What This Document Is
This document contains a set of practice exercises for STAT 5102: Theory of Statistics II, offered at the University of Minnesota Twin Cities. Specifically, it’s a homework assignment designed to reinforce understanding of core statistical concepts covered in the course. The assignment focuses on applying theoretical knowledge to practical problems involving likelihood functions, estimation, and asymptotic properties of estimators. It builds upon foundational statistical theory and introduces more advanced techniques.
Why This Document Matters
This assignment is invaluable for students enrolled in a graduate-level theoretical statistics course. Successfully completing these exercises will solidify your grasp of key concepts like maximum likelihood estimation, Fisher information, and asymptotic distributions. It’s best utilized *after* attending lectures and reviewing related course materials, serving as a crucial step in mastering the subject matter. Working through these problems will prepare you for more complex statistical modeling and inference tasks, and is excellent preparation for exams. Students aiming for a strong understanding of statistical theory will find this particularly beneficial.
Common Limitations or Challenges
This assignment focuses on the *application* of statistical theory, and does not provide a comprehensive review of the underlying concepts themselves. It assumes a solid foundation in probability theory, statistical inference, and calculus. The problems require independent problem-solving skills and a willingness to engage with potentially challenging mathematical derivations. It does not offer step-by-step solutions or detailed explanations of the reasoning behind each answer.
What This Document Provides
* A series of problems centered around deriving likelihood functions for various probability distributions.
* Exercises focused on calculating derivatives of log-likelihood functions.
* Opportunities to determine Fisher information for different statistical models.
* Problems designed to test your ability to find Maximum Likelihood Estimators (MLEs).
* Tasks requiring you to assess the properties of MLEs, including local maximization and concavity.
* Exercises involving the derivation of asymptotic distributions for estimators.
* Problems connecting MLEs to method of moments estimators.
* Practice constructing asymptotic confidence intervals.
* A challenge problem involving the sample median as an MLE.
* Exercises related to Poisson distributions and MLE estimation.
