Statistical inference for difference and differential equations
This is given as a guided self-study course leading to the Kalman filter and more general prediction and inference methods for difference and differential equations. The contents will be selected topics from the reading list given below.
Brockwell and Davis (1991): Time Series: Theory and Methods, Springer
Brockwell and Davis (2016): Introduction to Time Series and Forecasting, Springer
Øksendal (2013): Stochastic Differential Equations, Springer
Schervish (1995): Theory of statistics, Springer
Reading history and plan:
27.11.2018: EXAM in room 1138 Sentralbygg 2. Filip 0915-10, Marcus 1015-11.
20.11.2018: Schervish: Chap 5, Estimation
13.11.2018: Øksendal: Chap 6, The filtering problem, Theorem 6.2.8 (The 1-dimensional Kalman-Bucy filter)
06.11.2018: Brockwell and Davis, 2016, Chap 9: State-Space Models, the Kalman filter and estimation
30.10.2018: Brockwell and Davis, 2016, Chap 11.5 Continuous-Time ARMA Processes
23.10.2018: Brockwell and Davis, 2016, Chap 7.5-.6 Time Series Models for Financial Data
16.10.2018: No meeting. Self-study of Øksendal: Chap 3, Itô Integrals
09.10.2018: Brockwell and Davis, 2016, Appendix D, Lévy Processes, Brownian Motion and Itô Calculus
02.10.18: Øksendal: Chap 6, The filtering problem, Outline, Example 6.2.1 and Lemma 6.2.2
25.09.18: B&D: Ch 3 and Ch 5.7. The Wold decomposition. Existence and uniqueness for ARMA.
18.09.18: Brockwell and Davis, Chap 3, ARMA processes. Homogeneous Linear Difference Equations with Constant
Coefficients. The Autocovariance Generating Function.
11.09.18: Brockwell and Davis, Chap 2, Hilbert space theory for time series. Linear Regression and the General Linear Model.
Mean Square Convergence, Conditional Expectation and Best Linear Prediction
04.09.18: Brockwell and Davis, Chap 2, Elementary Hilbert space theory.
Beware that it is assumed that basic concepts needed for an understanding of the curriculum is known a priori, and if not you must prepare for this in addition. For Chap 5 in Schervish this means in particular that you must be familiar with basic probability and statistics including for instance the concept of a random variable and a sufficient statistic as defined by Schervish.
1) Brockwell and Davis (2016): Chap 7.5-7.6, 9.1-9.5
2) Brockwell and Davis (1991): Chap 2, 3, 5.1, 5.5, 5.7, 5.8
3) Øksendal (2013): Chap 3, 6
4) Schervish (1995): Chap 5 minus 5.1.2, 5.1.5, 5.2.3, 5.2.5