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Convergence of random variables: convergence in probability and distribution. Chebyshev's inequality. Central Limit Theorem. Order Statistics. Distribution of order statistics. Properties of estimators The sample, parametric models, definition of a statistic; Estimators: unbiasedness, consistency, sufficiency, the factorisation criterion, mean squared error. Minimum variance unbiased estimators, Cramer-Rao lower bound without proof, attainment by the exponential family. Maximum likelihood estimation The likelihood function for one and two parameters. Finding MLE's, the Newton-Raphson methods. General properties: uniqueness, sufficiency, turning points are maxima for exponential family. Asymptotic properties without proof: consistency,unbiasedness, efficiency, normality. Hypothesis testing and confidence intervals Hypotheses, significance, power. Neyman-Pearson lemma. Uniformly most powerful tests, two-sided tests. Confidence Intervals Calculation of Conf
idence Intervals - The Pivotal Quantity Method. Relationship between tests and Confidence intervals Bayesian Inference Bayes' theorem for one or more parameters. Comparison of Normal means. Prior distribution and their specification. Non-informative and Improper Priors. Subjectively assessed priors. Conjugate Priors.
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