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LTCC Advanced Course: Likelihood Inference
November/December, 2012
Course Outline
Also available as a pdf.
- Asymptotic theory for likelihood; likelihood root, maximum likelihood estimate, score function, pivotal quantities, exact and approximate ancillary.
Laplace approximations for Bayesian inference.
- Higher order approximations for non-Bayesian inference. Marginal, conditional and adjusted log-likelihoods for inference in the presence of nuisance parameters. Examples: regression models with non-normal error; logistic regression.
- Sample space differentiation and approximate ancillary; tangent exponential models;
Examples: contingency tables; risk difference and risk ratio; nonlinear regression
- Likelihood inference for complex data structure: time series, spatial models, space-time models, extremes. Composite likelihood: definition, summary statistics, asymptotic theory. Examples: longitudinal binary data; Gaussian random fields; Markov chains
- Semi-parametric likelihoods for point process data; empirical likelihood.
If you are taking the course for a mark
The homework has been graded, and the grades submitted to my contact at UCL, Nisha Jones. The papers may be picked up from Dr. Russell Evans, in Room 120 of the Statistical Science Department, until the end of term.
In each of weeks 1-4 some problems will be assigned, due the following week. Please bring hard copy to the class. These solutions will be the basis of your mark in the course. Problem solutions will be discussed in Week 5. The problems for Week 1, due November 12, are at the end of the Week 1 handout. The ``exercises'' embedded in the Week 1 slides are not required.
Hint for Week 1 Problems: The solution to 1(a) requires establishing properties about the constrained maximum likleihood estimator. The steps will be similar to those outlined in this moderately rigorous proof of the results in the scalar parameter case.
Running list of references and background reading
Review Papers
Likelihood Basics
- Davison, A.C. (2003) Statistical Models (SM) Cambridge University Press. -- Ch 4
- Barndorff-Nielsen, O.E. and Cox, D.R. (1994) Inference and Asymptotics (BNC) Chapman and Hall. -- Ch 2.2
- Cox, D.R. and Hinkley, D.V. (1974) Theoretical Statistics (CH) Chapman and Hall. -- Ch 2.1 (i), (ii)
- Cox, D.R. (2006). Principles of Statistical Inference (Cox) -- Ch.2.1
Likelihood and models
- Stochastic models: SM -- Ch 6
- Transformation models: SM -- Ch 5.3, BNC -- Ch 2.8
- Pivots: Brazzale, A.R., Davison, A.C. and Reid, N. (2007). Applied Asymptotics. (BDR) -- Ch. 2
Approximations
- Various: Barndorff-Nielsen, O.E. and Cox, D.R. (1989). Asymptotic Techniques for Use in Statistics. Good introduction to Laplace, Edgeworth, and saddlepoint expansions.
- Laplace approximation: SM -- Ch 11.3
- Approximate posteriors:
Tierney, L.J. and Kadane, J.B. (1986). Accurate approximation to posterior moments and marginal densities. JASA 81, 82-86
Johnson, R.A. (1970).
Asymptotic expansions associated with posterior
distributions.
Ann. Math. Statist. 41, 851--864.
Datta, G.S. and Mukerjee, R. (2004). Probability
Matching Priors: Higher Order Asymptotics. Lecture
Notes in Statistics 178, New York: Springer-Verlag
- Saddlepoint approximation: SM -- Ch 12.3
- Examples: BDR -- Ch 3, 4, 5
- Adjusted profile likelihood: SM -- Ch 12.4, BNC -- Ch 8
- Tangent exponential model: BDR -- CH 8.3
Composite Likelihood
Week 1
Week 2
Week 3
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