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E-raamat: Finite Form Representations for Meijer G and Fox H Functions: Applied to Multivariate Likelihood Ratio Tests Using Mathematica(R), MAXIMA and R

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  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Statistics 223
  • Ilmumisaeg: 13-Dec-2019
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030287900
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Finite Form Representations for Meijer G and Fox H Functions: Applied to Multivariate Likelihood Ratio Tests Using Mathematica(R), MAXIMA and RSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Statistics 223
  • Ilmumisaeg: 13-Dec-2019
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030287900
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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.

1 Setting the Scene
1(4)
References
3(2)
2 The Meijer G and Fox H Functions
5(14)
2.1 The Meijer G and Fox H Functions as Inverse Mellin Transforms
5(5)
2.2 Straightforward Instances of Meijer G and Fox H Functions with Alternative Finite Representations
10(6)
References
16(3)
3 Multiple Products of Independent Beta Random Variables with Finite Form Representations for Their Distributions
19(10)
3.1 A First Multiple Product and Its Particular Case of Interest
19(5)
3.2 A Second Multiple Product
24(2)
3.3 Applications of the Results in Theorems 3.1--3.3
26(1)
References
27(2)
4 Finite Form Representations for Extended Instances of Meijer G and Fox H Functions
29(42)
4.1 Two Corollaries Based on Theorems 3.1 and 3.2 in the Previous
Chapter
29(6)
4.2 A Third Corollary Based on Theorem 3.3 in the Previous
Chapter
35(3)
4.3 Further Corollaries Based on Combinations of Theorems 3.1--3.3
38(4)
4.4 Huge Gains in Computation Times
42(27)
References
69(2)
5 Application of the Finite Form Representations of Meijer G and Fox H Functions to the Distribution of Several Likelihood Ratio Test Statistics
71(382)
5.1 Likelihood Ratio Test Statistics Whose Distributions Correspond to the Products in Theorems 3.1 and 3.2 and That Have p.d.f. and c.d.f. Given by Corollary 4.1 or 4.2
72(287)
5.1.1 The Likelihood Ratio Statistic to Test the Equality of Mean Vectors for Real Random Variables [ EqMean VecR]
73(27)
5.1.2 The Test for Simultaneous Nullity of Several Mean Vectors (Real r.v.'s) [ NullMean VecR]
100(3)
5.1.3 The Parallelism Test for Profiles (Real r.v.'s) [ ProfParR]
103(9)
5.1.4 The Likelihood Ratio Statistic to Test Hypotheses on an Expected Value Matrix (Real r.v.'s) [ MatEVR]
112(34)
5.1.5 The Likelihood Ratio Statistic to Test the Equality of Mean Vectors for Complex Random Variables [ EqMean VecC]
146(4)
5.1.6 The Test for Simultaneous Nullity of Several Mean Vectors (Complex r.v.'s) [ NullMean VecC]
150(2)
5.1.7 The Parallelism Test for Profiles (Complex r.v.'s) [ ProfParC]
152(3)
5.1.8 The Likelihood Ratio Statistic to Test Hypotheses on an Expected Value Matrix (Complex r.v.'s) [ MatEVC]
155(8)
5.1.9 The Likelihood Ratio Statistic to Test the Independence of Two Groups of Real Random Variables [ Ind2R]
163(75)
5.1.10 The Likelihood Ratio Statistic to Test the Independence of Two Groups of Complex Variables [ Ind2C]
238(9)
5.1.11 The Likelihood Ratio Statistic to Test the Independence of Several Groups of Real Variables [ IndR]
247(21)
5.1.12 The Likelihood Ratio Statistic to Test the Independence of Several Groups of Complex Variables [ IndC]
268(3)
5.1.13 A Test for Outliers (Real r.v.'s) [ OutR]
271(7)
5.1.14 A Test for Outliers (Complex r.v.'s) [ OutC]
278(2)
5.1.15 Testing the "Symmetrical Equivalence" of Two Sets of Real Random Variables [ SymEqR]
280(6)
5.1.16 Testing the "Complete Symmetrical Equivalence" of Two Sets of Real Random Variables [ CompSymEqR]
286(8)
5.1.17 Testing for "Symmetrical Spherical Equivalence" or Independent Two-Block Compound Symmetry [ SymSphEq]
294(10)
5.1.18 Testing for "Complete Symmetrical Spherical Equivalence" [ CompSymSphEq]
304(8)
5.1.19 Testing the "Symmetrical Equivalence" of Two Sets of Complex Random Variables [ SywEqC]
312(1)
5.1.20 Testing the "Complete Symmetrical Equivalence" of Two Sets of Complex Random Variables [ CompSymEqC]
313(3)
5.1.21 The Likelihood Ratio Statistic to Test Scalar Block Sphericity for Blocks of Two Variables [ BSSph]
316(7)
5.1.22 The Likelihood Ratio Test for Equality of Mean Vectors, Under the Assumption of Circularity of the Covariance Matrices [ EqMean VecCirc]
323(7)
