BEA529 Probability and statistical inference

• Topics

  This course provides a rigorous foundation in probability and statistical inference for doctoral students at NHH. The emphasis is on the logical chain from probabilistic modeling assumptions to distributions, estimation, testing, and uncertainty quantification, supported by rigorous proofs in the language of mathematical analysis. This leads to insights into the conditions under which standard methods are valid and where they break down. 

  A central task is to establish the link between finite-sample behavior and large-sample approximations. Students study exact distributional results for key statistics in classical parametric settings and then develop limit theory tools that justify much of modern empirical practice.

  The course also introduces statistical principles that support empirical work, such as sensitivity to assumptions and the distinction between inferential and predictive goals (and whether these goals are really different). We will, for the sake of completeness, shed some light on different and no less important statistical paradigms than the frequentist parametric approach, such as Bayesian and nonparametric estimation, but these will not be treated in any detail. 

  The aim of this course is to equip students to read and evaluate technical arguments in applied research and to build theoretically grounded inferential approaches in their own work. 

  Core topics are:

  • Probability foundations
    • Sets, sigma-algebras, probability spaces, random variables
    • Conditional probability, Bayes' rule, independence and expectations
  • Distributions and parametric models
    • Distribution/density/probability mass/moment generating functions
    • The logical construction of parametric families of distributions
    • The exponential family and sufficient statistics
  • Random vectors
    • Joint, marginal, and conditional distributions
    • Transformations and Jacobians
    • Covariance/correlation, covariance matrices, conditional expectation
  • Sampling theory
    • Order statistics and distributional results for functions of samples
    • i.i.d. sampling and sampling distributions
    • Normal sampling theory leading to the t-, chi-squared-, and F-distributions.
  • Limit theory
    • Modes of convergence, the law of large numbers, and applications
    • A central limit theorem with proof, large-sample approximations
    • Continuous mapping, Slutsky, and the  Delta method
  • Maximum likelihood estimation
    • Likelihood, score, information, invariance, reparameterization
    • Consistency and asymptotic normality under regularity conditions
    • Likelihood-based uncertainty quantification
  • More estimation theory
    • Sufficiency, completeness, Rao-Blackwell, and Lehman-Scheffé theorems
    • Fisher information, Cramér-Rao lower bound, efficiency concepts
  • Statistical thinking and prediction
    • Identification, misspecification, robustness, and interpretation of uncertainty
    • Estimation versus prediction: Two sides of the same coin?
    • Principles of model evaluation

• Learning outcome

  Knowledge

  • Define the set- and measure-theoretic foundations of probability required for modern statistical inference
  • Recognize common univariate and multivariate distributions, including exponential-family structure
  • Explain convergence concepts and the main limit theorems that underpin large-sample statistical reasoning
  • State core optimality concepts in estimation (sufficiency, completeness, information, efficiency) and the conditions under which they apply

  Skills

  • Prove basic and central results in probability and statistical inference
  • Derive sampling distributions for key statistics (including those leading to t-, chi-squared-, and F-distributions), and use them to justify classical inference procedures
  • Prove a central limit theorem and use limit theory tools (Slutsky/continuous mapping/Delta method) to obtain asymptotic distributions
  • Formulate likelihoods, compute scores/information, derive MLEs (analytically when possible), and implement numerical MLE when necessary
  • Analyze estimator performance both exactly (when feasible) and asymptotically when necessary (consistency, asymptotic normality, efficiency)
  • Translate inferential outputs into prediction targets, construct prediction intervals, and evaluate predictive performance using principled rules

  General competence

  • Critically assess statistical arguments in empirical business research: detect hidden assumptions, identifiability gaps, and unjustified asymptotics
  • Communicate uncertainty and limitations clearly, without overclaiming from models or data

• Teaching

  The teaching consists of plenary lectures, exercise- and problem solving. 

• Restricted access

  • PhD candidates at NHH
  • PhD candidates at Norwegian institutions
  • PhD candidates at other institutions
  • PhD candidates from the ENGAGE.EU alliance
  • Motivated master’s students at NHH may be admitted after application, but are subject to the approval from the course responsible on a case-by-case basis

• Recommended prerequisites

  Business statistics or higher, undergraduate course in mathematics, including linear algebra. 

• Compulsory Activity

  Individual or group project (1-2 members) with oral presentation in class.

  Compulsory activities (work requirements) are valid for one semester after the semester they were obtained.

• Assessment

  Written school exam with pen and paper, 4 hours.

• Grading Scale

  A-F

• Literature

  Casella, G., & Berger, R. (2024).  "Statistical inference". Chapman and Hall/CRC.

  We will use material from the scientific literature as needed.

• Permitted Support Material

  One bilingual dictionary (Category I)

  All in accordance with Supplementary provisions to the Regulations for Full-time Study Programmes at the Norwegian School of Economics Ch.4 Permitted support material https://www.nhh.no/en/for-students/regulations/ and https://www.nhh.no/en/for-students/examinations/examination-support-materials/

• Retake

  Re-take is offered the semester after the course was offered for students with valid compulsory activities (work requirements).  Additionally, the students must fulfill one of the two requirements listed below in order to be eligible for re-take:

  • Students who, at the original exam failed or got a grade below C
  • Students who were sick on the day of the exam and has provided a valid sick note ("sykemelding")