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Statistical Inference

Year 2 · 6 sections · 55 concepts · 112 practice questions

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What's covered

free1 · Estimation11 concepts

Overview 1 · The plan of this section
Definition 1.1 · Likelihood and maximum likelihood
Example 1.2 · MLEs for the normal distribution
Proposition 1.3 · Invariance of the MLE
Definition 1.4 · Bias, variance and mean squared error
Proposition 1.5 · MSE = variance + bias²
Definition 1.6 · Sufficient statistics
Theorem 1.7 · The factorisation theorem
Definition 1.8 · Fisher information
Theorem 1.9 · The Cramér–Rao lower bound
Check 1.10 · Check: finding and assessing an estimator

2 · Sampling distributions8 concepts

Overview 2 · The plan of this section
Definition 2.1 · The chi-squared distribution
Proposition 2.2 · Properties of chi-squared
Definition 2.3 · Student's t distribution
Definition 2.4 · The F distribution
Theorem 2.5 · Sample mean and variance of a normal sample
Corollary 2.6 · The one-sample t statistic
Check 2.7 · Check: which sampling distribution?

3 · Confidence intervals9 concepts

Overview 3 · The plan of this section
Definition 3.1 · Confidence intervals and pivots
Proposition 3.2 · The pivotal method
Example 3.3 · The t interval for a normal mean
Example 3.4 · Interval for a normal variance
Proposition 3.5 · Difference of two normal means
Theorem 3.6 · Asymptotic normality of the MLE
Example 3.7 · Interval for a proportion
Check 3.8 · Check: constructing a confidence interval

4 · Hypothesis testing9 concepts

Overview 4 · The plan of this section
Definition 4.1 · Tests, errors, size and power
Definition 4.2 · p-values
Theorem 4.3 · The Neyman–Pearson lemma
Example 4.4 · Uniformly most powerful tests
Theorem 4.5 · Generalised likelihood ratio tests and Wilks' theorem
Proposition 4.6 · Pearson's chi-squared goodness-of-fit test
Proposition 4.7 · Duality of tests and confidence intervals
Check 4.8 · Check: carrying out a test

5 · Linear regression9 concepts

Overview 5 · The plan of this section
Definition 5.1 · The linear model
Theorem 5.2 · Least squares and the normal equations
Example 5.3 · Simple linear regression
Proposition 5.4 · Mean and covariance of the least squares estimator
Theorem 5.5 · The Gauss–Markov theorem
Theorem 5.6 · Inference for coefficients in the normal linear model
Proposition 5.7 · Analysis of variance and R²
Check 5.8 · Check: fitting and interpreting a regression

6 · Bayesian inference9 concepts

Overview 6 · The plan of this section
Definition 6.1 · Prior, posterior and predictive distributions
Proposition 6.2 · Posterior ∝ likelihood × prior
Example 6.3 · The beta–binomial model
Proposition 6.4 · Normal prior and normal data
Example 6.5 · The gamma–Poisson model
Proposition 6.6 · Bayes estimators
Definition 6.7 · Credible intervals
Check 6.8 · Check: a Bayesian analysis

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Proposition 1.3 · Invariance of the MLE

MLE of $g(\theta)$ is $g(\hat\theta)$ .

Proof

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