The Gradient Test - 1st Edition - ISBN: 9780128035962, 9780128036136

The Gradient Test

1st Edition

Another Likelihood-Based Test

Authors: Artur Lemonte
eBook ISBN: 9780128036136
Paperback ISBN: 9780128035962
Imprint: Academic Press
Published Date: 2nd February 2016
Page Count: 156
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The Gradient Test: Another Likelihood-Based Test presents the latest on the gradient test, a large-sample test that was introduced in statistics literature by George R. Terrell in 2002. The test has been studied by several authors, is simply computed, and can be an interesting alternative to the classical large-sample tests, namely, the likelihood ratio (LR), Wald (W), and Rao score (S) tests.

Due to the large literature about the LR, W and S tests, the gradient test is not frequently used to test hypothesis. The book covers topics on the local power of the gradient test, the Bartlett-corrected gradient statistic, the gradient statistic under model misspecification, and the robust gradient-type bounded-influence test.

Key Features

  • Covers the background of the gradient statistic and the different models
  • Discusses The Bartlett-corrected gradient statistic
  • Explains the algorithm to compute the gradient-type statistic


The short book regarding the gradient test can be used for a graduate course. Additionally, it can be used by researchers who are interested in developing research in likelihood-based theory. The short book will link theory and practice and hence a lot of researchers of many areas can use the book as a reference for the gradient test.

Table of Contents

  • Dedication
  • List of Figures
  • List of Tables
  • Preface
  • Chapter 1: The Gradient Statistic
    • Abstract
    • 1.1 Background
    • 1.2 The Gradient Test Statistic
    • 1.3 Some Properties of the Gradient Statistic
    • 1.4 Composite Null Hypothesis
    • 1.5 Birnbaum-Saunders Distribution Under Type II Censoring
    • 1.6 Censored Exponential Regression Model
  • Chapter 2: The Local Power of the Gradient Test
    • Abstract
    • 2.1 Preliminaries
    • 2.2 Nonnull Distribution up to Order O(n−1/2)
    • 2.3 Power Comparisons Between the Rival Tests
    • 2.4 Nonnull Distribution Under Orthogonality
    • 2.5 One-Parameter Exponential Family
    • 2.6 Symmetric Linear Regression Models
    • 2.7 GLM With Dispersion Covariates
    • 2.8 Censored Exponential Regression Model
  • Chapter 3: The Bartlett-Corrected Gradient Statistic
    • Abstract
    • 3.1 Introduction
    • 3.2 Null Distribution up to Order O(n−1)
    • 3.3 The One-Parameter Case
    • 3.4 Models With Two Orthogonal Parameters
    • 3.5 Birnbaum-Saunders Regression Models
    • 3.6 Generalized Linear Models
  • Chapter 4: The Gradient Statistic Under Model Misspecification
    • Abstract
    • 4.1 Introduction
    • 4.2 The Robust Gradient Statistic
    • 4.3 Examples
    • 4.4 Numerical Results
  • Chapter 5: The Robust Gradient-Type Bounded-Influence Test
    • Abstract
    • 5.1 Introduction
    • 5.2 The Robust Gradient-Type Test
    • 5.3 Robustness Properties
    • 5.4 Algorithm to Compute the Gradient-Type Statistic
    • 5.5 Closing Remarks
  • Bibliography


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© Academic Press 2016
Academic Press
eBook ISBN:
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About the Author

Artur Lemonte

Artur J. Lemonte is a professor at Federal University of Pernambuco, Department of Statistics, Recife/PE, Brazil. He works on higher order asymptotics, mathematical statistics, regression models, parametric inference, and distribution theory. In the last years, he has published more than 60 papers in refereed

statistical journals (most of them about the gradient test).

Affiliations and Expertise

Department of Statistics, Federal University of Pernambuco, Recife/PE, Brazil


"...carrying out the asymptotic chi-square test as a gradient test based on ST is computationally less involved, because it avoids the computation or the estimation, respectively, of the Fisher information matrix of the model." --Zentralblatt MATH, The Gradient Test

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