Topics for statistics 306 Statistical models Spring 2017 Part I: 1. 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 2. 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,... 3. 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 4. Point processes The Poisson point process Poisson superposition processes Doubly stochastic Poisson processes (Cox processes) Boson point process Moment and cumulant measures Janossy measures 5. 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 II: Random discrete structures 6. 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? 7. Trees Definitions: root, leaves, edges, branches,... Rooted and unrooted trees Tree-structured covariance matrices Tree estimation by maximum likelihood Marginal likelihood and distance matrices 8. Random trees Tree processes: the coalescent tree Markov fragmentation trees Gaussian nested cluster processes 9. Exchangeable processes Real-valued processes Partition-valued processes Cluster processes Rank-valued processes Permutation processes 10. Regression models Definition: exchangeability of units: Effects associated with covariates and relationships The lack of interference assumption (Kolmogorov, Cox,...) Parameter estimation Prediction 11. 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