Basics of modern mathematical statistics This textbook provides a unified and self contained presentation of the main approaches to and ideas of mathematical statistics. It collects the basic mathematical ideas and tools needed as a basis for more serious studies or even independent research in statistics. The majority of existing textbooks in mathematical statistics follow the classical asymptotic framework. Yet as modern statistics has changed rapidly in recent years new methods and approaches have appeared. The emphasis is on finite sample behavior large parameter dimensions and model misspecifications. The present book provides a fully self contained introduction to the world of modern mathematical statistics collecting the basic knowledge concepts and findings needed for doing further research in the modern theoretical and applied statistics. This textbook is primarily intended for graduate and postdoc students and young researchers who are interested in modern statistical methods Basic notions Parameter Estimation for an i.i.d. Model Regression Estimation Estimation in Linear Models Bayes Estimation Testing a Statistical Hypothesis Testing in Linear Models Some other Testing Methods Deviation Probability for Quadratic Forms