F71SM Statistical Methods

Course co-ordinator(s): Prof George Streftaris (Edinburgh).

Aims:

This course aims to provide postgraduate students taking the MSc in Actuarial Science, the MSc in Financial Mathematics, and other courses with a broad knowledge of the principal areas of mathematical statistics and statistical methods widely used in insurance and finance.

This course partially covers the material of CT3.

Summary:

• Data summary
• Basic probability concepts
• Random variables and their distributions
• Joint distributions
• Central limit theorm
• Parameter estimation
• Statistical inference
• Linear regression

Detailed Information

Course Description: Link to Official Course Descriptor.

Pre-requisites: none.

Location: Edinburgh.

Semester: 1.

Syllabus:

• Summarising and displaying data

• Probability and random variables
  • Random experiments, sample spaces, events
  • Probability axioms
  • Conditional probability
  • Independent events
  • Random variables
  • Density and distribution functions
  • Expected values
  • Moments and generating functions
  • Functions of a random variable
  • Some special discrete distributions
    • uniform
    • bernoulli
    • binomial
    • geometric
    • negative binomial
    • hypergeometric
    • Poisson
  • Some special continuous distributions
    • uniform
    • exponential, gamma
    • normal, chi-square
  • Joint distribution of several random variables
    • joint, marginal distributions
    • conditional distributions
  • Conditional expectation
  • Markov and Chebyshev (Tchebyshev) inequality, laws of large numbers
  • Central limit theorem
  • Sampling distributions
    • sampling distribution of the mean (normal, t-distribution)
    • sampling distribution of the variance (χ²)
    • sampling distribution of a proportion
    • ratio of 2 sample variances (F-distribution)
• Statistical inference
  • Estimation
    • by method of moments
    • by maximum likelihood
    • Properties of estimators
      • unbiasedness
      • efficiency
      • Cramèr-Rao lower bound
      • consistency
  • Confidence intervals: definition and construction
    • CIs for population mean and variance
    • CI for Poisson mean λ
    • CI for a proportion
    • CIs for difference between 2 population means μ₁-μ₂
    • CIs for ratio of 2 population variances σ₁²/σ₂²
  • Hypothesis testing
    • null and alternative hypotheses
    • test statistics and relation to confidence interval pivotal quantities
    • decision errors, significance level, p-values
    • power of tests
    • likelihood ratio
    • tests for
      • population mean and variance
      • equality of 2 populations means μ₁-μ₂
      • equality of 2 population variances σ₁² /σ₂²
• Linear regression
  • response and explanatory variables
  • linear regression model
  • least squares estimation
  • sums of squares, coefficient of determination R²
  • inference on regression parameters and tests for significance of regression
  • predicting a mean response and an actual response

Learning Outcomes: Subject Mastery

At the end of studying this course, students should be able to:

• Summarise and display data.
• Perform basic probability calculations.
• Calculate moments and the expected values of other functions of random variables.
• Apply the central limit theorem.
• Obtain estimators of parameters of certain common distributions.
• Determine properties of estimators: efficiency, Cramèr-Rao lower bound, (approx. large-sample) distribution.
• Perform inference on parameter estimates: obtain confidence intervals and carry out hypothesis testing.
• Fit a linear regression model.

Reading list:

The required sets of tables (provided) are:

• D V Lindley & W F Scott: New Cambridge Statistical Tables, Second edition, CUP 1995.
• Formulae and Tables for Examinations of the The Faculty of Actuaries and the Institute of Actuaries, 2002

Some students have found the following books helpful. The first book (Miller and Miller) is the required text-book. The second book (Rees) is an elementary introduction to some topics and is recommended for students with little or no previous study of statistics.

• Miller and Miller: John E. Freund’s Mathematical Statistics with Applications (7th or later edition), Pearson Prentice-Hall.
• Rees: Essential Statistics (4th or later edition) Chapman and Hall/CRC
• H. J. Larson: Introduction to Probability Theory and Statistical Inference (3rd Ed.), Wiley.
• W. Mendenhall and R. J. Beaver: Introduction to Probability and Statistics (8th or later edition.), Brooks/Cole.

SCQF Level: 11.

Other Information

Help: If you have any problems or questions regarding the course, you are encouraged to contact the course leader.

Canvas: further information and course materials are available on Canvas