Advanced principles of statistics* (24 lectures) Statistical methods (i)* (8 lectures & 8 classes) Algorithms for operational research (16 lectures) Case studies in S-Plus (4 lectures) (non-examinable) Survival data analysis (10 lectures & 2 classes) Statistical genetics of human disease (7 lectures & 1 class) Time series (8 lectures)
Lent Term
Actuarial statistics (16 lectures) Design of experiments* (10 lectures & 2 classes) Applied multivariate analysis* (14 lectures/classes) Statistics in medical practice (7 lectures & 1 class) Monte Carlo inference (16 lectures)
Easter Term
Sequential methods (8 lectures) Statistical methods (ii)* (4 lectures & 4 classes) * = BASIC
The above courses, together with the Applied Project, form the one-year Diploma course. For two-year Diploma students, the first year consists of those Part IIA and Part IIB courses of the Mathematical Tripos given by members of the Statistical Laboratory, and the second year consists of the courses listed above and the Applied Project.
See Appendix 1 for the appropriate Part IIA and Part IIB course schedules.
Students who wish (and who can reasonably spare the time) may also offer the following additional courses given by the staff of the Laboratory for Part III of the Mathematical Tripos:
Advanced probability (24) Advanced financial models (24) Stochastic networks (16) Dynamics of one-dimensional maps (16) Entropy and complexity (16) Ergodic theory (16) Large deviations with applications (16) Stochastic calculus and applications (24)
1. Candidates must submit their Applied Project (report and poster) by 30 April.
2. Written papers will fall into three groups:
Group 1
Group 2
Group 3
3. All candidates must offer the three papers in Group 1 and at least two of those in Group 2. Maximum credit on the written papers may be obtained by taking the Group 1 papers and any two of those in Group 2.
Candidates may in addition offer one and, by special request to the Examiners, more than one paper from Group 3.
Candidates are advised to offer a maximum of three papers in total from Groups 2 and 3.
4. Some candidates may be required to attend an Oral Examination, on a pre-announced date in mid-June. This date will be announced by 30 April, and candidates will be informed by 10 a.m. on the date of the Oral Examination whether or not they are required to attend.
5. Paper A will be in two sections, A and B. In Section A there will be six questions on the course Principles of Statistics and in Section B there will be four questions testing competence in basic statistical theory. No more than six questions may be attempted, of which at least two must be from Section A and at least two from Section B.
Paper B will contain six questions on the courses Design of Experiments and Applied Multivariate Analysis, of which candidates will be asked to attempt four.
Paper C will contain four questions on the courses Statistical Methods and Applied Multivariate Analysis; candidates will be asked to attempt three of these questions. Candidates will receive this paper at 9.00 a.m.\ on the Monday following the conclusion of the other written papers and must submit their solutions by 1.00 p.m.\ on the following Thursday.
6. Each candidate shall send to the Chairman of the Diploma Examiners, so as to arrive before 28 April, the titles of the papers that he or she wishes to offer.
In the examination for the Diploma in Mathematical Statistics to be held in 1998 the papers in Group 2 will be
Each of these papers will contain four questions of which candidates will be asked to attempt three.
Paper D will contain questions on the courses Survival Data, Statistics in Medical Practice and Statistical Genetics.
Paper E will contain questions on the courses Monte Carlo Inference and Time Series.
Paper F will contain questions on the courses Algorithms for Operational Research and Sequential Methods.
The papers in Group 3 will be
These papers will be based on the Part III lectures with the same titles.
1. The general linear model: estimation and hypothesis testing. The Gauss-Markov theorem, Cochran's theorem. Orthogonal parameterisations. Bayesian analysis of linear model.
2. Classical non-parametric statistical methods. Order statistics, empirical distribution functions. General theory of non-parametric tests by invariance. U statistics: estimation and links with hypothesis testing.
3. Outline proof of Wilks' theorem. Examples.
Appropriate books
G.A.F. Seber, Linear Regression Analysis, Wiley (1997). G.E.P. Box & G.C. Tiao, Bayesian Inference in Statistical Analysis, Addison-Wesley (1980). R.H. Randles & D.A. Wolfe, Introduction to the Theory of Non-parametric Statistics, Wiley (1975). R.J. Serfling, Approximation theorems of Mathematical Statistics, Wiley (1980). A. Agresti, Categorical Data Analysis, Wiley (1990).
Overview of statistical packages, e.g. Minitab, Genstat, SAS, BMDP. Introduction to UNIX and LaTeX.
Revision of methods of classical statistics, e.g. t-tests. Presentation of results, graphical, written, spoken. Linear regression. Residuals, q-q plots, leverages, Box-Cox transformations, Orthogonal polynomials.
