HANDBOOK OF APPLICABLE MATHEMATICS

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1 HANDBOOK OF APPLICABLE MATHEMATICS Chief Editor: Walter Ledermann Volume VI: Statistics PART A Edited by Emlyn Lloyd University of Lancaster A Wiley-Interscience Publication JOHN WILEY & SONS Chichester - New York - Brisbane - Toronto - Singapore

2 PART A INTRODUCTION xvii 1. INTRODUCTION TO STATISTICS The meaning of the word 'statistics' Sampling distribution; statistic; estimate The subject-matter of this book Sampling distributions; distribution-free methods Estimates, tests, decisions Bayesian inference Multivariate analysis Time series Bibliography and references Appendix: statistical tables Some topics that are not covered Conventions and notations Mathematical conventions Statistical and probabilistic conventions and abbreviations 15 (i) Abbreviations; (ii) Nomenclature of standard distributions; (iii) capital letter convention for random variables; (iv) Notation for moments and related quantities; (v) Non-standard usages: induced r.v., statistical copy; (vi) An ambiguity in the meaning of P(A/K): 'the probability of A on K'; (vii) Nomenclature for tabular values: percentage points 1.5. Further reading SAMPLING DISTRIBUTIONS Moments and other statistics Statistic Moments 22 (a) Population moments; (b) Sample moments; (c) Sample variance and sample standard deviation; (d) Bivariate samples vii

3 viii 2.2. Sampling distributions; introductory definitions and examples Sampling moments of a statistic Lower sampling moments of the sample mean Lower sampling moments of the sample variance Sampling covariance between the sample mean x and the sample variance v Sampling moments of higher sample moments Sampling moments of the sample standard deviation Sampling moments of the sample skewness Distributions of sums of independent and identically distributed variables Sampling distributions of functions of Normal variables. 33' The Normal distribution Effect of a linear transformation; standardization Linear functions of Normal variables (a) Linear functions of independent Normals (b) Sampling distribution of the sample mean (c) Linear function of correlated Normals (d) Several linear functions of correlated Normals.. 37 (e) Independent linear functions of correlated Normals 38 (f) Independent linear functions of i.i.d. Normals Quadratic functions of Normal variables (a) The chi-squared distribution. Sum of squares of independent standard Normals. Quadratic forms in Normal variables 39 (b) Independence of the sample sum of squares and the sample mean in Normal samples 42 (c) Tables of the x 2 distribution 42 (d) Sampling distribution of the sample variance. 44 (e) Sampling distribution of the sample standard deviation Student's distribution (the 't distribution') The variance ratio (F) distribution 50 (a) F as a weighted ratio of X 2 variables (b) Relation between the F distribution and the Beta distribution 52 (c) Approximation to the F distribution when one degree of freedom is very much larger than the other The sample correlation coefficient The independence of quadratic forms: the Fisher- Cochran theorem; Craig's theorem The range and the Studentized range Approximate sampling distribution of x and of non-linear functions of x Approximation to sampling expectation and variance of

4 ix non-linear statistics; variance-stabilizing transformations; Normalizing transformations Approximations 61 (a) Functions of a single r.v. 61 (b) Functions of two random variables Variance-stabilizing transformations 64 (a) A general formula 64 (b) Poisson data: the square root transformation. 65 (c) Binomial data: the arc sine (or angular) transformation 66 (d) Stabilizing the variance of a regressed variable by weighting Normalizing transformations 67 (a) The log transform: positively skewed positive variables 67 (b) The log transform for variables bounded above and below; the inverse hyperbolic tangent transformation for the correlation coefficient 67 (c) Normalizing transforms of x 2 68 (d) The probability integral transform: probits Curve-straightening transformations A Student-distributed transform of the sample correlation coefficient when p = A x 2 -distributed transform of a uniformly distributed variable Non-central sampling distributions Non-central chi-squared Non-central F The non-central Student distribution The multinomial distribution in sampling distribution theory Binomial, Trinomial and multinomial (m) distributions (i) Binomial; (ii) Trinomial; (iii) Multinomial Properties of the Multinomial (m) 76 (a) Low-order moments 76 (b) Marginal distributions The multinomial as a conditional of the joint distribution "~ of independent Poisson variables Frequency tables Further reading and references ESTIMATION: INTRODUCTORY SURVEY The estimation problem Intuitive concepts and graphical methods Introduction 84

