Index

• alternative hypothesis Definition 22.1
• ascent property Proposition 20.2
• asymptotic normality of the MLE Theorem 17.1
• asymptotically efficient estimator Definition 17.2
• asymptotically unbiased estimator Definition 2.4
• Bayes estimator §28.3
  • posterior mean Lemma 28.7
• Bayes factor §29.2
• Bayes’ theorem Proposition A.20
  • for the parameter Lemma 28.2
• Bayesian test Definition 29.4
• bias Definition 2.1
• bias-variance decomposition Lemma 2.7
• Cauchy–Schwarz inequality Lemma A.8
• central limit theorem Theorem A.15
• change of variable Proposition A.19
• Chebyshev’s inequality Lemma A.6
• chi-squared distribution Definition A.2
  • mean and variance Lemma A.3
• chi-squared test for a variance Proposition 31.3
• complete data §19.1
• complete statistic Definition 10.3
  • examples Proposition 10.6
• complete-data log-likelihood §19.1
• completeness Definition 10.3
• composite hypothesis Definition 22.1
• conditional expectation
  • law of total variance Proposition A.11
  • taking out what is known Proposition A.10
  • tower property Proposition A.9
• confidence bound §26.5
• confidence interval Definition 26.1
  • by inverting a test Lemma 26.6
  • one-sided §26.5
  • Wald Definition 26.7
• confidence level Definition 26.1
• confidence set §26.1
• conjugate prior Definition 28.3
  • exponential family Proposition 28.4
  • normal-normal Proposition 28.6
• conservative test §22.5
• consistent estimator Definition 2.9
  • and asymptotic unbiasedness Corollary 2.13
  • and mean squared error Lemma 2.11
  • maximum likelihood estimator Theorem 16.3
• continuous mapping theorem Proposition A.17
• convergence
  • in distribution Definition A.13
  • in probability Definition A.12
• convergence in distribution Definition A.13
• convergence in probability Definition A.12
• covariance Proposition A.1
• coverage probability Definition 26.1
• Cramér–Rao inequality Theorem 13.2
  • attainment Proposition 13.5
  • for the parameter Corollary 13.3
  • multiparameter Proposition 14.5
• credibility Definition 29.1
• credible interval Definition 29.1
  • equal-tailed Definition 29.1
  • highest posterior density Definition 29.2
• critical region Definition 22.2
• critical value §22.2
• cumulant §7.3
• cumulant function Definition 7.9
  • moments of the natural statistic Lemma 7.10
• cumulant generating function §7.3
• curved exponential family §7.2
• delta method Proposition A.18
• density §1.1
• efficiency Definition 13.4
  • asymptotic Definition 17.2
• efficient estimator Definition 13.4
  • exponential family Corollary 13.6
• EM algorithm Definition 19.2
  • ascent property Proposition 20.2
  • complete-data log-likelihood §19.1
  • decomposition of the log-likelihood Lemma 20.1
  • fixed points Proposition 20.3
  • observed-data log-likelihood §19.1
• estimate Definition 1.4
• estimator Definition 1.4
  • asymptotically efficient Definition 17.2
  • asymptotically unbiased Definition 2.4
  • Bayes §28.3
  • consistent Definition 2.9
  • efficient Definition 13.4
  • family of §1.2
  • maximum likelihood Definition 5.1
  • method of moments Definition 4.6
  • unbiased Definition 2.1
  • uniformly minimum variance unbiased Definition 10.4
• expectation
  • linearity Proposition A.1
• exponential family Definition 7.1
  • completeness Proposition 10.6
  • conjugate prior Proposition 28.4
  • curved §7.2
  • efficient estimator Corollary 13.6
  • full rank Definition 7.8
  • $k$-parameter Definition 7.6
  • mean and variance Lemma 7.10
  • one-parameter Definition 7.1
  • sample from Lemma 7.5
  • shape §7.1—§7.1
  • sufficient statistic Corollary 8.5
• $F$ distribution Definition A.5
• $F$-test §31.4
• factorisation theorem Proposition 8.3
• family of estimators §1.2
• Fisher information Definition 11.2
  • additivity Proposition 11.4
  • information identity Lemma 11.3
  • matrix Definition 14.2
  • reparametrisation Proposition 11.9
• Fisher information matrix Definition 14.2
  • information identity Proposition 14.3
  • properties Proposition 14.4
• Fisher–Neyman factorisation theorem Proposition 8.3
• flat prior §28.4
• full rank Definition 7.8
• generalised likelihood ratio Definition 25.1
• generalised likelihood-ratio test Definition 25.1
• highest posterior density interval Definition 29.2
• HPD interval, see highest posterior density interval
• hyperparameter Definition 28.1
• hypothesis Definition 22.1
  • alternative Definition 22.1
  • composite Definition 22.1
  • null Definition 22.1
  • simple Definition 22.1
• identifiability Definition 4.10
• improper prior §28.4
• index, see index
• indicator function §8.2
• inequality
  • Cauchy–Schwarz Lemma A.8
  • Chebyshev’s Lemma A.6
  • Jensen’s Lemma A.7
  • Markov’s Lemma 2.10
• information identity Lemma 11.3
• invariance of the MLE Proposition 5.6
• Jeffreys prior Definition 28.8
  • invariance Proposition 28.9
  • location and scale models Proposition 28.11
• Jensen’s inequality Lemma A.7
