Topics for statistics 306	Statistical models	Fall 2013

Part I:
1. Multivariate Gaussian distribution
	Definition by density, Definition by MGF
	Linear transformation
	Marginal distributions, conditional distributions: Partitioned matrices
	Continuous random functions

2. Structured covariance models (linear Gaussian only)
	Block factor models
	Spatial models
	Temporal models
	Brownian motion and FBM
	Spline-smoothing models
	BLUP and Kriging
	Prediction

3. Estimation using residual likelihood (Gaussian only)
	The kernel subspace
	REML and variance-components estimation
        Regression effects estimation
	Likelihood ratio statistics: fixing the kernel
	Computation

4. Strategies for statistical modelling
	Exchangeability; covariates, relationships,...
	Example 1: Rats
	Example 2: Growth curves: plants and animals
	Example 3: Factors affecting female fulmar fitness
	Example 4: Out of Africa?
	Example 5: Mortality and health sequences

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Part II:
5. Moments and cumulants: (Tensor Methods, chapters 2, 3)
	Generating functions:  MGF  PGF  CGF
	Ordinary GFs;  exponential GFs
	Operations on GFs: sum, scalar multiplication,
	Composition of GFs
	
6. Moments and cumulants: multivariate case
	Index notation for linear combinations
	Partitions and the partition lattice: Mobius inversion
	Relation between moments and cumulants
	Cumulants as a linear transformation (Seq, *) to (Seq, *):
	Generalized cumulants: connected partitions
	Examples: k-statistics, quadratic forms,...

7. Some theory of simple random sampling (TM chapter 4, notes)
	Units, samples and values
	Inheritance and U-statistics
	k-statistics and polykays
	Finite-population variances

8. Point processes
	The Poisson point process
	Poisson superposition processes
	Doubly stochastic Poisson processes (Cox processes)
	Boson point process
	Moment and cumulant measures
	Janossy measures

9. Factors and factorial models
	Permutation, selection and injective maps
	Linear representations: preservation of structure
	Factorial subspaces: marginality
	Homologous factors
	Non-linear representations: projective structures

Part III: Random discrete structures
10. Random partitions
	Exchangeable partition process
	Dirichlet random partitions
	The Ewens process and the CRP
	The Gauss-Ewens cluster process: multi-level GECP
	Prediction and classification
	Counting clusters: how many species?
	
11. Trees
	Definitions: root, leaves, edges, branches,...
	Rooted and unrooted trees
	Tree-structured covariance matrices
	Tree estimation by maximum likelihood
	Marginal likelihood and distance matrices

12. Random trees
	Tree processes: the coalescent tree
	Markov fragmentation trees
	Gaussian nested cluster processes

13. Exchangeable processes
	Real-valued processes
	Partition-valued processes
	Cluster processes
	Rank-valued processes
	Permutation processes

14. Regression models
	Definition: exchangeability of units:
	Effects associated with covariates and relationships  
	The lack of interference assumption (Kolmogorov, Cox,...)
	Parameter estimation
	Prediction

15. Likelihood and the likelihood principle
	Parametric inference versus non-parametric inference
	Bartlett identities
	Asymptotic theory for regular models
	Prediction and $p$-values


TM available at www.stat.uchicago.edu/~pmcc/tensorbook