Analysis of particular experimental designs, e.g. randomized block, complete factorial experiment, also analysis of data from an unbalanced factorial experiment. Interpretation of interactions between factors.
Statistical methods for discrete data, contingency tables etc. Conditional tests, bio-assay, fitting a response curve to binomial data. Non-parametric methods. Graphics for multivariate data.
Modern regression, e.g. generalized additive models. Methods for survival data analysis and time series. Construction of fractional designs.
The above methods will be put into practice via S-Plus.
In these practical classes, emphasis is placed on the importance of clear presentation of analysis, so students are required to submit their solutions to the lecturer. Diploma students will be asked to give brief talks on their Applied Projects in March/April.
Appropriate books
J.M. Chambers & T.J. Hastie (eds.) Statistical Models in S, Wadsworth & Brooks (1992). Aitkin, Anderson, Francis & Hinde, Statistical Modelling in GLIM (1989). W.N. Venables & B.D. Ripley, Modern Applied Statistics with S-Plus, (1994). Glim 4 Manual (ed. Francis et al.), (1993).
This course introduces methods of solving some practical problems in operations research. It is essential to have an understanding of the Lagrangian methods, the simplex algorithm and related duality theory, although these will be reviewed briefly in the course.
Review of Lagrangian methods, the simplex algorithm and duality theory.
Complexity of algorithms: typical behaviour and worst-case behaviour. *NP-Completeness*.
Exponential complexity of the simplex algorithm. Polynomial-time algorithms for linear programming. The ellipsoid algorithm. *The projective algorithm*.
Linear problems that can be put in network form. Minimal spanning trees, maximal flow, problems of transportation type; general circulation problems. *The `out of kilter' algorithm*. *Shortest and longest paths; critical paths; project cost-time functions*.
Integer programming and tree searching. The branch and bound method.
The travelling salesmen problem.
Other topics may be included, for example: searching and sorting, quadratic programming, geometric programming, dynamic programming, decision trees, simulated annealing.
Appropriate books
M.S. Bazaraa, J.J. Jarvis & H.D. Sherali, Linear Programming and Network Flows, Wiley (2nd ed. 1990). F.S. Hillier & G.J. Lieberman, Introduction to Operation Research, Holden-Day (3rd ed. 1980). G.L. Nemhauser & L.A. Wolsey, Integer and Combinatorial Optimization, Wiley (1988). C. Papadimitriou & K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall (1982).
This short course will comprise a series of case studies which demonstrate the effective use of the S-PLUS statistical package within a X-windows environment on UNIX workstations. Particular emphasis will be given to the graphics capabilities of the package and to the writing of functions in the S language.
Appropriate books
R.A. Becker, J.M. Chambers & A.R. Wilks, The New S Language, Chapman and Hall, London (1988). J.M. Chambers & T.J. Hastie (eds.) Statistical Models in S, Chapman and Hall, London (1992). B. Everitt, A Handbook of Statistical Analyses using S-PLUS, Chapman and Hall, London (1994). W.N. Venables & B.D. Ripley, Modern Applied Statistics with S-PLUS, Springer, New York, (1994).
Characteristics of survival data; censoring. Definition and properties of the survival function, hazard and integrated hazard. Examples.
Review of inference using likelihood. Estimation of survival function and hazard both parametrically and non-parametrically (including discussion of life tables).
Explanatory variables: accelerated life and proportional hazards models. Special case of two groups (including discussion of standardised mortality ratios, the log-rank test). Goodness of fit.
Case studies and recent developments.
Principal book
D.R. Cox & D. Oakes, Analysis of Survival Data, London: Chapman & Hall (1984).
Other books
P. Armitage & G. Berry, Statistical Methods in Medical Research (2nd ed), Oxford: Blackwell (1987) [Chapter 14 for preliminary reading]. J.D. Kalbfleisch & R.L. Prentice, The Statistical Analysis of Failure Time Data, New York: Wiley (1980).
Time series analysis refers to problems in which observations are collected at regular time intervals and there are correlations among successive observations. Applications cover virtually all areas of Statistics but some of the most important include economic and financial time series, and many areas of environmental or ecological data.
In this course, I shall cover some of the most important methods for dealing with these problems. In the case of time series, these include the basic definitions of autocorrelations etc., then time-domain model fitting including autoregressive and moving average processes, spectral methods, and some discussion of the effect of time series correlations on other kinds of statistical inference, such as the estimation of means and regression coefficients.
Appropriate books
C. Chatfield, The Analysis of Time Series: Theory and Practice, Chapman and Hall (1975). Good general introduction, especially for those completely new to time series. P.J. Brockwell & R.A. Davis, Time Series: Theory and Methods, Springer Series in Statistics (1986). Comprehensive text; this will be my main reference for the mathematics of time series analysis. B.D. Ripley & W.N. Venables, Modern Applied Statistics with S-Plus, Springer (1994). Chapter 14 is a good introduction to time series.