• joint density §4.1
• Kullback–Leibler divergence §16.2
• label switching §20.2
• latent variable §19.1
• law of large numbers Proposition A.14
• law of total variance Proposition A.11
• Lehmann–Scheffé theorem Theorem 10.5
• level of a test Definition 22.5
• likelihood equation §5.1
• likelihood function Definition 4.1
• likelihood ratio §23.2
  • at the true parameter Lemma 16.2
• likelihood-ratio test Definition 25.1
  • Wilks’ theorem Theorem 25.3
• log-likelihood Definition 4.1
• log-odds Example 7.2
• log-partition function, see cumulant function
• Markov’s inequality Lemma 2.10
• maximum likelihood estimator Definition 5.1
  • asymptotic normality Theorem 17.1
  • consistency Theorem 16.3
  • invariance Proposition 5.6
  • standard error Corollary 17.3
• mean squared error Definition 2.6
  • bias-variance decomposition Lemma 2.7
• method of moments Lecture 4
• method of moments estimator Definition 4.6
• minimal sufficient statistic Definition 8.10
  • criterion Proposition 8.11
• mixture model Example 19.1
• MLE, see maximum likelihood estimator
• moment
  • population §4.3
  • sample §4.3
• moment generating function §7.3
• most powerful test Definition 23.1
• natural parameter Definition 7.1
• natural parameter space Definition 7.9
• natural parametrisation Definition 7.9
• natural statistic Definition 7.1
• nested models §25.1
• Neyman–Pearson lemma Theorem 23.2
• normal distribution
  • conjugate prior Proposition 28.6
  • sample mean and sample variance Proposition A.4
  • test for the variance Proposition 31.3
• nuisance parameter §14.4
  • effect on the Cramér–Raobound Lemma 14.7
• null hypothesis Definition 22.1
• observed data §19.1
• observed information §17.3
• observed-data log-likelihood §19.1
• one-sided test Proposition 23.4
• order statistics
  • maximum and minimum Proposition 8.8
• orthogonal parameters Definition 14.8
• $p$-value Definition 22.8
  • uniform distribution Lemma 22.9
• parameter Definition 1.1
• parameter space Definition 1.1
• parametric model Definition 1.1
• pivot, see pivotal quantity
• pivotal quantity Definition 26.2
• posterior density Definition 28.1
  • Bayes’ theorem Lemma 28.2
• posterior distribution Lecture 28
• posterior mean Lemma 28.7
• posterior odds Definition 29.6
• posterior probability Definition 29.4
• power Definition 22.4
• power function Definition 22.4
• prior density Definition 28.1
• prior distribution Lecture 28
  • conjugate Definition 28.3
  • flat §28.4
  • improper §28.4
  • Jeffreys Definition 28.8
• prior odds Definition 29.6
• probability generating function §7.3
• probability mass function, see probability weights
• probability weights §1.1
• random sample §1.1
• Rao–Blackwell theorem Proposition 10.1
  • Rao–Blackwellisation §10.1
• Rao–Blackwellisation §10.1
• regularity conditions Definition 16.1
• sample mean
  • consistency Example 2.12
  • mean and variance Lemma 2.2
• sample variance §2.1
  • normal sample Proposition A.4
  • unbiasedness Lemma 2.3
• sampling distribution §1.2
• score function Definition 4.4
  • covariance with a statistic Lemma 13.1
  • mean zero Lemma 11.1
• score vector Definition 14.1
• significance level §22.3
• significant §22.5
• simple hypothesis Definition 22.1
• size of a test Definition 22.5
• Slutsky’s theorem Proposition A.16
• standard error §17.3
  • of the MLE Corollary 17.3
• statistic Definition 1.3
  • complete Definition 10.3
  • minimal sufficient Definition 8.10
  • natural Definition 7.1
  • sufficient Definition 8.1
  • test §22.2
• sufficient statistic Definition 8.1
  • factorisation theorem Proposition 8.3
  • in an exponential family Corollary 8.5
  • minimal Definition 8.10
• super-efficiency Example 13.7
• $t$ distribution Definition A.5
• $t$-interval Example 26.4
• $t$-test Proposition 31.1
  • one-sample Proposition 31.1
  • paired §31.4
  • two-sample §31.4
• taking out what is known Proposition A.10
• test Definition 22.2
  • Bayesian Definition 29.4
  • conservative §22.5
  • likelihood-ratio Definition 25.1
  • most powerful Definition 23.1
  • one-sided Proposition 23.4
  • uniformly most powerful §23.4
• test function Definition 22.2
• test inversion Lemma 26.6
• test statistic §22.2
• tower property Proposition A.9
• trigamma function Example 14.10
• true parameter value §1.1
• type I error Definition 22.3
• type II error Definition 22.3
• UMVUE, see uniformly minimum variance unbiased estimator
• unbiased estimator Definition 2.1
  • asymptotically Definition 2.4
• uniformly minimum variance unbiased estimator Definition 10.4
  • Lehmann–Scheffé theorem Theorem 10.5
• uniformly most powerful test §23.4
  • non-existence for two-sided alternatives Proposition 23.5
  • one-sided alternative Proposition 23.4
• variance
  • of a linear function Proposition A.1
• Wald interval §17.3, Definition 26.7
  • coverage Proposition 26.8
• weak law of large numbers Proposition A.14
• Wilks’ theorem Theorem 25.3