Statistics I*

obligatory courses

The aim of the course is to present a background to statistical theory. The course covers: population characteristics and their sample counterparts, structure of statistical models from classical and Bayesian perspectives, problems of parametric and nonparametric estimation, testing statistical hypotheses and an introduction to linear models and logistic regression.

The aim of the course is to present basic ideas of statistical analysis.

The course covers:

1) Properties of a random sample: empirical distributions, population characteristics and their sample counterparts, statistical models and exponential families.

2) Principles of data reduction: the sufficiency principle and likelihood.

3) Parameter estimation: methods of estimation, methods of evaluating estimators, consistency, efficiency.

4) Hypothesis testing: likelihood ratio tests, Bayesian tests, methods of evaluating tests, most powerful tests, the Neyman-Pearson lemma, loss function optimality, asymptotic distribution of LRT.

5) Interval estimation: pivotal quantities, Bayesian intervals, evaluation of intervals, approximate maximum likelihood intervals.

6) Analysis of variance and regression, logistic regression.

The program is in principle the same as for the basic lecture. However, topics will be treated more deeply. The lecture is addressed to students with deeper interest in the subject.

Bickel, P.J.; Doksum; K.A. [2007] Mathematical Statistics vol. 1 (Basic Ideas and Selected Topics) Pearson Education, Inc.

Knowledge and skills:

1) Knows the basic parameters which characterise the population and their sample equivalents: average, variance, standard deviation, moments, skewness, kurtosis; quantiles; knows basic properties of the empirical distribution and the Kolmogorov Smirnov theorem.

2) Can build simple statistical models describing actual phenomenon; in particular, families of probability distributions and exponential families.

3) Understands the concept of sufficiency and can determine sufficient statistics for an exponential family. Can estimate parameters using method of moments, maximum likelihood, Bayesian methods, the EM algorithm; knows concepts such as unbiased, best unbiased estimator and can evaluate the performance of estimators using mean squared error and loss function optimality.

4) Is aware of the Cramer-Rao lower bound and definition of consistency of an estimator. Can calculate the asymptotic distribution of the estimator when the hypotheses for the Cramer-Rao lower bound hold.

5) Is aware of the concepts related to testing a null hypothesis versus an alternative: significance, power, p-value. Understands likelihood ratio tests, Bayesian tests, uniformly most powerful tests, the Neyman-Pearson lemma, the asymptotic distribution of the LRT.

6) Understands the basic concepts of Interval estimation: how to use pivotal quantities, how to compute Bayesian intervals, methods of evaluating an intervals, approximate maximum likelihood intervals.

7) Can use the chi-square test for homogeneity in simple situations.

8) Understands the principles of Analysis of Variance.

9) Understands basic linear models, linear regression and logistic regression.