5 ; Frequency tables, histograms and the empirical c.d.f.. 84 (a) Discrete data 84 (b) Bar chart and histogram of discrete data (c) Continuous data 90 (d) Histogram of continuous data 92 (e) Sample analogue of the c.d.f.; Probability graph papers Some general concepts and criteria for estimates Introduction: Dimension, exchangeability, consistency, concentration 94 (a) Dimensions 95 (b) Exchangeability 95 (c) Consistency 96 (d) Concentration (high local probability) Unbiased estimates and minimum-variance unbiased estimates Efficiency: the Cramer-Rao ('C-R') bound (a) The C-R inequality: random sample from a univariate one-parameter distribution (b) Attainment of the C-R bound. Efficiency of an estimate 106 (c) Regularity conditions 108 (d) The C-R inequality for independent vector observations on a 1-parameter multivariate distribution. 109 (e) The C-R inequality for observations that are not independent and/or not identically distributed. Ill (f) A generalization of the C-R inequality to the case of several parameters. The information matrix. Ill 3.4. Sufficiency Definition of sufficiency The Factorization criterion and the exponential family Sufficiency and minimum-variance unbiased estimates 121 (a) The Rao-Blackwell theorem 121 (b) Minimum-variance unbiased estimates and sufficiency Sufficiency in the multi-parameter case Practical methods of constructing estimates: an introduction Graphical methods Minimum variance unbiased estimates and the method of least squares The method of moments The method of maximum likelihood Normal linear models in which maximum likelihood and least squares estimates coincide Further reading 136

6 xi 4. INTERVAL ESTIMATION Introduction: the problem An intuitive approach Standard error Probability intervals 140 (a) Probability intervals for continuous random variables 140 (b) Probability intervals for discrete random variables Confidence intervals and confidence limits Constructing a confidence interval by using a pivot Factors affecting the interpretation of a confidence interval as a measure of the precision of an estimate Confidence intervals when there are several parameters Individual confidence intervals Confidence intervals for functions of two parameters, including the difference between parameters and the ratio of parameters. (Fieller's theorem) Construction of confidence intervals without using pivots Approximate confidence intervals when the underlying distribution is discrete 168 (a) A crude approximation; (b) A better approximation Distribution-free confidence intervals for quantiles Multiparameter confidence regions Exact confident regions Elliptical confidence regions for the expectation vector of a bivariate Normal distribution; and approximate confidence regions for a pair of maximum likelihood estimates Confidence interval obtainable from a large sample, using the likelihood function The likelihood function Approximate confidence intervals, using the derivative of the log-likelihood Confidence intervals obtained with the aid of an (approximately) Normalizing transformation Confidence band for an unknown continuous cumulative distribution function The empirical c.d.f The Kolmogorov-Smirnov distance between the c.d.f. and the empirical c.d.f The sampling distribution of the Kolmogorov-Smirnov statistic; confidence limits for the c.d.f Tolerance intervals Likelihood intervals Likelihood 196 (a) Likelihood and log-likelihood: definitions and examples 196 (b) Likelihood and sufficiency 199

7 xii Plausible values and likelihood intervals (a) Equiplausible values of (b) One value of 6 more plausible than another. 200 (c) Conventionally implausible values of 6: likelihood intervals 201 (d) Likelihood intervals for 6 and for g{6) The two-parameter case Bayesian intervals Further reading and references STATISTICAL TESTS What is meant by a significance test? Introduction to tests involving a simple null hypothesis in discrete distributions A two-sided (Binomial) test: ingredients, procedure and interpretation 210 (a) Probability model 210 (b) Condensing the data: the test statistic (c) Null hypothesis, null distribution 211 (d) Alternative hypothesis 212 (e) Compatibility of sample with H (f) Significance set, significance level (significance probability). Critical region. 213 (f]) distance ordering 214 (f 2 ) probability ordering 214 (f 3 ) likelihood ratio ordering 216 (g) Interpretation of the significance level (h) Strength of evidence Conventional interpretation of significance levels; practices used in specifying significance levels; Critical region A one-sided (Binomial) test Tests with Poisson distributions Tests with continuous distributions Choosing a test statistic Criteria for a test Sensitivity function Sensitivity function of the one-tailed test when a is unknown, in a sample from a Normal (0, a-) distribution Sensitivity function of a two-tailed test Tests involving a composite null hypothesis Conditional tests: equality of Binomial proportions; ratio of Poisson parameters Tests for independence in 2x2 contingency tables; Fisher's 'Exact Test' 241

