# STATISTICS II

# STATISTICS II

Knowledge of elementary mathematics. Basic notions of differential and integral calculus.

Good knowledge of the basic tools of descriptive statistics and of the basic concepts of probability theory.

The aim of the course is to enhance the knowledge of the main statistical-inferential methods adopted in economic, social and financial analysis. At the end of the course, students will be able to properly deal with the main statistical inferential techniques, such as estimators, confidence intervals and parametric and non-parametric hypothesis testing.

Probability, probability space, random variables, cumulative distribution function and its properties. Discrete and continuous random variables. Moments. Expectation and variance. Linear transformations of random variables.

Bivariate random variables. Joint and marginal cumultaive distribution functions. Bivariate discrete and continuous random variables. Joint, marginal and conditional probability density functions. Some comments on k-variate random vectors. Independent random variables. Linear combinations of random variables.

Markov inequality, Chebyshev inequality and its generalization.

Sequence of random variables, almost sure convergence, convergence in probability, mean squared convergence, convergence in distribution.

Weak law of large numbers by Chebychef, strong law of large numbers by Kolmogorov, Fundamental Limit Theorem.

Sample, random sample. Statistical model, parametric statistical model. Aims of inference.

Sample statistics: sample mean and its properties, sample variance and its properties, difference of sample means, ratios of sample mean and variances.

Point estimators. Mean squared error. Relative efficiency. Uniformly most efficient estimators. Unbiased estimators. Rao-Cramer inequality. Efficient estimators. Estimating the precision. Relative root mean squared error. Large sample properties of point estimators: asymptotic unbiasedness, weak and strong consistency and mean squared error consistency.

Sufficient statistic. Factorization criterion. Minimal sufficient statistic. Rao-Blackwell Theorem. Complete statistics. Lehmann-Scheffè theorem. Exponential family of distributions and sufficient and complete statistics.

Likelihood function. Maximum likelihood estimators and their properties. Some important applications.

Estimators based on the method of moments.

Interval estimator. Pivotal quantity technique. Large sample confidence intervals. Some important applications. Large sample confidence intervals based on the asymptotic properties of maximum likelihood estimators.

Testing hypotheses. Parametric tests. Test, test statistic, power function, type I and II errors, significance level. Test properties: unbiased tests, uniformly most powerful tests, consistent tests. P-value. Neyman-Pearson Lemma. Uniformly most powerful tests for composite hypotheses. Some widely applied tests. Generalized likelihood ratio test.

Testing means for paired data. One-way analysis of variance.

Mood., M., Graybill, F.A., Boes, D.C.

Introduzione alla statistica

McGraw-Hill

Supplementary material (slides, exercises) is available on the moodle platform.

Lectures.

The exam conssts in a written and oral tests. The written test with exercises is aimed to evaluate the theoretical knowledge and the capability of applying the inferential techniques in economic, business and financial contexts.

When the written test is satisfactory, the student will face an oral test, based on questions on the main course topics, aimed to evaluate the knowledge of the theoretical basis of the inferential procedures and the ability of correctly interpreting the inferential conclusions.

Two written midterms formed by exercises are scheduled. The topics of the first are random variables and vectors, probability inequalities, sequence of random variables, asymptotic theorems, point estimators and their properties. The second midterm deals with the likelihood function, interval estimation and testing hypothesis.

If both midterms are satisfactory, the student has to pass only the oral test to conclude the exam.

None.