Statistical Methods in Experimental Physics (2nd Edition) Book

Preface to the Second Edition     v
Preface to the First Edition     vii
Introduction     1
Outline     1
Language     2
Two Philosophies     3
Notation     4
Basic Concepts in Probability     9
Definitions of Probability     9
Mathematical probability     10
Frequentist probability     10
Bayesian probability     11
Properties of Probability     12
Addition law for sets of elementary events     12
Conditional probability and independence     13
Example of the addition law: scanning efficiency     14
Bayes theorem for discrete events     15
Bayesian use of Bayes theorem     16
Random variable     17
Continuous Random Variables     18
Probability density function     19
Change of variable     20
Cumulative, marginal and conditional distributions     21
Bayes theorem for continuous variables     22
Bayesian use of Bayes theorem for continuous variables     22
Properties of Distributions     24
Expectation, mean and variance     24
Covariance andcorrelation     26
Linear functions of random variables     28
Ratio of random variables     30
Approximate variance formulae     32
Moments     33
Characteristic Function     34
Definition and properties     34
Cumulants     37
Probability generating function     38
Sums of a random number of random variables     39
Invariant measures     41
Convergence and the Law of Large Numbers     43
The Tchebycheff Theorem and Its Corollary     43
Tchebycheff theorem     43
Bienayme-Tchebycheff inequality     44
Convergence     45
Convergence in distribution     45
The Paul Levy theorem     46
Convergence in probability     46
Stronger types of convergence     47
The Law of Large Numbers     47
Monte Carlo integration     48
The Central Limit theorem     49
Example: Gaussian (Normal) random number generator     51
Probability Distributions     53
Discrete Distributions     53
Binomial distribution     53
Multinomial distribution      56
Poisson distribution     57
Compound Poisson distribution     60
Geometric distribution     62
Negative binomial distribution     63
Continuous Distributions     64
Normal one-dimensional (univariate Gaussian)     64
Normal many-dimensional (multivariate Gaussian)     67
Chi-square distribution     70
Student's t-distribution     73
Fisher-Snedecor F and Z distributions     77
Uniform distribution     79
Triangular distribution     79
Beta distribution     80
Exponential distribution     82
Gamma distribution     83
Cauchy, or Breit-Wigner, distribution     84
Log-Normal distribution     85
Extreme value distribution     87
Weibull distribution     89
Double exponential distribution     89
Asymptotic relationships between distributions     90
Handling of Real Life Distributions     91
General applicability of the Normal distribution     91
Johnson empirical distributions     92
Truncation     93
Experimental resolutions     94
Examples of variable experimental resolution      95
Information     99
Basic Concepts     100
Likelihood function     100
Statistic     100
Information of R.A. Fisher     101
Definition of information     101
Properties of information     101
Sufficient Statistics     103
Sufficiency     103
Examples     104
Minimal sufficient statistics     105
Darmois theorem     106
Information and Sufficiency     108
Example of Experimental Design     109
Decision Theory     111
Basic Concepts in Decision Theory     112
Subjective probability, Bayesian approach     112
Definitions and terminology     113
Choice of Decision Rules     114
Classical choice: pre-ordering rules     114
Bayesian choice     115
Minimax decisions     116
Decision-theoretic Approach to Classical Problems     117
Point estimation     117
Interval estimation     118
Tests of hypotheses     118
Examples: Adjustment of an Apparatus     121
Adjustment given an estimate of the apparatus performance      121
Adjustment with estimation of the optimum adjustment     123
Conclusion: Indeterminacy in Classical and Bayesian Decisions     124
Theory of Estimators     127
Basic Concepts in Estimation     127
Consistency and convergence     128
Bias and consistency     129
Usual Methods of Constructing Consistent Estimators     130
The moments method     131
Implicitly defined estimators     132
The maximum likelihood method     135
Least squares methods     137
Asymptotic Distributions of Estimates     139
Asymptotic Normality     139
Asymptotic expansion of moments of estimates     141
Asymptotic bias and variance of the usual estimators     144
Information and the Precision of an Estimator     146
Lower bounds for the variance - Cramer-Rao inequality     147
Efficiency and minimum variance     149
Cramer-Rao inequality for several parameters     151
The Gauss-Markov theorem     152
Asymptotic efficiency     153
Bayesian Inference     154
Choice of prior density     154
Bayesian inference about the Poisson parameter     156
Priors closed under sampling     157
