Early undergraduateProbability & StatisticsLesson available
Statistical inference
Estimators and their properties, likelihood and maximum likelihood, confidence intervals, hypothesis tests and p-values, linear regression, and an introduction to Bayesian inference.
math.prob.statistics
Builds on
Required prerequisites, all the way down: 45
You will learn to
- Derive maximum likelihood estimators and assess their bias and variance.
- Construct confidence intervals and perform hypothesis tests, interpreting p-values correctly.
- Fit linear regression models and check their assumptions using residuals.
Definition
- An estimator is a function of the data used to estimate an unknown parameter.
- A p-value is the probability, computed assuming the null hypothesis, of a result at least as extreme as the one observed.
- The likelihood L(θ) is the probability or density of the observed data, regarded as a function of the parameter θ.
Theorem
- For an independent sample from a population with mean μ and variance σ², the sample mean is unbiased for μ and has variance σ²/n.
- Least-squares regression coefficients satisfy the normal equations XᵀXβ = Xᵀy.
- Performing many tests inflates the chance of false positives unless the analysis corrects for multiplicity.
Common misconceptions
- Reading a p-value as the probability that the null hypothesis is true.
- Treating a non-significant result as proof that there is no effect.
Optional discoveries
- Computational reproducibility Statistical analyses should be reproducible, with code, data and random seeds recorded so that others can rerun them.
- ← Statistics: centre, spread and distributions
- ← Conditional probability and independence
- ← Optimisation
Learning pathways
Further reading
- George Casella and Roger L. Berger, Statistical Inference
- Larry Wasserman, All of Statistics (2004)