
Course unit
THEORY AND METHODS OF INFERENCE
SCP4063246, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2016/17
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Educational activities in elective or integrative disciplines 
SECSS/01 
Statistics 
9.0 
Mode of delivery (when and how)
Period 
Second semester 
Year 
2nd Year 
Teaching method 
frontal 
Organisation of didactics
Type of hours 
Credits 
Hours of teaching 
Hours of Individual study 
Shifts 
Lecture 
9.0 
64 
161.0 
No turn 
Start of activities 
26/02/2018 
End of activities 
01/06/2018 
Examination board
Examination board not defined
Prerequisites:

First year Masters courses, especially Probability Theory and Statistics (Advanced). 
Target skills and knowledge:

The course aims at offering students a deep understanding of the likelihood based frequentist theory of statistical inference, inviting them as well to personal work on recent research in the field. An introduction to Bayesian inference will also be provided. 
Examination methods:

1/3 homework, 1/3 final written exam, 1/3 written and oral presentation rewiewing one or two recent research papers. 
Assessment criteria:

Student’s assessment will consider how the topics presented are mastered in problems and applications, how finely the main methodologies are appreciated in their strenghths and weaknesses, how deep the personal interaction with recent research papers appears. PhD students can sit the exam only once, in the exam session following the course. 
Course unit contents:

 Statistical models and uncertainty in inference. Statistical models. Paradigms of inference: the Bayesian and frequentist paradigms. Prior specification. Model specification (data variability). Levels of model specification. Problems of distribution (variability of statistics). Simulation. Asymptotic approximations and delta method.
 Generating functions, moment approximations, transformations. Moments, cumulants and their generating functions. Generating functions of sums of independent random variables. Edgeworth and CornishFisher expansions. Notations Op(·) and op(·). Approximations of moments and transformations. Laplace approximation.
 Likelihood: observed and expected quantities, exact properties. Dominated statistical models. Sufficiency. Likelihood: observed quantities. Examples: a twoparameter model, grouped data,
censored data, sequential sampling, Markov chains, Poisson processes. Likelihood and sufficiency. Invariance properties. Expected likelihood quantities and exact sampling properties.
Reparameterizations.
 Likelihood inference: firstorder asymptotics. Likelihood inference procedures. Consistency of the maximum likelihood estimator. Asymptotic distribution of the maximum likelihood estimator. Asymptotic distribution of the loglikelihood ratio: simple null hypothesis, likelihood confidence regions, asymptotically equivalent forms, nonnull asymptotic distributions, composite null hypothesis (nuisance parameters), profile likelihood, asymptotically equivalent forms and onesided versions, testing constraints on the components of the parameter. Nonregular models.
 Bayesian Inference. Noninformative priors. Inference based on the posterior distribution. Point estimation and credibility regions. Hypothesis testing and the Bayes factor. Linear models.
 Likelihood and Bayesian inference: numerical and graphical aspects in R. Scalar and vector parameter examples. EM algorithm.
 Estimating equations and pseudolikelihoods. Misspecification. Estimating equations. Quasi likelihood. Composite likelihood. Empirical likelihood.
 Data and model reduction by marginalization and conditioning. Distribution constant statistics. Completeness. Ancillary statistics. Data and model reduction with nuisance parameters:
lack of information with nuisance parameters, pseudolikelihoods. Marginal likelihood. Conditional likelihood. Profile and integrated likelihoods.
 The frequencydecision paradigm. Statistical decision problems. Optimality in estimation: Cram´erRao lower bound, asymptoticefficiency, Godambe efficiency, RaoBlackwellLehmannScheffe theorem. Optimal tests: NeymanPearson lemma, composite hypotheses: families with monotone likelihood ratio, locally most powerful tests, twosided alternatives, other constraint criteria. Optimal confidence regions.
 Exponential families, Exponential dispersion families, Generalized linear models. Exponential families of order 1. Mean value mapping and variance function. Multiparameter exponential
families. Marginal and conditional distributions. Sufficiency and completeness. Likelihood and exponential families: likelihood quantities, conditional likelihood, profile likelihood and mixed parameterization. Procedures with finite sample optimality properties. Firstorder asymptotic theory. Exponential dispersion families. Generalized linear models.
 Group families. Groups of transformations. Orbits and maximal invariants. Simple group families and conditional inference. Composite group families and marginal inference. 
Planned learning activities and teaching methods:

Lectures, homework, students' written and oral presentations. 
Additional notes about suggested reading:

Course material will be available on the course web page. 
Textbooks (and optional supplementary readings) 

Davison, Anthony Christopher, Statistical Models. New York: Cambridge University Press, 2003.

Pace, Luigi; Salvan, Alessandra, Principles of Statistical Inference, from a NeoFisherian Perspective. Singapore: World Scientific, 1997.

Severini, Thomas A., Likelihood Methods in Statistics. Oxford: Oxford University Press, 2000.

Severini, Thomas A., Elements of Distribution Theory. Cambridge: Cambridge University press, 2005.

Young, G. A.; Smith, R. L., Essentials of Statistical Inference. Cambridge: Cambridge University Press, 2005.