Bayesian inference about the mean, when the variance is known     157
Bayesian inference about the variance, when the mean is known     159
Bayesian inference about the mean and the variance     161
Summary of Bayesian inference for Normal parameters     162
Point Estimation in Practice     163
Choice of Estimator     163
Desirable properties of estimators     164
Compromise between statistical merits     165
Cures to obtain simplicity     166
Economic considerations     168
The Method of Moments     170
Orthogonal functions     170
Comparison of likelihood and moments methods     172
The Maximum Likelihood Method     173
Summary of properties of maximum likelihood     173
Example: determination of the lifetime of a particle in a restricted volume     175
Academic example of a poor maximum likelihood estimate     177
Constrained parameters in maximum likelihood     179
The Least Squares Method (Chi-Square)     182
The linear model     183
The polynomial model     185
Constrained parameters in the linear model     186
Normally distributed data in nonlinear models     190
Estimation from histograms; comparison of likelihood and least squares methods     191
Weights and Detection Efficiency     193
Ideal method maximum likelihood     194
Approximate method for handling weights     196
Exclusion of events with large weight     199
Least squares method     201
Reduction of Bias     204
Exact distribution of the estimate known     204
Exact distribution of the estimate unknown     206
Robust (Distribution-free) Estimation     207
Robust estimation of the centre of a distribution     208
Trimming and Winsorization     210
Generalized p[superscript th]-power norms     211
Estimates of location for asymmetric distributions     213
Interval Estimation     215
Normally distributed data     216
Confidence intervals for the mean     216
Confidence intervals for several parameters     218
Interpretation of the covariance matrix     223
The General Case in One Dimension     225
Confidence intervals and belts     225
Upper limits, lower limits and flip-flopping     227
Unphysical values and empty intervals      229
The unified approach     229
Confidence intervals for discrete data     231
Use of the Likelihood Function     233
Parabolic log-likelihood function     233
Non-parabolic log-likelihood functions     234
Profile likelihood regions in many parameters     236
Use of Asymptotic Approximations     238
Asymptotic Normality of the maximum likelihood estimate     238
Asymptotic Normality of [Characters not reproducible] In L/[Characters not reproducible theta]     238
[Characters not reproducible]L/[Characters not reproducible theta] confidence regions in many parameters     240
Finite sample behaviour of three general methods of interval estimation     240
Summary: Confidence Intervals and the Ensemble     246
The Bayesian Approach     248
Confidence intervals and credible intervals     249
Summary: Bayesian or frequentist intervals?     250
Test of Hypotheses     253
Formulation of a Test     254
Basic concepts in testing     254
Example: Separation of two classes of events     255
Comparison of Tests     257
Power     257
Consistency     259
Bias      260
Choice of tests     261
Test of Simple Hypotheses     263
The Neyman-Pearson test     263
Example: Normal theory test versus sign test     264
Tests of Composite Hypotheses     266
Existence of a uniformly most powerful test for the exponential family     267
One- and two-sided tests     268
Maximizing local power     269
Likelihood Ratio Test     270
Test statistic     270
Asymptotic distribution for continuous families of hypotheses     271
Asymptotic power for continuous families of hypotheses     273
Examples     274
Small sample behaviour     279
Example of separate families of hypotheses     282
General methods for testing separate families     285
Tests and Decision Theory     287
Bayesian choice between families of distributions     287
Sequential tests for optimum number of observations     292
Sequential probability ratio test for a continuous family of hypotheses     297
Summary of Optimal Tests     298
Goodness-of-Fit Tests     299
GOF Testing: From the Test Statistic to the P-value     299
Pearson's Chi-square Test for Histograms      301
Moments of the Pearson statistic     302
Chi-square test with estimation of parameters     303
Choosing optimal bin size     304
Other Tests on Binned Data     308
Runs test     308
Empty cell test, order statistics     309
Neyman-Barton smooth test     311
Tests Free of Binning     313
Smirnov-Cramer-von Mises test     314
Kolmogorov test     316
More refined tests based on the EDF     317
Use of the likelihood function     317
Applications     318
Observation of a fine structure     318
Combining independent estimates     323
Comparing distributions     327
Combining Independent Tests     330
Independence of tests     330
Significance level of the combined test     331
References     335
Subject Index     341