SH501:Advanced Statistical Inference

Course Code: SH501

Course Title: Advanced Statistical Inference

Level: Master's / Ph.d

Instructor: Mr. Rahul

Intensive 5-week course

Recordings will be available till August'2025

Course Description

Week 1:

Day 1- Introduction to Statistical Inference

Overview of statistical inference

Review of probability theory and random variables 

Day 2- Sufficiency and Minimal Sufficiency

Definition and concept of sufficient statistics

Factorization Theorem

Minimal Sufficiency 

Day 3- Completeness and Exponential Families

Completeness of a statistic

Completeness in exponential families

Properties of exponential families

Day 4- Ancillary Statistic and Basu's Theorem

Definition and examples of ancillary statistics

Basu’s Theorem and applications in inference 

Day 5- Problem set 1

Week 2:

Day 1- Unbiased Estimation

Definition and properties of unbiased estimators

Uniformly minimum variance unbiased estimation (UMVUE) 

Day 2- Rao-Blackwell Theorem

Statement and proof of the Rao-Blackwell theorem

Application of Rao-Blackwellization 

Day 3- Lehmann-Scheffe's Theorem

Statement and proof of the Lehmann-Scheff ́e theorem

Construction of UMVUE using Lehmann-Scheffe 

Day 4- Cramer-Rao Inequality

Derivation and interpretation of the Cramer-Rao lower bound

Efficiency of estimators 

Day 5- Problem set 2

Week 3:

Day 1- Consistent Estimators

Definition and properties of consistent estimators

Methods for proving consistency 

Day 2- Method of Moments Estimators

Introduction to method of moments

Examples and applications 

Day 3- Maximum Likelihood Estimators

Derivation and properties of MLEs

Asymptotic properties of MLEs 

Day 4- MLE vs Method of Moments

Comparison of MLE and Method of Moments

Examples where each method is preferable 

Day 5- Problem Set 3

Week 4:

Day 1- Interval Estimation and Pivotal Quantities

Confidence intervals and coverage probability

Construction of intervals using pivotal quantities 

Day 2- Neyman-Pearson Lemma

Statement and proof of the Neyman-Pearson lemma

Application to hypothesis testing 

Day 3- Most Powerful Tests

Construction of most powerful tests

Examples and applications in hypothesis testing 

Day 4- Monotone Likelihood Ratio (MLR) Property

Definition and properties of MLR

Connection between MLR and UMP tests 

Day 5- Problem Set 4

Week 5:

Day 1- Uniformly Most Powerful (UMP) and UMP Unbiased Tests

Construction of UMP tests for families with MLR property

UMP Unbiased tests for exponential families 

Day 2- Likelihood Ratio Tests

Likelihood ratio tests and their properties

Application of likelihood ratio tests to real-world problems 

Day 3- Large Sample Tests

Asymptotic distribution of test statistics

Examples of large sample tests 

Day 4- Course Review

Review of key concepts from the entire course

Open discussion and Q&A on difficult topics 

Day 5- Exam