8 xiii 5.5. Tests involving more than one parameter; generalized likelihood ratio tests Large sample approximation to the significance level of a likelihood ratio test Randomization tests Standard tests with the Normal distribution model The significance of a sample mean when the variance is known The significance of a mean when the variance is unknown): Student's /-test The Fisher-Behrens test for the significance of the difference between two means The f-test for the significance of the difference between two means (variances being equal) The f-test for the significance of a regression coefficient in simple linear regression Test for the equality of two variances Tests for the equality of several means: introduction to the analysis of variance 266 (a) Introduction 266 (b) Comparison of two means as analysis of variance. 266 (c) The case of k samples Tests for Normality Test for the equality of k variances (Bartlett's test) Combining the results of several tests Neyman-Pearson theory Sampling inspection schemes The Neyman-Pearson theory of tests Further reading and references MAXIMUM-LIKELIHOOD ESTIMATES Introduction The method of maximum likelihood The likelihood and log-likelihood functions "The maximum likelihood estimate (MLE) MLE: intuitive justification MLE: kinds of maxima Theoretical justification: for the MLE properties of the likelihood function 297 (a) Asymptotic distribution of the derivative of the loglikelihood function 297 (b) Confidence interval for an estimated parameter: first method 299

9 xiv (c) The principal practical approximation for the sampling distribution of the MLE 302 (d) Confidence intervals and confidence regions: second method Estimating a function of 6. Invariance Examples of ML estimation in one-parameter cases Examples of MLE in multi-parameter cases Maximum likelihood estimation of linear regression coefficients The meaning of'regression' The MLE procedure for linear regression with weighted data Linear regression with equally weighted data (i) The estimate; (ii) Sampling properties of the estimates; (iii) Confidence intervals for a and j3; (iv) Significance tests for a and /3; (v) Confidence interval for y(x) 6.6. Estimation of dosage-mortality relationship Quantal responses Probability model The likelihood surface The maximum likelihood estimator Reliability of the estimates Dose required for a specified response Maximum likelihood estimates from grouped, censored or truncated data Discrete data 348 (i) Truncation; (ii) Grouping, censoring Continuous data: grouping. Sheppard's correction Further reading CHI-SQUARED TESTS OF GOODNESS OF FIT AND OF INDEPENDENCE AND HOMOGENEITY Validity of the model Introduction Adequacy of the linear regression model Karl Pearson's distance statistic: the \ 2 test Amalgamating low frequency cells: Cochran's criteria The x 2 test for continuous distributions Cross-classified frequency tables (contingency tables). Tests of independence and of homogeneity x2 tables; the special case of a single degree of freedom kxm tables Index of dispersion Index of dispersion for Binomial samples Index of dispersion for Poisson samples Further reading and references 381

10 xv 8. LEAST SQUARES ESTIMATION AND THE ANALYSIS OF VARIANCE Least squares estimation for general models Least squares estimation for full-rank linear models, Normal equations. The Gauss-Markov theorem Examples of Least Squares estimation Matrix formulation for linear models Properties of Least Squares estimates Residuals Orthogonal designs and orthogonal polynomials Modifications for observations of unequal accuracy; weighted Least Squares Modification for non-independent observations Analysis of variance and tests of hypotheses for full-rank designs Normal theory Confidence regions Testing hypotheses The basic identities Tests of hypotheses in orthogonal designs Hypothesis testing in non-orthogonal designs Group orthogonality The general linear hypothesis Further reading THE DESIGN OF COMPARATIVE EXPERIMENTS Historical introduction The tea-tasting lady A more elaborate example: Darwin's experiment (i) Blocking; (ii) Replication; (Hi) Balance; (iv) Randomization Complete randomized block designs Treatments at one level, and at several levels The need for devices to reduce block size Balanced incomplete blocks for comparing single-level treatments The complete factorial design Main effects and interaction: two factors at two levels Main effects and interactions: three factors at two levels Main effects and interactions in a 2' design: Incomplete blocks; confounding Partial confounding Factors at three or more levels The Latin square The graeco-latin square Further reading and references 457

11 xvi 10. LEAST SQUARES AND THE ANALYSIS OF STATIS- TICAL EXPERIMENTS: SINGULAR MODELS; MULTI- PLE TESTS Singular models Introduction Estimation. Estimable functions Tests of hypotheses The two-way hierarchical classification The two-way cross-classification Higher order classifications Analysis of covariance Multiple tests and comparisons Introduction Combination of tests, and overall size Multiple comparisons The assumptions about the errors The basic assumptions Residual analysis Further reading and references 498 Appendix Index Al xxiii Part B contains Chapters

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