Loss distributions
Standard loss distributions; parameter estimation.
Reinsurance
Types of reinsurance; moments of claim amounts paid by insurer and
reinsurer.
Aggregate claims
Risk models; compound distributions; exact and approximate calculation
of compound distributions; effect of reinsurance.
Ruin theory
Ruin probabilities over finite and infinite time horizons; compound
Poisson process; the adjustment coefficient; the Lundberg inequality;
the Cramer-Lundberg approximation; effect on ruin probability of
changing parameter values; effect of reinsurance.
Credibility theory
The credibility factor; Bayesian credibility theory; empirical Bayes
credibility theory.
Run-off triangles
Incurred but not reported and outstanding reported claims; methods for
projecting claims.
No Claims Discount (NCD) Systems
Discount categories; transition matrices; NCD projections.
Appropriate books
C.D. Daykin, T. Pentikainen & E. Pesonen, Practical Risk Theory for Actuaries and Insurers, Chapman and Hall (1993). J. Grandell, Aspects of Risk Theory, Springer (1991). R.V. Hogg & S.A. Klugman, Loss Distributions, Wiley (1984). I.B. Hossack, J.H. Pollard & B. Zehnwirth, Introductory Statistics with Applications in General Insurance, CUP (1983).
This course covers most of the syllabus for Paper C2 of the Institute of Actuaries.
Principles of design of experiments, natural variation and comparison of treatments, randomization, replication and control of error. The randomization theory.
Particular designs; completely randomized, paired comparisons, randomized blocks, complete factorial.
Fractional replications. Taguchi methods. Confounding. Latin squares. Split plots. Balanced incomplete blocks. Covariance analysis.
Appropriate books
Hinkelmann & Kempthorne, Design and Analysis of Experiments. Logothetis & Wynn, Quality through Design (1989). P.W.M. John, Statistical Design and Analysis of Experiments. Cochran & Cox, Experimental Design. Finney, Introduction to the Theory of Experimental Design. John & Quenouille, Experiments: Design and Analysis.
Multivariate data, graphical presentation. The multivariate normal distribution; partial correlations. Maximum likelihood estimates, likelihood ratio tests. Regression and MANOVA. Tests about covariance matrices. Canonical correlations. Principal components. Cluster analysis, discriminant analysis. Tree-based methods.
The course will use S-Plus.
Appropriate books
W.J. Krzanowski, Principles of Multivariate Analysis, Oxford Science (1990). K.V. Mardia, J.T. Kent & J.M. Biddy, Multivariate Analysis, London: Academic Press (1979). G.A.F. Seber, Multivariate Observations, (1984). Digby, Galwey & Lane, Genstat 5: A Second Course, (1989). W.N. Venables & B.D. Ripley, Modern Applied Statistics with S-Plus, (1994).
Elementary theory of Mendelian inheritance in relation to human disease. Single gene and polygenic disorders. Application of likelihood methods to mapping the human genome. Computer algorithms for probability calculations for large pedigrees and for multiple markers. Linkage and association mapping of disease suceptibility genes.
Appropriate books
B.S. Weir, Genetic Data Analysis, Sinauer Associates (1990). J. Ott, Analysis of Human Genetic Linkage (2nd ed.), Johns Hopkins University Press, Baltimore (1991). R.C. Elandt-Johnson, Probability Models and Statistical Methods in Genetics, Wiley (1971). K. Lange, Mathematical and Statistical Methods for Genetic Analysis, Springer (1997).
Each lecture will be a self-contained study of a recent problem in biostatistics. Topics will include AIDS forecasting, clinical trials, survival analysis, diagnostic testing and computer-aided diagnosis, illustrating the use of techniques such as sequential analysis, non-proportional hazards models, Bayesian prediction, latent class models, and shrinkage estimation.
Appropriate books
There are no appropriate books but relevant handouts will be provided.
Uses and aims of simulation in statistical inference.
Pseudo-random numbers. Generation of random variables; principles, techniques and examples.
Monte Carlo tests and their properties.
The bootstrap and jackknife. Bootstrap for estimation. Bootstrap confidence sets and hypothesis tests. Asymptotic theory. Iterated bootstrap. Bootstrap and dependent data.
Empirical and bootstrap likelihood.
Markov chain Monte Carlo in Bayesian and classical inference.
