Preface

1 Introduction

    1.1 The Problem Defined

    1.2 A Brief History of Smooth Tests

    1.3 Monograph Outline

    1.4 Examples

2 Pearson’s X 2 Test

    2.1 Introduction

    2.2 Foundations

    2.3 The Pearson X 2 Test - an Update

        2.3.1 Notation, Definition of the Test, and Class Construction

        2.3.2 Power related properties

        2.3.3 The Sample Space Partition Approach

    2.4 X 2 Tests of Composite Hypotheses

    2.5 Examples

3 Asymptotically Optimal Tests

    3.1 Introduction

    3.2 The Likelihood Ratio, Wald, and Score Tests

    3.3 The Likelihood Ratio, Wald and Score Tests

    3.4 Generalised Score Tests

4 Neyman Smooth Tests For Simple Null Hypotheses

    4.1 Neyman’s Ψ2 test

    4.2 Neyman Smooth Tests for Uncategorised Simple Null Hypotheses

    4.3 The Choice of Order

    4.4 Examples

    4.5 EDF Tests

5 Neyman Smooth Tests for Categorised Simple Null Hypotheses

    5.1 Smooth tests for completely specified multinomials

    5.2 X 2 Effective Order

    5.3 Components of X 2

        5.3.1 Construction of the Components

        5.3.2 Power Study

        5.3.3 Diagnostic Tests

        5.3.4 Cressie and Read Tests

    5.4 Examples

    5.5 Class Construction

        5.5.1 The Alternatives

        5.5.2 Results of the Simulation Study

        5.5.3 Discussion

    5.6 A More Comprehensive Class of Tests

    5.7 Overlapping Cells Tests

6 Smooth Tests for Uncategorised Composite Null Hypotheses

    6.1 Smooth Tests for Uncategorised Composite Null Hypotheses

    6.2 Smooth Tests for the Univariate Normal Distribution

        6.2.1 The Construction of the Smooth Test

        6.2.2 Simulation Study

        6.2.3 Examples

        6.2.4 Relationship with a Test of Thomas and Pierce

    6.3 Smooth Tests for the Exponential Distribution

    6.4 Smooth Tests for Multivariate Normal Distribution

    6.5 Smooth Tests for the Bivariate Poisson Distribution

        6.5.1 Definitions

        6.5.2 Score Tests for the Bivariate Poisson Model

        6.5.3 A Smooth Covariance Test

        6.5.4 Variance Tests

        6.5.5 A Competitor for the Index of Dispersion Test

        6.5.6 Revised Index of Dispersion and Crockett Tests

    6.6 Components of the Rao-Robson X 2 Statistic

7 Smooth Tests for Categorised Composite Null Hypotheses

    7.1 Neyman Smooth Tests for Composite Multinomials

    7.2 Components of the Pearson-Fisher Statistic

    7.3 Composite Overlapping Cells and Cell Focusing X 2 Tests

    7.4 Comparison of Pearson-Fisher and Rao-Robson X 2 Tests

8 Smooth Tests for Certain Discrete Distributions

    8.1 Smooth Tests for Discrete Composite Null Hypotheses

    8.2 Smooth and EDF Tests for the Univariate Poisson Distribution

        8.2.1 Definitions

        8.2.2 Size and Power Study

        8.2.3 Examples

    8.3 Smooth and EDF Tests for the Binomial Distribution

        8.3.1 Definitions

        8.3.2 Size and Power Study

        8.3.3 Examples

    8.4 Smooth Tests for the Geometric Distribution        

        8.4.1 Definitions

        8.4.2 Size and Power Study

        8.4.3 Examples

9 Generalised Smooth Tests: Theoretical Contributions

    9.1 Introduction

    9.2 Smooth Test Statistics with Informative Decompositions

        9.2.1 Sufficient condition for ‘convenient’ test statistics

        9.2.2 Testing for an exponential family of distributions

        9.2.3 Testing for distributions not from an exponential family of distributions

    9.3 Generalised Smooth Tests with Informative Decompositions

        9.3.1 Uncategorized Distributions

        9.3.2 Categorized Distributions

        9.3.3 A Note on the Efficient Score Test

    9.4 Efficiency

    9.5 Diagnostic Component Tests

        9.5.1 Are Smooth Tests and Their Components Diagnostic?

        9.5.2 Properly Rescaled Tests

        9.5.3 Rescaling Outside Exponential Families

        9.5.4 A Simulation Study

10 Smooth Modelling

    10.1 Introduction

    10.2 Model Selection through Hypothesis Testing

        10.2.1 Forward selection and Backward elimination

        10.2.2 Smooth tests for improved models

        10.2.3 Examples

    10.3 Model Selection based on Loss Functions

        10.3.1 Loss Functions and Expected Loss

        10.3.2 AIC and BIC

    10.4 Goodness of Fit Testing after Model Selection

        10.4.1 Motivation

        10.4.2 Theory

        10.4.3 Examples

        10.4.4 A final note

    10.5 Correcting the Barton density

11 Generalised Smooth Tests for Composite Null Hypotheses

    11.1 Introduction

    11.2 Generalised Smooth Tests For The Logistic Distribution

    11.3 Generalised Smooth Tests For The Laplace Distribution

    11.4 Generalised Smooth Tests For The Extreme Value Distribution

    11.5 Generalised Smooth Tests for the Negative Binomial Distribution

    11.6 Generalised Smooth Tests For The ZIP Distribution

    11.7 Generalised Smooth Tests For The Generalised Pareto Distribution

A Orthonormal Polynomials and Recurrence Relations

B Parametric Bootstrap p-Values Appendix

C Some Details for Particular Distributions

    C.1 The One Parameter Logistic Distribution

    C.2 The Two Parameter Logistic Distribution

    C.3 The Zero-Inflated Poisson Distribution

    C.4 The Laplace Distribution

    C.5 The Extreme Value Distribution

    C.6 The Negative Binomial Distribution

    C.7 The Generalised Pareto Distribution

Bibliography