Table Of ContentIntermediate
Mathematica I
Statistics
Intermediate
Mathematica I
Statistics
G.P. BEAUMONT
Senior Lecturer, Department of Statistics and Computer Science,
Royal Holloway College. London
CHAPMAN AND HALL
LONDON AND NEW YORK
First published 1980
by Chapman and Hall Ltd
11 New Fetter Lane, London EC4P 4EE
© G. P. Beaumont 1980
ISBN-13: 978-0-412-15480-5 e-ISBN-13: 978-94-009-5794-7
DOl: 10.1007/978-94-009-5794-7
Published in the USA
by Chapman and Hall
in association with Methuen, Inc.
733 Third Avenue, New York, NY 10017
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British Library Cataloguing in Publication Data
Beaumont, G P
Intermediate mathematical statistics.
I. Mathematical statistics
I. Title
519.5 QA276 79-4061
Contents
Preface Page IX
Acknowledgements XI
Notation XII
Standard Distributions XIV
Introduction
1 Sufficiency
1.1 Introduction 9
1.2 Factorization criterion 13
1.3 Distribution of statistics conditional on
a sufficient statistic 16
1.4 Joint sufficiency 16
1.5 Minimal sufficiency 19
2 Unbiased point estimators
2.1 Introduction 23
2.2 Rao-Blackwell theorem 27
2.3 The role of sufficient statistics 29
2.4 Completeness 31
2.5 Joint completeness 33
2.6 Sufficiency, completeness and independence 36
2.7 Minimum-variance bounds 40
2.8 Computation of a minimum-variance bound 44
2.9 Minimum attainable variance 45
2.10 Mean square error 47
2.11 Two parameters 48
VI Contents
3 Elementary decision theory and
Bayesian methods
3.1 Comments on classical techniques 52
3.2 Loss functions 53
3.3 Decision theory 55
3.4 Bayes decisions 59
3.5 Using data 63
3.6 Computing posterior distributions 67
3.7 Conjugate distributions 72
3.8 Distribution of the next observation 74
3.9 More than one parameter 75
3.10 Decision functions 77
3.11 Bayes estimators 80
3.12 Admissibility 82
4 Methods of estimation
4.1 Introduction 85
4.2 Maximum likelihood estimation 86
4.3 Locating the maximum likelihood estimator 87
4.4 Estimation of a function of a parameter 88
4.5 Truncation and censoring 90
4.6 Estimation of several parameters 92
4.7 Approximation techniques 94
4.8 Large-sample properties 97
4.9 Method of least squares 100
4.10 Normal equations 104
4.11 Solution of the normal equations (non-singular
case) 106
4.12 Use of matrices 107
4.13 Best unbiased linear estimation 108
4.14 Covariance matrix 111
4.15 Relaxation of assumptions 113
5 Hypothesis testing I
5.1 Introduction 116
5.2 Statistical hypothesis 118
5.3 Simple null hypothesis against simple alternative 120
5.4 Applications of the Neyman-Pearson theorem 125
5.5 Uniformly most powerful tests for a single
parameter 130
5.6 Most powerful randomized tests 132
Contents vii
5.7 Hypothesis testing as a decision process 133
5.8 Minimax and Bayes tests 138
6 Hypothesis testing II
6.1 Two-sided tests for a single parameter 140
6.2 Neyman-Pearson theorem extension (non-
randomized version) 141
6.3 Regular exponential family of distributions 145
6.4 Uniformly most powerful unbiased test of e = eo
e eo
against =1= 146
6.5 Nuisance parameters 149
6.6 Similar tests 150
6.7 Composite hypotheses-several parameters 155
6.8 Likelihood ratio tests 159
6.9 Bayes methods 165
6.10 Loss function for one-sided hypotheses 165
e eo e eo
6.11 Testing = against =1= 169
7 Interval estimation
7.1 One parameter, Bayesian confidence intervals 173
7.2 Two parameters, Bayesian confidence regions 175
7.3 Confidence intervals (classical) 177
7.4 Most selective limits 180
7.5 Relationship to best tests 181
7.6 Unbiased confidence intervals 182
7.7 Nuisance parameters 184
7.8 Discrete distributions 185
7.9 Relationship between classical and Bayesian
intervals 187
7.10 Large-sample confidence intervals 187
Appendix 1 Functions of random variables
A 1.1 Introduction 193
A1.2 Transformations: discrete distributions 194
A1.3 Continuous distributions 196
A 1.4 The order statistics 202
Appendix 2 The regular exponential family of
distributions
A2.1 Single parameter 207
A2.2 Several parameters 210
viii Contents
A2.3 The regular exponential family of bivariate
distributions 211
Further exercises 214
Brief solutions to further exercises 229
Further reading 242
Author index 243
Subject index 244
Preface
This book covers those basic topics which usually form the core of
intermediate courses in statistical theory; it is largely about estima
tion and hypothesis testing. It is intended for undergraduates
following courses in statistics but is also suitable preparatory read
ing for some postgraduate courses.
It is assumed that the reader has completed an introductory
course which covered probability, random variables, moments
and the sampling distributions. The level of mathematics required
does not go beyond first year calculus. In case the reader has not
acquired much facility in handling matrices, the results in least
squares estimation are first obtained directly and then given
an (optional) matrix formulation. If techniques for changing from
one set of variables to another have not been met, then the appendix
on these topics should be studied first. The same appendix contains
essential discussion of the order statistics which are frequently
used for illustrative purposes.
Introductory courses usually include the elements of hypothesis
testing and of point and interval estimation though the treatment
must perforce become rather thin since at that stage it is difficult to
provide adequate justifications for some procedures-plausible
though they may seem. This text discusses these important topics in
considerable detail, starting from scratch. The level is nowhere
advanced and proofs of asymptotic results are omitted. Methods
deriving from the Bayesian point of view are gradually introduced
and alternate with the more usual techniques.
Many illustrative examples have been included since the average
student typically grasps the import of a theorem by seeing how it is
Preface x
applied. Each chapter contains exercises which either give practice
in techniques or take a previous example a little further. At the
end of the book will be found a selection of typical questions for
which brief solutions are provided.
London, June 1979 G. P. B.
