.3.1 Definitions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.3.2 Size and Power Study<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.3.3 Examples <br /></p>
<p class="paragraph_style_1"> 8.4 Smooth Tests for the Geometric Distribution <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.4.1 Definitions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.4.2 Size and Power Study<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>8.4.3 Examples <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">9 Generalised Smooth Tests: Theoretical Contributions <span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.1 Introduction <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.2 Smooth Test Statistics with Informative Decompositions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.2.1 Sufficient condition for ‘convenient’ test statistics<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.2.2 Testing for an exponential family of distributions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.2.3 Testing for distributions not from an exponential family of distributions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.3 Generalised Smooth Tests with Informative Decompositions <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.3.1 Uncategorized Distributions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.3.2 Categorized Distributions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.3.3 A Note on the Efficient Score Test<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.4 Efficiency <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.5 Diagnostic Component Tests<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.5.1 Are Smooth Tests and Their Components Diagnostic?<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.5.2 Properly Rescaled Tests <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.5.3 Rescaling Outside Exponential Families<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>9.5.4 A Simulation Study<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">10 Smooth Modelling <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.1 Introduction <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.2 Model Selection through Hypothesis Testing<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.2.1 Forward selection and Backward elimination <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.2.2 Smooth tests for improved models <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.2.3 Examples <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.3 Model Selection based on Loss Functions<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.3.1 Loss Functions and Expected Loss<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.3.2 AIC and BIC <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.4 Goodness of Fit Testing after Model Selection<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.4.1 Motivation<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.4.2 Theory<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.4.3 Examples <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.4.4 A final note<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>10.5 Correcting the Barton density<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">11 Generalised Smooth Tests for Composite Null Hypotheses<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>11.1 Introduction <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>11.2 Generalised Smooth Tests For The Logistic Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>11.3 Generalised Smooth Tests For The Laplace Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>11.4 Generalised Smooth Tests For The Extreme Value Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>11.5 Generalised Smooth Tests for the Negative Binomial Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>11.6 Generalised Smooth Tests For The ZIP Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>11.7 Generalised Smooth Tests For The Generalised Pareto Distribution<br /></p>
<p class="paragraph_style_1">A Orthonormal Polynomials and Recurrence Relations<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1">B Parametric Bootstrap p-Values Appendix<span style="line-height: 14px; " class="style_4"> <br /></span></p>
<p class="paragraph_style_1">C Some Details for Particular Distributions <span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>C.1 The One Parameter Logistic Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>C.2 The Two Parameter Logistic Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>C.3 The Zero-Inflated Poisson Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>C.4 The Laplace Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"><span style="line-height: 14px; " class="style_4"> </span>C.5 The Extreme Value Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"> C.6 The Negative Binomial Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p class="paragraph_style_1"> C.7 The Generalised Pareto Distribution<span style="line-height: 14px; " class="style_4"><br /></span></p>
<p style="padding-bottom: 0pt; " class="paragraph_style_1">Bibliography</p>
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