Statistical Methods in Experimental Physics

The following sections are included:

• PREFACE TO THE SECOND EDITION

• PREFACE TO THE FIRST EDITION

• CONTENTS

The following sections are included:

• Outline

• Language

• Two Philosophies

• Notation

The following sections are included:

• Definitions of Probability
  • Mathematical probability
  • Frequentist probability
  • Bayesian probability

• Properties of Probability
  • Addition law for sets of elementary events
  • Conditional probability and independence
  • Example of the addition law: scanning efficiency
  • Bayes theorem for discrete events
  • Bayesian use of Bayes theorem
  • Random variable

• Continuous Random Variables
  • Probability density function
  • Change of variable
  • Cumulative, marginal and conditional distributions
  • Bayes theorem for continuous variables
  • Bayesian use of Bayes theorem for continuous variables

• Properties of Distributions
  • Expectation, mean and variance
  • Covariance and correlation
  • Linear functions of random variables
  • Ratio of random variables
  • Approximate variance formulae
  • Moments

• Characteristic Function
  • Definition and properties
  • Cumulants
  • Probability generating function
  • Sums of a random number of random variables
  • Invariant measures

The following sections are included:

• The Tchebycheff Theorem and Its Corollary
  • Tchebycheff theorem
  • Bienaymé–Tchebycheff inequality

• Convergence
  • Convergence in distribution
  • The Paul Levy theorem
  • Convergence in probability
  • Stronger types of convergence

• The Law of Large Numbers
  • Monte Carlo integration
  • The Central Limit theorem
  • Example: Gaussian (Normal) random number generator

The following sections are included:

• Discrete Distributions
  • Binomial distribution
  • Multinomial distribution
  • Poisson distribution
  • Compound Poisson distribution
  • Geometric distribution
  • Negative binomial distribution

• Continuous Distributions
  • Normal one-dimensional (univariate Gaussian)
  • Normal many-dimensional (multivariate Gaussian)
  • Chi-square distribution
  • Student's t-distribution
  • Fisher–Snedecor F and Z distributions
  • Uniform distribution
  • Triangular distribution
  • Beta distribution
  • Exponential distribution
  • Gamma distribution
  • Cauchy, or Breit–Wigner, distribution
  • Log-Normal distribution
  • Extreme value distribution
  • Weibull distribution
  • Double exponential distribution
  • Asymptotic relationships between distributions

• Handling of Real Life Distributions
  • General applicability of the Normal distribution
  • Johnson empirical distributions
  • Truncation
  • Experimental resolution
  • Examples of variable experimental resolution

The following sections are included:

• Basic Concepts
  • Likelihood function
  • Statistic

• Information of R.A. Fisher
  • Definition of information
  • Properties of information

• Sufficient Statistics
  • Sufficiency
  • Examples
  • Minimal sufficient statistics
  • Darmois theorem

• Information and Sufficiency

• Example of Experimental Design

The following sections are included:

• Basic Concepts in Decision Theory
  • Subjective probability, Bayesian approach
  • Definitions and terminology

• Choice of Decision Rules
  • Classical choice: pre-ordering rules
  • Bayesian choice
  • Minimax decisions

• Decision-theoretic Approach to Classical Problems
  • Point estimation
  • Interval estimation
  • Tests of hypotheses

• Examples: Adjustment of an Apparatus
  • Adjustment given an estimate of the apparatus performance
  • Adjustment with estimation of the optimum adjustment

• Conclusion: Indeterminacy in Classical and Bayesian Decisions

The following sections are included:

• Basic Concepts in Estimation
  • Consistency and convergence
  • Bias and consistency

• Usual Methods of Constructing Consistent Estimators
  • The moments method
  • Implicitly defined estimators
  • The maximum, likelihood method
  • Least squares methods

• Asymptotic Distributions of Estimates
  • Asymptotic Normality
  • Asymptotic expansion of moments of estimates
  • Asymptotic bias and variance of the usual estimators

• Information and the Precision of an Estimator
  • Lower bounds for the variance — Cramér–Rao inequality
  • Efficiency and minimum variance
  • Cramér–Rao inequality for several parameters
  • The Gauss–Markov theorem
  • Asymptotic efficiency