Appropriate reading
B. Efron & R.J. Tipshirani, An Introduction to the Bootstrap, New York: Chapman and Hall (1993). B.D. Ripley, Stochastic Simulation, London: Wiley (1987). A.E. Gelfand & A.F.M. Smith, `Sampling based approaches to calculating marginal densities.' Journal of the American Statistical Association 85, (1990), 398-409. J. Shao & D. Tu, The Jackknife and Bootstrap, New York: Springer (1995). A.F.M. Smith & G.O. Roberts, `Bayesian computation via the Gibbs sampler and related Markov Chain Monte Carlo methods.' Journal of the Royal Statistical Society B, 55, 1, (1993), 3-24.
Sequential Probability Ratio Test (SPRT).
Wald's identities and application to SPRT.
Choice of stopping barriers. Optimality of SPRT.
Composite hypotheses, characteristics of the SPRT.
Decision-theoretic formulation of sequential problems; Bayesian approach. Sequential testing; the DP equation of sequential analysis. Derivation of the SPRT as the optimal (Bayes) procedure in the case of two hypotheses. Asymptotics for the case of small sampling cost. Sequential testing (case when posterior expected loss is independent of observations).
Appropriate books
G.B. Wetherill & K.D. Glazebrook, Sequential Methods in Statistics, 3rd ed, Chapman and Hall (1986). H. Chernoff, Sequential Analysis and Optimal Design, SIAM (1972).
Michaelmas Term
After a brief review of foundational aspects of probability theory, the course will cover central topics of the modern theory together with many applications. Prior experience of intermediate-level probability would be useful but not indispensable (such as experience of some or all of the material of the Part IIB course on `Probability and Measure'). The course leads on to other Part 3 courses, especially `Stochastic Calculus and Applications'.
It is anticipated that the following principle topics will form the main elements of the course.
Appropriate books
G. Grimmett and D. Stirzaker, Probability and Random Processes, OUP, 1991. D. Williams, Probability and Martingales, CUP, 1991. P. Billingsley, Probability and Measure, Wiley, 1979. S. Kavrin and H. Taylor, A First Course in Stochastic Processes, Academic Press, 1975.
The aim of this course is to give an overview of the mathematical ideas underlying the pricing of derivative assets in financial markets. It touches on probabilistic notions such as martingales, Brownian motion and stochastic integration in an applied context. Topics covered will include:
Arbitrage-pricing theory; existence of equivalent martingale measures; hedging strategies; complete markets; incomplete markets and preference-based asset pricing.
The Black-Scholes model; pricing European options and the Black-Scholes formula; completeness of the model and the representation of martingales as stochastic integrals; pricing path-dependent and other exotic options.
Models for the term structure of interest rates; interest rates as a two-parameter stochastic process; mean reversion of spot rates; Gauss-Markov models; pricing interest-rate derivatives.
Appropriate book
To get a flavour of the material have a look at the book\nl D. Duffie, Dynamic Asset Pricing Theory, Princeton (1992).
For an elementary introduction obtain a copy of the printed lecture notes for the Part IIA course Stochastic Financial Models,
The emphasis will be on mathematical interesting and practically useful models of communication networks. Topics will be selected from amongst the following.
Recommended books
F.P. Kelly, S. Zachary and I. Ziedins (eds), Stochastic Networks: Theory and Applications, RSS Lecture Note Series 4, Oxford: Clarendon Press, 1996. Keith W. Ross, Multiservice Loss Models for Broadband Telecommunication Networks, London: Springer-Verlag, 1995. F.P. Kelly, Reversibility and Stochastic Networks, Chichester: Wiley, 1979. P. Whittle, Systems in Stochastic Equilibrium, Chichester: Wiley, 1986.
Lent Term
In this course we will study the topological dynamics of continuous mappings on the interval. For such maps $f$ we seek to understand the behaviour of orbits (sequences of points $x_0,x_1, x_2,\ldots$, where $x_{i+1}=f(x_i)$, and the way in which this behaviour changes as the map $f$ changes with a parameter. We will be concerned primarily with the topological properties of orbits and parameter space, rather than with measure-theoretic properties. For the problems we will study (restricting ourselves mainly to unimodal maps) there is a very full topological theory; much of this theory does not carry over immediately to dynamical systems of higher dimension, though there are very strong connections. As we go along, I will try to point out applications of our one-dimensional results to more complicated dynamical systems which are still the subject of active research.
Topics to be covered (time allowing) will include: families of continuous unimodal maps of the interval, bifurcations and chaotic behaviour, renormalisation, decomposition of the non-wandering set, topological entropy, kneading theory, and quantitative universality (Feigenbaum's $\delta$).
There are no specific prerequisites for the course, though those students who have taken a third-year course on Dynamical Systems or Nonlinear Dynamics will find these helpful.
