Mathematical Statistics
A second course in statistics that builds the theory of inference from first principles: populations, samples and the sampling distributions of statistics, the chi-square, t and F distributions, bias, variance and mean squared error of estimators, the method of moments and the delta method, maximum likelihood estimation with censored data and Newton's method, Fisher information and the asymptotic normality and efficiency of maximum likelihood estimators, sufficiency, the Rao–Blackwell and Lehmann–Scheffé theorems and the Cramér–Rao bound, confidence intervals from pivots, likelihood theory and the bootstrap, hypothesis tests, power and p-values with the Neyman–Pearson lemma, likelihood ratio, score, Wald and chi-square tests with Wilks's theorem, the linear model with least squares, the Gauss–Markov theorem, t and F tests and residual diagnostics, and a capstone on Bayesian inference and a complete likelihood analysis of reliability data, with examples from quality control, clinical trials, reliability engineering and traffic safety and runnable Python code throughout.
STAT-252 Year 2 3 credits 12 lessons · 4 h 31 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Distinguish populations, samples, parameters and statistics, describe the sampling distribution of a statistic, and use order statistics and simulation to study it.
- Derive and use the chi-square, t and F distributions as sampling distributions of normal-theory statistics, including the independence of the sample mean and variance.
- Evaluate point estimators by bias, variance, mean squared error and consistency, and explain the bias–variance trade-off.
- Construct method of moments estimators and approximate their variances with the delta method.
- Write likelihood and log-likelihood functions, including for censored data, and find maximum likelihood estimates analytically and by Newton’s method.
- Compute Fisher information, state and apply the asymptotic normality and invariance of maximum likelihood estimators, and compute standard errors.
- Identify sufficient statistics with the factorization theorem, improve estimators by Rao–Blackwellization, and apply the Cramér–Rao bound and the Lehmann–Scheffé theorem.
- Build confidence intervals from pivotal quantities, likelihood theory and the bootstrap, and interpret coverage correctly.
- Formulate hypothesis tests, compute error probabilities, power and p-values, and derive most powerful tests with the Neyman–Pearson lemma.
- Carry out generalized likelihood ratio, score, Wald and chi-square tests using Wilks’s theorem, and invert them to obtain intervals.
- Fit linear models by least squares, state the Gauss–Markov theorem, and use t and F tests, prediction intervals and residual diagnostics.
- Perform Bayesian inference with conjugate priors, compare credible and confidence intervals, and complete a full likelihood analysis of a real-world dataset.
Course content
12 lessons · 4 h 31 min total · final exam
- 1 From probability to statistics: samples, statistics and models 23 min · quiz
- 2 Sampling distributions: chi-square, t and F 23 min · quiz
- 3 Point estimators: bias, variance and mean squared error 23 min · quiz
- 4 The method of moments 23 min · quiz
- 5 Maximum likelihood estimation 23 min · quiz
- 6 Properties of maximum likelihood estimators 23 min · quiz
- 7 Sufficiency, the Rao–Blackwell theorem and the Cramér–Rao bound 23 min · quiz
- 8 Confidence intervals and pivotal quantities 21 min · quiz
- 9 Hypothesis testing and the Neyman–Pearson lemma 22 min · quiz
- 10 Likelihood ratio tests 22 min · quiz
- 11 The linear model and least squares 23 min · quiz
- 12 Capstone: Bayesian inference and a full likelihood analysis 22 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A second course in statistics that builds the theory of inference from first principles: populations, samples and the sampling distributions of statistics, the chi-square, t and F distributions, bias, variance and mean squared error of estimators, the method of moments and the delta method, maximum likelihood estimation with censored data and Newton's method, Fisher information and the asymptotic normality and efficiency of maximum likelihood estimators, sufficiency, the Rao–Blackwell and Lehmann–Scheffé theorems and the Cramér–Rao bound, confidence intervals from pivots, likelihood theory and the bootstrap, hypothesis tests, power and p-values with the Neyman–Pearson lemma, likelihood ratio, score, Wald and chi-square tests with Wilks's theorem, the linear model with least squares, the Gauss–Markov theorem, t and F tests and residual diagnostics, and a capstone on Bayesian inference and a complete likelihood analysis of reliability data, with examples from quality control, clinical trials, reliability engineering and traffic safety and runnable Python code throughout.
Part of the Science stream, typically taken in year 2.