5.1.23 The Likelihood Ratio Test for Simultaneous Nullity of Mean Vectors, Under the Assumption of Circularity of the Covariance Matrices [ NuUMean VecCirc]
330(5)
5.1.24 The Likelihood Ratio Test for Equality of Mean Vectors, Under the Assumption of Compound Symmetry of the Covariance Matrices [ EqMean VecCS]
335(5)
5.1.25 The Likelihood Ratio Test for Simultaneous Nullity of Mean Vectors, Under the Assumption of Compound Symmetry of the Covariance Matrices [ NuIlMean VecCS]
340(3)
5.1.26 Brief Note on the Likelihood Ratio Test for Equality of Mean Vectors, Under the Assumption of Sphericity of the Covariance Matrices [ EqMean VecSph]
343(3)
5.1.27 Brief Note on the Likelihood Ratio Test for Simultaneous Nullity of Mean Vectors, Under the Assumption of Sphericity of the Covariance Matrices [ NullMean VecSph]
346(2)
5.1.28 The Likelihood Ratio Test for Profile Parallelism Under the Assumption of Circularity of the Covariance Matrices [ ProfParCirc]
348(4)
5.1.29 Brief Note on the Likelihood Ratio Test for Profile Parallelism Under the Assumption of Compound Symmetry of the Covariance Matrices [ ProfParCS]
352(4)
5.1.30 Brief Note on the Likelihood Ratio Test for Profile Parallelism Under the Assumption of Sphericity of the Covariance Matrices [ ProfParSph]
356(3)
5.2 Likelihood Ratio Test Statistics Whose Distributions Correspond to the Product in Theorem 3.3 and That Have p.d.f. and c.d.f. Given by Corollary 4.3
359(18)
5.2.1 The Likelihood Ratio Statistic to Test Circularity of the Covariance Matrix [ CircOddp]
359(5)
5.2.2 The Likelihood Ratio Statistic to Test Simultaneously the Circularity of the Covariance Matrix and the Equality of the Means (for an Odd Number of Variables) [ CircMeanOddp]
364(4)
5.2.3 The Likelihood Ratio Statistic to Test the Simultaneous Circularity of m Covariance Matrices [ CircS]
368(4)
5.2.4 The Likelihood Ratio Statistic to Test Simultaneously the Circularity of the Covariance Matrices and the Equality of Means in m Subsets with an Odd Number of Variables [ CircMeansOddp]
372(5)
5.3 Likelihood Ratio Test Statistics Whose Distributions Correspond to a Multiplication of the Products in Theorem 3.1 or 3.2 and in Theorem 3.3 and That Have p.d.f. and c.d.f. Given by Corollary 4.4 or 4.5
377(71)
5.3.1 The Likelihood Ratio Statistic to Test Simultaneously the Circularity of the Covariance Matrix and the Equality of the Means (for an Even Number of Variables) [ CircMeanEvenp]
378(2)
5.3.2 The 1.r.t. Statistic to Test Simultaneously the Circularity of the Covariance Matrices and the Equality of Means in m Subsets with an Even Number of Variables [ CircMeansEvenp]
380(2)
5.3.3 The 1.r.t. Statistic to Test Simultaneously the Circularity of the Covariance Matrices and the Equality of Means in m Subsets of Variables, Some with an Odd and the Other with an Even Number of Variables [ CircMeans]
382(1)
5.3.4 The 1.r.t. Statistic to Test Simultaneously the Independence of m Sets of Variables, the Circularity of Their Covariance Matrices and the Equality of Means, When All Sets Have an Even Number of Variables [ IndCircMeans]
383(4)
5.3.5 The 1.r.t. Statistic to Test Simultaneously the Independence of m Sets of Variables, the Circularity of Their Covariance Matrices and the Equality of Means, When All but One of the Sets Have an Even Number of Variables [ IndCircMeans1Odd]
387(2)
5.3.6 Testing for a Two-Block Independent Circular-Spherical Covariance Structure [ IndCircSph]
389(6)
5.3.7 Testing for a Two-Block Independent Circular-Spherical Covariance Structure and Equality of Means [ IndCircSphEqMean]
395(53)
References
448(5)
6 Mathematica®, Maxima, and R Packages to Implement the Likelihood Ratio Tests and Compute the Distributions in the Previous
Chapter
453(38)
6.1 Introduction
453(4)
6.2 Loading the Packages and Getting Help
457(4)
6.3 Modules to Read Data Files with One or More Samples
461(10)
6.3.1 Modules ReadFileR and ReadFileC
462(4)
6.3.2 Modules ReadFiledifpR and ReadFiledifpC
466(1)
6.3.3 Modules ReadFilelsR and ReadFilelsC
466(5)
6.4 Computational End-User Functions and Modules
471(19)
6.4.1 Modules to Compute the GIG and EGIG p.d.f. and c.d.f.
471(4)
6.4.2 Modules to Assist the Implementation of the Tests in Chap. 5
475(15)
References
490(1)
7 Approximate Finite Forms for the Cases Not Covered by the Finite Representation Approach
491(16)
7.1 Upper Bounds on the Error of the Approximations for the Meijer G Functions
497(6)
References
503(4)
Index 507
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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