Books
R. Devaney, Introduction to Chaotic Dynamics, Benjamin-Cummings (1986). S. van Strien, `Smooth dynamics on the interval', an article in the book New Directions in Dynamical Systems, edited by T. Bedford & J. Swift, London Mathematical Society Lecture Notes Series 127, Cambridge University Press (1988). W. de Melo and S. van Strien, One-dimensional Dynamics, Springer-Verlag (1993).
The first of these gives a flavour of the problems from a naive point of view, and contains some material which we will cover very quickly in the course. The second will be hard reading before taking the course, but covers one part of the material in a very similar style (though with less explanation) to the course. The last is a massive (and expensive) and difficult book, containing most of what was known about one-dimensional maps three or four years ago. Don't buy it unless you think you may go on in this direction.
Reading
T.M. Cover & J.A. Thomas, Elements of Information Theory, Wiley (1991). G.J. Chaitin, Algorithmic Information Theory, Cambridge University Press (1987). M. Li & P. Vitanyi, An Introduction to Kolmogorov Complexity and its Applications, Springer (1993). T.M. Cover, P. Gacs & R.M. Gray, `Kolmogorov's contribution to information theory and algorithmic complexity', Ann. Prob. 17 (1989), 840-865.
Ergodic Theory is the mathematical study of the long-term average behaviour of systems. The collection of all states of a system forms a space $X$. The evolution of a system is represented by a transformation $T:X\to X$, where $Tx$ is taken as the state at time 1 of a system which at time 0 is in state $x$. If one prefers a continuous variable for time, one can consider a one-parameter family $\{ T_t : t\in\R\}$ of maps of $X$ into itself. When the laws governing the behaviour of the system are unchanged with time, it is natural to suppose that $T_{t+s}=T_s\circ T_t$,so that $\{ T_t : t\in\R\}$ is a flow, or group action of $\R$ on $X$. Similarly we think of the iterates $\{T^n : n\in\N\}$ of a single transformation as the action of $\Z_+$ on $X$ (or the action of $\Z$ if $T$ is invertible). (We will only deal with the iterates of a single map $T$ in this course.)
In order to analyze a system mathematically, one needs to have some structure on $X$. Two important cases are
This course is intended as an introduction to ergodic theory. Topics that will be covered:
The remainder of the time (if any) will be devoted to topics chosen by the lecturer.
Some familiarity with measure theory will be assumed (the Part II(B) course Probability and Measure is certainly sufficient), but there is a brief handout available which covers the necessary measure theory background.
There are strong connections between ergodic theory and other branches of mathematics, particularly with Analysis, Dynamical Systems and Probability Theory. Students interested in this course may also want to consider it in connection with relevant courses from these areas (e.g. Dynamics of One-Dimensional Maps (Sparrow, Lent 1998), or Advanced Probability (Grimmett, Michaelmas 1997) and Entropy and Complexity (Suhov, Lent 1998) in a different direction).
Books
Suitable books for the course are
P. Walters, An Introduction to Ergodic Theory, Springer. K. Petersen, Ergodic Theory, Cambridge University Press. W. Parry, Topics in Ergodic Theory, Cambridge University Press.
Many large systems, such as communication networks, exhibit failure behaviour which becomes exponentially rare as some parameter of the system varies. Large deviations is the study of such exponentially decaying probabilities. It has applications to performance analysis, entropy, simulation methodology and hypothesis testing, among other topics.
This course will provide an introduction to large deviations, illustrated and motivated by examples. The examples have yet to be determined, but the theory covered will bear some resemblance to the following.
Literature
The following papers provide readable introductions to the course.
A. Weiss (1995). An introduction to large deviations for communication networks. IEEE Journal on Selected Areas in Communications 13, 938-952. P. Heidelberger (1995). Fast simulation of rare events in queueing and reliability models. ACM Transactions on Modeling and Computer Simulation 5, 43-85.
There are very few books on large deviations, and many of those are hard to read. But the following are recommended.
A. Shwartz & A. Weiss (1995). Large Deviations for Performance Analysis: Queues, Communications and Computing, Chapman & Hall. A. Dembo & O. Zeitouni (1993). Large Deviations Techniques and Applications, Jones & Bartlett.
We will survey the theory of stochastic integration with respect to continuous semimartingales and stochastic differential equations driven by Brownian motion. Practical motivation will be drawn from problems in engineering, finance and economics. In particular I plan to cover
Preliminaries: Filtrations and stopping times. Inequalities for continuous time martingales. Submartingale convergence theorem. Doob-Meyer decomposition. Quadratic variation of local martingales.
Brownian motion: Construction. Strong Markov property. Reflection principle. Hitting probabilities, recurrence and transience. Properties of sample paths.
Stochastic integration: Predictable and optional processes. Integration with respect to Brownian motion. Change of variable formula. Covariation process, Kunita-Watanabe inequality. Integration with respect to continuous local martingales. Levy's characterisation of Brownian motion. Representation of continuous local martingales in terms of Brownian motion. The Girsanov theorem.
