Statistical Data Analysis, Contents
G. Cowan, Statistical Data Analysis
Table of Contents
- Preface v
- Notation xiii
- Chapter 1 Fundamental concepts 1
- 1.1 Probability and random variables 1
-
1.2 Interpretation of probability 4
- 1.2.1 Probability as a relative frequency 4
- 1.2.2 Subjective probability 5
- 1.3 Probability density functions 7
- 1.4 Functions of random variables 13
- 1.5 Expectation values 16
- 1.6 Error propagation 20
- 1.7 Orthogonal transformation of random variables 22
- Chapter 2 Examples of probability functions 26
- 2.1 Binomial and multinomial distributions 26
- 2.2 Poisson distribution 29
- 2.3 Uniform distribution 30
- 2.4 Exponential distribution 31
- 2.5 Gaussian distribution 32
- 2.6 Log-normal distribution 34
- 2.7 Chi-square distribution 35
- 2.8 Cauchy (Breit--Wigner) distribution 36
- 2.9 Landau distribution 37
- Chapter 3 The Monte Carlo method 40
- 3.1 Uniformly distributed random numbers 40
- 3.2 The transformation method 41
- 3.3 The acceptance--rejection method 42
- 3.4 Applications of the Monte Carlo method 44
- Chapter 4 Statistical tests 46
- 4.1 Hypotheses, test statistics, significance level, power 46
- 4.2 An example with particle selection 48
- 4.3 Choice of the critical region using the Neyman--Pearson lemma 50
-
4.4 Constructing a test statistic 51
- 4.4.1 Linear test statistics, the Fisher discriminant function 51
- 4.4.2 Nonlinear test statistics, neural networks 54
- 4.4.3 Selection of input variables 56
- 4.5 Goodness-of-fit tests 57
- 4.6 The significance of an observed signal 59
- 4.7 Pearson's chi^2 test 61
- Chapter 5 General concepts of parameter estimation 64
- 5.1 Samples, estimators, bias 64
- 5.2 Estimators for mean, variance, covariance 66
- Chapter 6 The method of maximum likelihood 70
- 6.1 ML estimators 70
- 6.2 Example of an ML estimator: an exponential distribution 72
- 6.3 Example of ML estimators: mu and sigma^2 of a Gaussian 74
- 6.4 Variance of ML estimators: analytic method 75
- 6.5 Variance of ML estimators: Monte Carlo method 76
- 6.6 Variance of ML estimators: the RCF bound 76
- 6.7 Variance of ML estimators: graphical method 78
- 6.8 Example of ML with two parameters 80
- 6.9 Extended maximum likelihood 83
- 6.10 Maximum likelihood with binned data 87
- 6.11 Testing goodness-of-fit with maximum likelihood 89
- 6.12 Combining measurements with maximum likelihood 92
- 6.13 Relationship between ML and Bayesian estimators 93
- Chapter 7 The method of least squares 95
- 7.1 Connection with maximum likelihood 95
- 7.2 Linear least-squares fit 97
- 7.3 Least squares fit of a polynomial 98
- 7.4 Least squares with binned data 100
- 7.5 Testing goodness-of-fit with chi^2 103
-
7.6 Combining measurements with least squares 106
- 7.6.1 An example of averaging correlated measurements 109
- 7.6.2 Determining the covariance matrix 112
- Chapter 8 The method of moments 114
- Chapter 9 Statistical errors, confidence intervals and
limits 118
- 9.1 The standard deviation as statistical error 118
- 9.2 Classical confidence intervals (exact method) 119
- 9.3 Confidence interval for a Gaussian distributed estimator 123
- 9.4 Confidence interval for the mean of the Poisson distribution 126
- 9.5 Confidence interval for correlation coefficient, transformation
of parameters 128
- 9.6 Confidence intervals using the likelihood function or chi^2 130
- 9.7 Multidimensional confidence regions 132
- 9.8 Limits near a physical boundary 136
- 9.9 Upper limit on the mean of Poisson variable with background 139
- Chapter 10 Characteristic functions and related examples 143
- 10.1 Definition and properties of the characteristic function 143
- 10.2 Applications of the characteristic function 144
- 10.3 The central limit theorem 147
-
10.4 Use of the characteristic function to find the p.d.f.
of an estimator 149
- 10.4.1 Expectation value for mean lifetime and decay constant 150
- 10.4.2 Confidence interval for the mean of an exponential
random variable 151
- Chapter 11 Unfolding 153
- 11.1 Formulation of the unfolding problem 154
- 11.2 Inverting the response matrix 159
- 11.3 The method of correction factors 164
- 11.4 General strategy of regularized unfolding 165
-
11.5 Regularization functions 167
- 11.5.1 Tikhonov regularization 167
- 11.5.2 Regularization functions based on entropy 169
- 11.5.3 Bayesian motivation for the use of entropy 170
- 11.5.4 Regularization function based on cross-entropy 173
- 11.6 Variance and bias of the estimators 173
- 11.7 Choice of the regularization parameter 177
- 11.8 Examples of unfolding 179
- 11.9 Numerical implementation 184
- Bibliography 188
- Index 194