Statistical Data Analysis (Oxford Science Publications)

Front cover......Page 1
Title page......Page 3
Copyright page......Page 4
Preface......Page 5
Contents......Page 9
Notation......Page 13
1.1 Probability and random variables......Page 15
1.2.1 Probability as a relative frequency......Page 18
1.2.2 Subjective probability......Page 19
1.3 Probability density functions......Page 21
1.4 Functions of random variables......Page 27
1.5 Expectation values......Page 30
1.6 Error propagation......Page 34
1.7 Orthogonal transformation of random variables......Page 36
2.1 Binomial and multinomial distributions......Page 40
2.2 Poisson distribution......Page 43
2.3 Uniform distribution......Page 44
2.4 Exponential distribution......Page 45
2.5 Gaussian distribution......Page 46
2.6 Log-normal distribution......Page 48
2.7 Chi-square distribution......Page 49
2.8 Cauchy (Breit-Wigner) distribution......Page 50
2.9 Landau distribution......Page 51
3.1 Uniformly distributed random numbers......Page 54
3.2 The transformation method......Page 55
3.3 The acceptance-rejection method......Page 56
3.4 Applications of the Monte Carlo method......Page 58
4.1 Hypotheses, test statistics, significance level, power......Page 60
4.2 An example with particle selection......Page 62
4.3 Choice of the critical region using the Neyman-Pearson lemma......Page 64
4.4.1 Linear test statistics, the Fisher discriminant function......Page 65
4.4.2 Nonlinear test statistics, neural networks......Page 68
4.4.3 Selection of input variables......Page 70
4.5 Goodness-of-fit tests......Page 71
4.6 The significance of an observed signal......Page 73
4.7 Pearson\'s $\\chi^2$ test......Page 75
5.1 Samples, estimators, bias......Page 78
5.2 Estimators for mean, variance, covariance......Page 80
6.1 ML estimators......Page 84
6.2 Example of an ML estimator: an exponential distribution......Page 86
6.3 Example of ML estimators: $\\mu$ and $\\sigma^2$ of a Gaussian......Page 88
6.4 Variance of ML estimators: analytic method......Page 89
6.6 Variance of ML estimators: the RCF bound......Page 90
6.7 Variance of ML estimators: graphical method......Page 92
6.8 Example of ML with two parameters......Page 94
6.9 Extended maximum likelihood......Page 97
6.10 Maximum likelihood with binned data......Page 101
6.11 Testing goodness-of-fit with maximum likelihood......Page 103
6.12 Combining measurements with maximum likelihood......Page 106
6.13 Relationship between ML and Bayesian estimators......Page 107
7.1 Connection with maximum likelihood......Page 109
7.2 Linear least-squares fit......Page 111
7.3 Least squares fit of a polynomial......Page 112
7.4 Least squares with binned data......Page 114
7.5 Testing goodness-of-fit with $\\chi^2$......Page 117
7.6 Combining measurements with least squares......Page 120
7.6.1 An example of averaging correlated measurements......Page 123
7.6.2 Determining the covariance matrix......Page 126
8 The method of moments......Page 128
9.1 The standard deviation as statistical error......Page 132
9.2 Classical confidence intervals (exact method)......Page 133
9.3 Confidence interval for a Gaussian distributed estimator......Page 137
9.4 Confidence interval for the mean of the Poisson distribution......Page 140
9.5 Confidence interval for correlation coefficient, transformation of parameters......Page 142
9.6 Confidence intervals using the likelihood function or $\\chi^2$......Page 144
9.7 Multidimensional confidence regions......Page 146
9.8 Limits near a physical boundary......Page 150
9.9 Upper limit on the mean of Poisson variable with background......Page 153
10.1 Definition and properties of the characteristic function......Page 157
10.2 Applications of the characteristic function......Page 158
10.3 The central limit theorem......Page 161
10.4 Use of the characteristic function to find the p.d.f. of an estimator......Page 163
10.4.1 Expectation value for mean lifetime and decay constant......Page 164
10.4.2 Confidence interval for the mean of an exponential random variable......Page 165
11 Unfolding......Page 167
11.1 Formulation of the unfolding problem......Page 168
11.2 Inverting the response matrix......Page 173
11.3 The method of correction factors......Page 178
11.4 General strategy of regularized unfolding......Page 179
11.5.1 Tikhonov regularization......Page 181
11.5.2 Regularization functions based on entropy......Page 183
11.5.3 Bayesian motivation for the use of entropy......Page 184
11.6 Variance and bias of the estimators......Page 187
11.7 Choice of the regularization parameter......Page 191
11.8 Examples of unfolding......Page 193
11.9 Numerical implementation......Page 198
Bibliography......Page 202
Index......Page 208
Back cover......Page 212