• Bayesian Inference
  • Choice of prior density
  • Bayesian inference about the Poisson parameter
  • Priors closed under sampling
  • Bayesian inference about the mean, when the variance is known
  • Bayesian inference about the variance, when the mean is known
  • Bayesian inference about the mean and the variance
  • Summary of Bayesian inference for Normal parameters

The following sections are included:

• Choice of Estimator
  • Desirable properties of estimators
  • Compromise between statistical merits
  • Cures to obtain simplicity
  • Economic considerations

• The Method of Moments
  • Orthogonal functions
  • Comparison of likelihood and moments methods

• The Maximum Likelihood Method
  • Summary of properties of maximum likelihood
  • Example: determination of the lifetime of a particle in a restricted volume
  • Academic example of a poor maximum likelihood estimate
  • Constrained parameters in maximum likelihood

• The Least Squares Method (Chi-Square)
  • The linear model
  • The polynomial model
  • Constrained parameters in the linear model
  • Normally distributed data in nonlinear models
  • Estimation from histograms; comparison of likelihood and least squares methods

• Weights and Detection Efficiency
  • Ideal method — maximum likelihood
  • Approximate method for handling weights
  • Exclusion of events with large weight
  • Least squares method

• Reduction of Bias
  • Exact distribution of the estimate known
  • Exact distribution of the estimate unknown

• Robust (Distribution-free) Estimation
  • Robust estimation of the centre of a distribution
  • Trimming and Winsorization
  • Generalized p^th-power norms
  • Estimates of location for asymmetric distributions

The following sections are included:

• Normally distributed data
  • Confidence intervals for the mean
  • Confidence intervals for several parameters
  • Interpretation of the covariance matrix

• The General Case in One Dimension
  • Confidence intervals and belts
  • Upper limits, lower limits and flip-flopping
  • Unphysical values and empty intervals
  • The unified approach
  • Confidence intervals for discrete data

• Use of the Likelihood Function
  • Parabolic log-likelihood function
  • Non-parabolic log-likelihood functions
  • Profile likelihood regions in many parameters

• Use of Asymptotic Approximations
  • Asymptotic Normality of the maximum likelihood estimate
  • Asymptotic Normality of ∂in L/∂θ
  • ∂L/∂θ confidence regions in many parameters
  • Finite sample behaviour of three general methods of interval estimation

• Summary: Confidence Intervals and the Ensemble

• The Bayesian Approach
  • Confidence intervals and credible intervals
  • Summary: Bayesian or frequentist intervals?

The following sections are included:

• Formulation of a Test
  • Basic concepts in testing
  • Example: Separation of two classes of events

• Comparison of Tests
  • Power
  • Consistency
  • Bias
  • Choice of tests

• Test of Simple Hypotheses
  • The Neyman–Pearson test
  • Example: Normal theory test versus sign test

• Tests of Composite Hypotheses
  • Existence of a uniformly most powerful test for the exponential family
  • One- and two-sided tests
  • Maximizing local power

• Likelihood Ratio Test
  • Test statistic
  • Asymptotic distribution for continuous families of hypotheses
  • Asymptotic power for continuous families of hypotheses
  • Examples
  • Small sample behaviour
  • Example of separate families of hypotheses
  • General methods for testing separate families

• Tests and Decision Theory
  • Bayesian choice between families of distributions
  • Sequential tests for optimum number of observations
  • Sequential probability ratio test for a continuous family of hypotheses

• Summary of Optimal Tests

The following sections are included:

• GOF Testing: From the Test Statistic to the P-value

• Pearson's Chi-square Test for Histograms
  • Moments of the Pearson statistic
  • Chi-square test with estimation of parameters
  • Choosing optimal bin size

• Other Tests on Binned Data
  • Runs test
  • Empty cell test, order statistics
  • Neyman–Barton smooth test

• Tests Free of Binning
  • Smirnov-Cramér-von Mises test
  • Kolmogorov test
  • More refined tests based on the EDF
  • Use of the likelihood function

• Applications
  • Observation of a fine structure
  • Combining independent estimates
  • Comparing distributions

• Combining Independent Tests
  • Independence of tests
  • Significance level of the combined test

The following sections are included:

• REFERENCES

• SUBJECT INDEX