Mathematical Statistics III
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Statistical methods used in practice are based on a foundation of statistical theory. One branch of this theory uses the tools of probability to establish important distributional results that are used throughout statistics. Another major branch of statistical theory is statistical inference. It deals with issues such as how do we define a "good" estimator or hypothesis test, how do we recognise one and how do we construct one? This course is concerned with the fundamental theory of random variables and statistical inference.
This course provides an introduction to fundamental distribution theory together with the underlying basics in statistical inference.
The ability to perform calculations with standard univariate and multivariate distributions, an understanding of the basic asymptotic theory for sequences of independent random variables, an understanding of the basic principles and key results of classical statistical inference and the ability to apply them to a range of statistical problems.
Topics covered are: calculus of distributions, moments, moment generating functions; multivariate distributions, marginal and conditional distributions, conditional expectation and variance operators, change of variable, multivariate normal distribution, exact distributions arising in statistics; weak convergence, convergence in distribution, weak law of large numbers, central limit theorem; statistical inference, likelihood, score and information; estimation, minimum variance unbiased estimation, the Cramer-Rao lower bound, exponential families, sufficient statistics, the Rao-Blackwell theorem, efficiency, consistency, maximum likelihood estimators, large sample properties; tests of hypotheses, most powerful tests, the Neyman-Pearson lemma, likelihood ratio, score and Wald tests, large sample properties.
This course is not recorded as prequisite for other courses.
Rice, J.A. Mathematical Statistics and Data Analysis, 3rd Edn.References: Casella, G. and Berger, R.L. Statistical Inference, 2nd Edn.