1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Mathematical Statistics

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Mathematical Statistics - opis przedmiotu

Cel przedmiotu

Theoretical background of statistical inference.

Wymagania wstępne

Zapisz zmiany

Passed lecture on probability theory.

Zakres tematyczny

 

Lecture

1. Normal distribution and related distributions. Random variable and its basic characteristics, random variable with normal distribution (repeat). Chi-square, Student's t, F-Snedecor distribution.
2. Statistical model. The purpose of statistical research, statistical space, the concept of sample, the theorem on the convergence of the empirical distribution function. Probability distributions of selected statistics from the sample, Fisher's theorem. Sufficient statistics, factorization theorem. Complete statistics. Exponential families of probability distributions, natural parameter space, theorem on the form of sufficient statistics, Lehmann's theorem.
3, Estimation theory. Unbiased estimators with minimum variance, Lehmann-Scheffe theorem, Rao-Blackwell theorem. Method of moments. Maximum likelihood method. Confidence intervals.
4. The theory of statistical hypothesis testing. Basic concepts. Uniformly most powerful tests, Neyman-Pearson lemma. Uniformly most powerful tests in models with monotonic likelihood ratio, Karlin-Rubin theorem.
 

Class

1. Repetition and supplementation of knowledge from probability theory. Normal distribution and its properties. Using statistical tables. Distribution of multidimensional random variables and its basic numerical characteristics. Functions of random variables and their distributions.
2. Independence of variables. The concept of a sample - determining the distribution in the case of a simple random sample. Checking whether given random variables are statistics. Determining, based on Fisher's theorem, the distributions of selected random variables.
3. Conditional distributions. Determining sufficient statistics by definition. Application of the factorization criterion to determine sufficient statistics.
4. Checking whether a given family of probability distributions is an exponential family. Using Lehmann's theorem to determine sufficient and complete statistics.
5.The concept of an estimator. Calculating the expected value and variance of selected estimators. Checking their load-bearing capacity.
6. Application of the Lehmann-Sheffe and Rao-Blackwell theorems to the construction of unbiased estimators with minimum variance.
7. Use of the method of moments and the maximum likelihood method to determine estimators of selected parameters.
8. Construction of confidence intervals for selected parameters. Determining interval scores based on observed values. Using appropriate statistical tables.
9. Calculating the probability of making a type I and type II error. Test power function.
10. Constructing uniformly strongest tests.
 

Laboratory

1. Introductory classes on the software used.
2. Calculating probabilities. Illustration of how the Central Limit Theorem works.
3. Basic distributions of mathematical statistics. Determining quantiles and critical values.
4. llustration of the operation of the empirical distribution function convergence theorem.
5. Point estimation.
6. Determining confidence intervals for normal distribution parameters. Testing the impact of the confidence level and sample size on the length of the intervals.
7. Basics of statistical hypothesis verification: type 1 and type 2 error. Critical area and p-value.
8. Testing statistical hypotheses in the normal model.

Metody kształcenia

Traditional lecture (chalk and blackboard only for the most important formulations, proofs of theorems), during classes solving previously given tasks (calculation tasks, carrying out proofs with simplifying assumptions), during the laboratory, using procedures of a selected statistical package to illustrate some of the concepts and theorems.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. The student's preparation for the exercises is verified by checking the knowledge (concepts, properties, theorems) necessary to solve the next task on the list (lack of preparation for the exercises is taken into account in the final grade for the exercises). Colloquia with tasks of varying difficulty, allowing for the assessment of whether the student has achieved the minimum learning outcomes.
2/ The grade for the laboratory is based on a colloquium with tasks of various levels of difficulty, allowing to determine the degree of mastery of statistical tools and the ability to draw correct conclusions based on the obtained analysis results.
3/ Written exam (1st term) with questions referring directly to concepts, statements, as well as questions checking the understanding of the acquired knowledge. Make-up exam in oral form, type of questions as above.

The grade for the course consists of the grade for exercises (30%), the grade for the laboratory (20%) and the grade for the exam (50%). The condition for taking the exam is a positive grade from the exercises. The condition for passing the course is positive grades from exercises, laboratory and exam.

Literatura podstawowa

1. G. Casella, R. L. Berger, Statistical Inference, Wadsworth, Belmont, California 1990.
2. N. H. De Grot. Optimal Statistical Decisions, New York 1970. 
3. E. L. Lehmann, Testing Statistical Hypothesis, Second edition. Wiley, New York 1986.
4. S. Zacks, The Theory of Statistical Inference, New York 1971. 

Literatura uzupełniająca

1. T. Ferguson, Mathematical Statistics - a Decision Theoretic Approach, New York 1967. 

Uwagi

Zmodyfikowane przez dr hab. Stefan Zontek, prof. UZ (ostatnia modyfikacja: 07-02-2024 11:55)