Stochastic differential equations: Strong solutions (It\^o theory). Comparison and approximation theorems. Weak solutions and the martingale problem. The one-dimensional case.
Applications will be selected from among the following:
Filtering theory: The linear filtering problem. Kalman-Bucy
filter.
Partial differential equations: The Dirichlet problem. The heat
equation. Feynman-Kac formula.
Optimal stopping: Snell envelope. Existence and uniqueness of
solutions.
Stochastic control: The Hamilton-Jacobi-Bellman equation. Markov
controls. Linear regulator.
References
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 3rd edition, Springer-Verlag (1992). I. Karatzas & S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer-Verlag (1991). R. Durrett, Stochastic Calculus, A Practical Introduction, CRC Press (1996). D. Revuz & M. Yor, Continuous Martingales and Brownian Motion, 2nd edition, Springer-Verlag (1994).
A second course on mathematical statistics which develops in a rigorous way the properties of classical procedures of estimation and hypothesis testing, as well as decision theoretic and Bayesian approaches to statistics. The key concerns are with defining criteria of optimality and examination of standard statistical procedures against these criteria.
Decision theory and Bayesian inference
Sample, parameter and action spaces. Loss and risk functions. Risk
set, decision rules, admissibility, minimax rules, randomised rules.
[3]
The Bayesian paradigm. Prior and posterior distributions. Bayes estimates and Bayes confidence regions. Bayes decision rules. Philosophy of Bayesian inference. Model selection, Bayes factors. Bayesian computation via Monte Carlo simulation. [5]
Classical inference
Sufficiency, minimal sufficiency and completeness. Exponential
families. [2]
Point estimation. Cramer-Rao lower bound, Rao-Blackwell theorem, Lehmann-Scheffe Theorem. [2]
Hypothesis testing. Monotone likelihood ratio, uniformly most powerful tests. Confidence bounds. Unbiasedness of tests. Uniformly most powerful unbiased tests in one-parameter exponential families. Nuisance parameters. Conditional tests: comparing two Poisson or binomial populations. The generalized likelihood ratio test (proof for simple null hypothesis). [6]
Maximum likelihood estimates and their asymptotic distribution. Wald, score and likelihood ratio tests. Multiparameter problems. Profile likelihood. [4]
Bootstrap inference. [2]
Appropriate books
D.R.Cox & D.V.Hinkley, Theoretical Statistics, Chapman and Hall, 1974. T.S.Ferguson, Mathematical Statistics: A Decision-Theoretic Approach, Academic Press, 1967. E.L.Lehmann, Theory of Point Estimation, Wiley, 1983. E.L.Lehmann, Testing Statistical Hypotheses, Wiley, 1986. B.W.Lindgren, Statistical Theory, Collier & Macmillan, 1976. J.A.Rice, Mathematical Statistics and Data Analysis, Wadsworth& Brooks/Cole, 1988. J.Q.Smith, Making Decisions: A Bayesian Approach, Chapman & Hall.
This course is examined in the Diploma in Mathematical Statistics (Paper A).
Review of likelihood function
Asymptotic distribution of the maximum likelihood estimator (outline
only). Approximate confidence intervals for parameters. Asymptotic
distribution of deviance, and of reduction in deviance when fitting
nested models. [4]
Linear models with normal error function
Construction of analysis of deviance for testing nested models; use of
$F$-tests. Orthogonality of sets of parameters, and analysis of
variance tables. [4]
Exponential families and generalized linear models
Iterative solution of likelihood equations. Regression for binomial
data; use of logit and probit link functions. The Poisson error
function, contingency tables. [6]
Model criticism
*Examination of residuals and leverages*. [2]
[The suggested division of the 16 hours is 10 hours of lectures and 6 hours of practical classes, the order to be decided by the lecturer during the term. The practical classes will be given using the statistical package GLIM on CATAM.]
Appropriate books
M.Aitkin, D.Anderson, B.Francis & J.Hinde, Statistical Modelling in GLIM, Oxford Science Publications, 1989. A.J.Dobson, An Introduction to Generalized Linear Models, Chapman and Hall, 1990.
This course is examined in the Diploma in Mathematical Statistics (Paper A).
