# Theory and Methods of Statistical Inference

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1 PhD School in Statistics cycle XXIX, 2014 Theory and Methods of Statistical Inference Instructors: B. Liseo, L. Pace, A. Salvan (course coordinator), N. Sartori, A. Tancredi, L. Ventura Syllabus Some prerequisites: Empirical distribution function. Convergence of sums of r.v.s. Order statistics. Density function and Radon-Nikodym theorem. Scale and location families. Exponential families. Multivariate normal distributions. Parametric inference: basics. Statistical models: data variability and uncertainty in inference: Statistical models. Paradigms of inference. The Fisherian paradigm. Model specification (data variability). Levels of specification. Problems of distribution (statistical variability). 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 Cornish-Fisher expansions. Notations O p ( ) and o p ( ). Approximations of moments and transformations. Likelihood: observed and expected quantities, exact properties: Dominated statistical models. Sufficiency. Likelihood: observed quantities. Examples: a two-parameter 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 (first-order asymptotics): Likelihood inference procedures. Consistency of the maximum likelihood estimator. Asymptotic distribution of the maximum likelihood estimator. Asymptotic distribution of the log-likelihood ratio: simple null hypothesis, likelihood confidence regions, comparisons among asymptotically equivalent forms, non-null asymptotic distributions, composite null hypothesis (nuisance parameters), profile likelihood, asymptotically equivalent forms and one-sided versions, testing constraints on the components of the parameter. Non-regular models. Likelihood: numerical and graphical aspects in R Scalar and vector parameters: A scalar parameter example: log likelihood, plot of the log likelihood, MLE and observed/expected information, Wald confidence intervals, deviance confidence regions, simulation, numerical optimization methods, significance function. A vector parameter example: plot of the log likelihood, parameter estimates, simulation. Likelihood: numerical and graphical issues in R - Parameter of interest and profile likelihood: Parameter of interest and profile likelihood. Examples in the Weibull model. Deviance intervals: simulation. Stratified models. 1

2 Estimating equations and pseudolikelihoods: Misspecification. Estimating equations. Quasi likelihood. Pairwise 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, pseudo-likelihoods. Marginal likelihood. Conditional likelihood. The frequency-decision paradigm: Statistical decision problems. Efficient estimators: Cramér- Rao lower bound, asymptotic efficiency, Godambe efficiency, Rao-Blackwell-Lehmann-Scheffé theorem, other constraints for point estimation. Optimal tests: Neyman-Pearson lemma, composite hypotheses: families with monotone likelihood ratio, locally most powerful tests, two-sided 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. First-order 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. Higher order asymptotics: Laplace expansions. Approximation of marginal likelihood. Bayesian interpretation of approximate conditional likelihood. Saddlepoint expansion for density functions. Improvements over classical asymptotic theory. The p formula. Tail probabilities and modified directed likelihood. Bartlett adjustment. Modified profile likelihood. Bayesian Inference: Statistical models and prior information. Inference based on the posterior distribution. Choice of the prior distribution. Point and interval estimation. Hypothesis testing and the Bayes factor. Linear models. Model selection. Monte Carlo. MCMC. Generalized linear models. Hierarchical models and excheangeability. 2

3 References Barndorff-Nielsen, O.E. and Cox, D.R. (1994). Inference and Asymptotics. Chapman and Hall, London. Casella, G. and Berger, R.L. (1990, 2002). Statistical Inference. Wadsworth publishing Co., Belmont, CA. Cox, D.R. (2006). Principles of Statistical Inference. Cambridge University Press, Cambridge. Cox, D.R. and Hinkley, D.V. (1974). Theoretical Statistics. Chapman and Hall, London. Davison, A.C. (2003). Statistical Models. Cambridge University Press, Cambridge. Lehmann, E.L. (1983). Theory of Point Estimation. Wiley, New York. Lehmann, E.L. (1986). Testing Statistical Hypotheses. 2-nd ed.. Wiley, New York. McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models. 2-nd ed.. Chapman and Hall, London. Pace, L. and Salvan, A. (1996). Teoria della Statistica: Metodi, Modelli, Approssimazioni Asintotiche. Cedam, Padova. Pace, L. and Salvan, A. (1997). Principles of Statistical Inference from a neo-fisherian Perspective. World Scientific, Singapore. Severini, T.A. (2000). Likelihood Methods in Statistics. Oxford University Press, Oxford. Severini, T.A. (2005). Elements of Distribution Theory. Cambridge University Press, Cambridge. van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge University Press, Cambridge. Wasserman, L. (2004). All of Statistics. Springer, New York. Welsh, A.H. (1996). Aspects of Statistical Inference. Wiley, New York. Young, G.A. and Smith, R.L. (2005). Essentials of Statistical Inference. Cambridge University Press, Cambridge. 3

4 References for Bayesian Inference Albert, J. (2009). Bayesian Computation with R. (Second edition). Springer, New York. Ghosh, J.K., Delampady, M., Tapas, S. (2006). An Introduction to Bayesian Analysis. Springer, New York. Hoff, P.D. (2009). A First Course in Bayesian Statistical Methods. Springer, New York. Lee, P. (2004). Bayesian Statistics: an Introduction. Oxford University Press, New York. Liseo, B. (2007). Introduzione alla Statistica Bayesiana. O Hagan, A. and Forster, J. (2004) Bayesian Inference. 2-nd ed.. Edward Arnold, London. Robert C.P. and Casella G. (2004) Monte Carlo Statistical Methods (Second edition). Springer, New York. 4

5 Lectures and topics date topic instructor 13/03/ course presentation, some prerequisites AS 14/03/ Sweave: mixing R and L A TEX NS 17/03/ statistical models: data variability and sampling variability AS 18/03/ generating functions, approximation of moments, transformations AS 20/03/ likelihood: observed and expected quantities, exact sampling properties AS 26/03/ likelihood: observed and expected quantities, exact sampling properties AS 28/03/ likelihood inference (first-order asymptotics) NS 01/04/ likelihood inference (first-order asymptotics) NS 02/04/ lab. on likelihood: graphical and numerical techniques with R, I NS 08/04/ estimating equations and pseudolikelihoods NS 09/04/ lab. on likelihood: graphical and numerical techniques with R, II NS 15/04/ data and model reduction by marginalization and conditioning AS 17/04/ data and model reduction by marginalization and conditioning AS 29/04/ Bayesian inference. Introduction: prior and posterior distributions LV 06/05/ choice of priors; noninformative priors LV 08/05/ linear models; model selection LV 09/05/ the frequency decision paradigm AS 13/05/ the frequency decision paradigm AS 14/05/ exchangeability and hierarchical models AT 15/05/ Monte Carlo Integration for Bayesian inference AT 15/05/ lab session AT 20/05/ the frequency decision paradigm AS 21/05/ MCMC methods, Metropolis-Hastings Algorithms AT 22/05/ Gibbs sampler AT 22/05/ lab session AT 27/05/ exponential families AS 29/05/ exponential families AS 03/06/ exponential dispersion models, generalized linear models AS 05/06/ group families AS 06/06/ group families AS (Autumn) higher-order likelihood theory NS 24/06/ written exam 25/06/ practical exam 26/06/ Writing papers and reports + paper assignment AS+NS??/09/ oral presentations Instructors: AS is Alessandra Salvan, NS is Nicola Sartori, LV is Laura Ventura, AT is Andrea Tancredi. Luigi Pace (LP) collaborates in lecture material, in the final exam and in paper assignment and evaluation. Brunero Liseo (BL) collaborates in lecture material for the Bayesian part. 5

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