### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title THEORY OF STATISTICAL INFERENCE Code MATH361 Coordinator Dr Y Zhang Mathematical Sciences Yi.Zhang@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2021-22 Level 6 FHEQ Second Semester 15

### Aims

To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options.

### Learning Outcomes

(LO1) To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

(LO2) To acquire an understanding of the blossoming area of Bayesian approach to inference.

(S1) Problem solving skills

(S2) Numeracy

### Syllabus

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.

### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH263 Statistical Theory and Methods I; MATH264 STATISTICAL THEORY AND METHODS II; MATH365 MEASURE THEORY AND PROBABILITY

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment open book and remote There is a resit opportunity. Standard UoL penalty applies for late submission. This is an anonymous assessment. Assessment Schedule (When) :Second seme  1 hour time on task    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Homework 4 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Second semester  equivalent to 2-5 si    10
Homework 5 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Second semester  equivalent to 2-5 si    10
Homework 3 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Second semester  equivalent to 2-5 si    10
Homework 2 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Second semester  equivalent to 2-5 si    10
Homework 1 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Second semester  equivalent to 2-5 si    10