Early undergraduateProbability & StatisticsPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
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)