**Aims & Learning Objectives:**
**Aims: **To develop a formal basis for methods of statistical inference including criteria for the comparison of procedures. To give an in depth description of the asymptotic theory of maximum likelihood methods and hypothesis testing.
**Objectives: **
On completing the course, students should be able to:
* calculate properties of estimates and hypothesis tests
* derive efficient estimates and tests for a broad range of problems, including applications to a variety of standard distributions.
**Content: ** Revision of standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and normal, and their interrelationships.
Sufficiency and Exponential families.
Point estimation: Bias and variance considerations, mean squared error. Rao-Blackwell theorem. Cramer-Rao lower bound and efficiency. Unbiased minimum variance estimators and a direct appreciation of efficiency through some examples. Bias reduction. Asymptotic theory for maximum likelihood estimators.
Hypothesis testing: Hypothesis testing, review of the Neyman-Pearson lemma and maximisation of power. Maximum likelihood ratio tests, asymptotic theory. Compound alternative hypotheses, uniformly most powerful tests. Compound null hypotheses, monotone likelihood ratio property, uniformly most powerful unbiased tests. Nuisance parameters, generalised likelihood ratio tests.
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