Discrete-time chains
Definition and basic properties, the transition matrix. Calculation of
$n$-step transition probabilities. Communicating classes, closed
classes, absorption, irreducibility. Calculation of hitting
probabilities and mean hitting times; survival probability for birth
and death chains. Stopping times and statement of the strong Markov
property. [5]
Recurrence and transience; equivalence of transience and summability of $n$-step transition probabilities; equivalence of recurrence and certainty of return. Recurrence as a class property, relation with closed classes. Simple random walks in dimensions one, two and three. [3]
Invariant distributions, statement of existence and uniqueness up to constant multiples. Mean return time, positive recurrence; equivalence of positive recurrence and the existence of an invariant distribution. Convergence to equilibrium for irreducible, positive recurrent, aperiodic chains *and proof by coupling*. Long-run proportion of time spent in given state. [3]
Time reversal, detailed balance, reversibility; random walk on a graph. [1]
Continuous-time chains
The Poisson process; definition by holding times, definition by
stationary independent increments, definition by infinitesimal
transition probabilities; equivalence of these definitions. the Markov
property and relation by the memoryless property of the exponential
distribution. Conditional distribution of a single jump in an
interval. [3]
Birth processes; definition by holding times, forward equations and definition by transition probabilities, definition by infinitesimal transition probabilities; *equivalence of these definitions*. Necessary and sufficient conditions for explosion. [2]
$Q$-matrices, backward and forward equations. Existence and uniqueness of the semi-group in the case of finite state-space. Calculation of transition probabilities. [1]
General continuous-time chains; definition by jump chain and holding times, definition by transition probabilities, definition by infinitesimal transition probabilities; *equivalence of these definitions in the case of these definitions in the case of finite state-space*. (The minimal process only will be considered.) Simple conditions for non-explosion. Stopping times and statement of the strong Markov property. [3]
Class structure. Calculation of hitting probabilities. Recurrence and transience. Invariant distributions, relation with distributions invariant for the jump chain. Positive recurrence; equivalence of positive recurrence and the existence of an invariance distribution in the non-explosive case. Statement of convergence to equilibrium for irreducible, positive-recurrent chains. [2]
Birth and death chains. Application to queueing theory. [1]
Appropriate books
W.Feller, An Introduction to Probability Theory and its Applications, Vol.I, 3rd edition, Wiley, 1968 . E.Cinlar, Introduction to Stochastic Processes, Prentice-Hall, 1975. P.G.Hoel, S.C.Port & C.J.Stone, Introduction to Stochastic Processes, Houghton Mifflin, 1972. *G.R.Grimmett & D.R.Stirzaker, Probability and Random Processes, 2nd edition, Oxford University Press, 1992. S.Karlin & H.M.Taylor, A First Course in Stochastic Processes, 2nd edition, Academic Press, 1975.
Networks
Oriented networks, flows, circulations, potentials and differentials;
paths and cuts; Painted Network Algorithm. [3]
Min path Max tension Theorem; Max flow Min cut Theorem (review); conditions for the existence of feasible differentials and feasible distributions, and algorithms to find them. Examples and applications. [3]
Optimal distribution problems. Kilter conditions for optimality. Simplex-0n-a-graph algorithm, and relationship with the simplex algorithm. Out-of-kilter algorithm, and termination in simple cases. Examples and applications. [4]
Convex optimization
Brief introduction to convex optimization. Strong Lagrangian problems;
sufficient conditions for convexity of the optimal value function.
[The supporting hyperplane theorem may be stated but not proved.]
Karush-Kuhn-Tucker conditions. [2]
Quadratic programming with linear constraints. Examples and applications. [2]
Geometric programming. Examples and applications. [2]
Appropriate books
M.S.Bazaraa, J.J.Jarvis & H.D.Sherali, Linear Programming and Network Flows, 2nd edition, Wiley, 1990. R.Rockafeller, Network Flows and Monotropic Optimization, Wiley, 1984. A.L.Peressini, F.E.Sullivan & J.J.Uhl, Mathematics of Nonlinear Programming, Springer, 1988.
Poisson process: infinite server queue, shot-noise process, nonhomogeneous and compound Poisson processes. [3]
Renewal theory: limit theorems, equilibrium distribution and inspection paradox, renewal reward processes, inventory and maintenance models, Little's formula. Regenerative processes: asymptotic normality and simulation. [4]
Markov chains: embedded M/G/1 and G/M/1 queues. Dams and insurance ruin problems, duality. Heavy traffic in queues. [3]
Continuous-time Markov chains: Erlang loss formula, networks of queues, epidemic thresholds. [3]
Brownian motion: definition, Brownian Bridge. Hitting times, absorption and reflection, geometric Brownian motion. Value of a stock option. [3]
Appropriate books
W.Feller, An Introduction to Probability Theory and Its Applications Vol.II, 2nd edition, Wiley, New York, 1971. G.R.Grimmett & D.R.Stirzaker, Probability and Random Processes, 2nd edition, Oxford University Press, 1992. S.M.Ross, Stochastic Processes, Wiley, New York, 1983. R.W.Wolff, Stochastic Modelling and the Theory of Queues, Prentice-Hall, New Jersey, 1989.
