Finite Form Representations for Meijer G and Fox H Functions

1. Front Matter

   Pages i-xviii

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2. Setting the Scene

   • Carlos A. Coelho, Barry C. Arnold

   Pages 1-4

3. The Meijer G and Fox H Functions

   • Carlos A. Coelho, Barry C. Arnold

   Pages 5-18

4. Multiple Products of Independent Beta Random Variables with Finite Form Representations for Their Distributions

   • Carlos A. Coelho, Barry C. Arnold

   Pages 19-27

5. Finite Form Representations for Extended Instances of Meijer G and Fox H Functions

   • Carlos A. Coelho, Barry C. Arnold

   Pages 29-70

6. Application of the Finite Form Representations of Meijer G and Fox H Functions to the Distribution of Several Likelihood Ratio Test Statistics

   • Carlos A. Coelho, Barry C. Arnold

   Pages 71-452

7. Mathematica^Ⓡ, Maxima, and R Packages to Implement the Likelihood Ratio Tests and Compute the Distributions in the Previous Chapter

   • Carlos A. Coelho, Barry C. Arnold

   Pages 453-490

8. Approximate Finite Forms for the Cases Not Covered by the Finite Representation Approach

   • Carlos A. Coelho, Barry C. Arnold

   Pages 491-505

9. Back Matter

   Pages 507-515

   PDF

This book depicts a wide range of situations in which there exist finite form representations for the Meijer G and the Fox H functions. Accordingly, it will be of interest to researchers and graduate students who, when implementing likelihood ratio tests in multivariate analysis, would like to know if there exists an explicit manageable finite form for the distribution of the test statistics. In these cases, both the exact quantiles and the exact p-values of the likelihood ratio tests can be computed quickly and efficiently.

The test statistics in question range from common ones, such as those used to test e.g. the equality of means or the independence of blocks of variables in real or complex normally distributed random vectors; to far more elaborate tests on the structure of covariance matrices and equality of mean vectors. The book also provides computational modules in Mathematica^®, MAXIMA and R, which allow readers to easily implement, plot and compute the distributions of any of these statistics, or any other statistics that fit into the general paradigm described here.

• tests on mean values and covariance structures
• likelihood ratio test statistics
• products of Beta distributed random variables
• finite representations of distributions for l.r.t. statistics
• Meijer G function
• Fox H function
• special functions
• Mathematica
• MAXIMA
• R
• exact distribution
• exact quantile
• exact p-value
• multivariate likelihood ratio tests
• multivariate analysis

• Department of Mathematics, Universidade NOVA de Lisboa, Caparica, Portugal

  Carlos A. Coelho

• Department of Statistics, University of California, Riverside, USA

  Barry C. Arnold

Carlos A. Coelho is a Full Professor at the Mathematics Department of Nova University of Lisbon, Portugal. He obtained his Ph.D. in Biostatistics from the University of Michigan, USA. An Elected Member of the International Statistical Institute, he is primarily pursuing research in the fields of mathematical statistics and distribution theory, e.g. exact and near-exact distributions for likelihood ratio statistics used in multivariate analysis.

Barry C. Arnold is a Distinguished Professor in the Department of Statistics at the University of California, Riverside, USA. He holds a Ph.D. in Statistics from Stanford University. He is a Fellow of both the American Statistical Association and the Institute of Mathematical Statistics and a former Elected Member of the International Statistical Institute. His research interests include multivariate models, inequality measurement and ordered data.

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