Group A (Marks: 30)
(Related Course - STAT-401: Classical and Bayesian Statistical Inference)
The aim of this course is to provide a strong mathematical and conceptual foundation in the methods of statistical inference, with an emphasis on practical aspects of the interpretation and communication of statistically-based conclusions in statistical research.
Course Objectives:
The objectives of the course are to:
- Estimate the point and interval for parameters,
- Estimate the parameters by Classical and Bayesian approaches,
- Draw samples from different distributions,
- Test several hypotheses,
- Develop LR test, GLR test, MLR test, SPRT, and Bayesian test of hypothesis.
Learning Outcomes:
After completing this part, students will be able to:
- Estimate the Bayes' Statistics,
- Find point and interval estimation,
- Test the different testing approaches,
- Execute uniformly minimum variance unbiased estimators,
- Compute best asymptotically normal estimators, asymptotically efficient estimators, consistent asymptotically normal estimators,
- Calculate loss and risk function, conjugate and Non-informative prior, etc.,
- Calculate Bayesian and Fiducial confidence intervals,
- Perform MP, UMP, UMPU, LUMP, LUMPU, LM, and Wald tests,
- Calculate prior and posterior odds and Bayes factor,
- Use SPRT techniques, draw OC curve, and calculate ASN function for different distributions,
- Solve different mathematical problems related to the contents using various methods.
Contents:
- Drawing samples from bivariate normal, multivariate normal, gamma, beta, and other distributions.
- Estimation of population parameters of different distributions by different methods.
- Inference about the mean vector and variance-covariance matrix of multivariate populations.
- Comparison of several multivariate means, estimation of confidence intervals for mean and variance.
- Test of simple hypotheses for mean and variance, test of multiple regression coefficients, test of multiple correlation coefficients, test of the mean vector.
- Most powerful tests, uniformly most powerful tests, uniformly most powerful unbiased tests, locally uniformly most powerful unbiased tests.
- Optimal tests in different situations, randomized tests, consistent tests, unbiased tests, similar regions, likelihood ratio tests, generalized likelihood ratio tests, monotone likelihood ratio tests.
- Test of homogeneity in parallel samples, LM test, Wald test, SPRT, OC function, Bayesian test of hypothesis, Bayesian contingency table analysis.
Rationale:
Group B (Marks: 20)
(Related Course - STAT-404: Stochastic Process)
This course introduces the idea of a stochastic process and shows how simple probability and matrix theory can be used to build this notion into a useful piece of applied mathematics.
Course Objectives:
The objectives of this course are to:
- Generate Markov chains, transition probability matrices with real-world phenomena,
- Develop homogeneous and non-homogeneous Poisson processes,
- Analyze queuing theory and apply the theory to real-world problems,
- Test Markov chains.
Learning Outcomes:
At the end of this course, students will be able to:
- Understand the notion of a Markov chain, and how simple ideas of conditional probability and matrices can be used to give a thorough and effective account of discrete-time Markov chains,
- Understand notions of long-time behavior including transience, recurrence, and equilibrium,
- Solve the Gambler's Ruin mathematical problem,
- Estimate the transition probabilities,
- Test the transition probabilities,
- Apply queuing theory to real-world problems,
- Determine the steady-state probabilities for different queuing systems.
Contents:
- Markov chain, closed sets, classification of states, properties of states, Chapman-Kolmogorov equations, first entrance decomposition formula, ergodic properties of irreducible chains, higher-order and secondary probabilities, recurrent events, delayed recurrent events, periodic chains, transient and recurrent states, Gambler's ruin problem.
- Estimation and hypothesis testing: Transition probabilities of Markov chains, asymptotic behavior of TPM, determination of different properties of transition probability matrices, homogeneous and non-homogeneous Poisson processes, determination of steady-state probabilities for different queuing systems.