The standard model of a communication channel. Concepts of coding and decoding. Error detection and error correction. Noise-free coding: decipherability, the Kraft inequality. [2]
Entropy, conditional entropy, mutual entropy. [3]
Rate of symbol generation and information rate of a discrete source $u_t$. Information rates of Bernoulli and Markov sources. [2]
Reliable transmission through a discrete channel. Reliable transmission rate and channel capacity. Relation of channel capacity to maximized mutual entropy rate. Random coding. Calculation of capacity and optimal channel coding for the discrete memoryless symmetric channel. [5]
Linear codes: syndrome decoding. Examples of codes. The Singleton, Hamming and Gilbert-Varshamov bounds. *Cyclic codes and BCH codes*. [4]
Appropriate books
R.B.Ash, Information Theory, Dover, 1990. R.E.Blahut, Principles and Practice of Information Theory, Addison-Wesley, 1987. *C.M.Goldie & R.G.E.Pinch, Communication Theory, Cambridge University Press, 1991. R.W.Hamming, Coding and Information Theory, 2nd edition, Prentice-Hall, 1986. J.H.van Lint, Introduction to Coding Theory, Springer-Verlag, 1982 (currently out of print). D.Welsh, Codes and Cryptography, Oxford University Press, 1988.
Dynamic programming
Markov decision processes in discrete time, the dynamic programming
equation for finite-horizon problems, the principle of optimality and
the optimality equation. Examples. [5]
LQG systems
The LQG case (linear systems, quadratic costs, Gaussian noise). The
Riccati recursion. Controllability and observability. Limiting results
in the infinite horizon case. Imperfect process observation. Certainty
equivalence. The Kalman filter. Examples. [5]
Continuous-time models
Formal treatment of the continuous-time case for deterministic
processes and jump processes. Formal derivation of
the Pontryagin maximum principle; its relation to Hamiltonian and
path-integral concepts. Examples. [6]
Appropriate books
D.P.Bertsekas, Dynamic Programming, Prentice-Hall, 1987. S.Ross, Introduction to Stochastic Dynamic Programming, Academic Press, 1983. P.Whittle, Optimization Over Time, Vols. I,II, Wiley, 1982, 1983.
The elements of measure theory are developed in tandem with their application as the rigorous basis for probability. This course provides essential skills needed in advanced probability and analysis, culminating in the Strong Law of Large Numbers, the Pointwise Ergodic Theorem and Central Limit Theorem.
Measure spaces, $\sigma$-algebras, $\pi$-systems and uniqueness of extension, statement of Carath\'eodory's extension theorem. *Construction of Lebesgue measure on $\R$*. Borel $\sigma$-algebra of $\R$, Lebesgue-Stieltjes measures and probability distribution functions. Independence of events, independence of $\sigma$-algebras. Borel-Cantelli lemmas. Kolmogorov's zero-one law. [6]
Measurable functions, random variables, independence of random variables. Construction of the integral, expectation. Convergence in measure and convergence almost everywhere. Fatou's lemma, monotone and dominated convergence, uniform integrability, differentiation under the integral sign. Discussion of product measure and statement of Fubini's theorem. [5]
Chebychev's inequality, tail estimates. Jensen's inequality. Completeness of $L^p$ for $1\leqslant p<\infty$. H\"older's and Minkowski's inequalities. [5]
The Hilbert space $L^2$, variance and covariance, Gaussian case. The multivariate normal distribution. [2]
The strong law of large numbers, proof for random variables with fourth moments. Measure preserving transformations, Bernoulli shifts. Maximal ergodic theorem and Birkhoff's almost everywhere ergodic theorem, proof of the strong law. [4]
The Fourier transform of a finite measure, characteristic functions, statement of uniqueness. Weak convergence, statement of Levy's convergence theorem for characteristic functions. The central limit theorem. Cramer's theorem on large derivations (in $\R^1$). [2]
Appropriate books
P.Billingsley, Probability and Measure, 2nd edition, Wiley, 1987. R.M.Dudley, Real Analysis and Probability, Wadsworth, 1989. D.Williams, Probability with Martingales, Cambridge University Press, 1991
Utility and asset pricing: Utility theory; risk aversion and risk neutrality. The capital asset pricing model. [3]
Dynamic models: Introduction to dynamic programming; optimal stopping and optimal portfolio selection. [4]
Brownian motion: Introduction to Brownian motion; hitting-time distributions. [4]
Option pricing: Arbitrage pricing theory; prices as martingales.
Option pricing; the random-walk case; the Black-Scholes formula. [5]
Appropriate books
R.A. Jarrow, Finance Theory, Prentice-Hall, 1988. *J. Hull, Options, Futures and Other Derivative Securities, Prentice-Hall, 1989.