Glossary and index
Every named idea, A to Z: 532 entries. The 211 definitions, theorems, remarks and distributions each have an article of their own; sections link into their chapter, questions link to their page, and redirects to the entry they stand for. Hover over a link to preview it.
A
| Term | Meaning | Where |
|---|---|---|
| Acceptance and rejection section | Simulation | |
| Acceptance–rejection method definition | Suppose the target density f is zero outside a finite interval [a,b] and bounded by c=\max f. Draw X uniformly from [a,b] and, independently, Y uniformly from [0,c]. If Y\le f(X), return X. Otherwise repeat. | Simulation |
| Accidental deaths section | Forecasting | |
| ACF and PACF question | Dependence along a series | |
| Addiction categories section | Categorical responses | |
| Adjusted R-squared definition | For a regression with p explanatory variables on n observations, the adjusted R^2 is … | Regression examples |
| Advertising-response table section | Categorical data and the Dirichlet distribution | |
| Affine images of normal vectors theorem | If X_i\sim\operatorname{N}(\mu_i,\sigma_i^2) are independent and a,b_1,\ldots,b_n are constants, then … | Functions of random variables and limit theorems |
| Analysis of deviance section | Paired comparisons, and how far a fit is from the data | |
| Ancillary statistic definition | A statistic A({\boldsymbol{X}}) is ancillary if its sampling distribution does not depend on \theta. | Exponential families and conjugacy |
| ANOVA F distribution theorem | In the one-way model, \mathrm{SS}_{\mathrm{error}}/\sigma^2 is \chi^2_{n-k}, and it is independent of \mathrm{SS}_{\mathrm{treatment}}. | Analysis of variance |
| ANOVA sum of squares decomposition proposition | With … | Analysis of variance |
| AR, MA and ARMA orders question | Dependence along a series | |
| ARCH and GARCH question | Dependence along a series | |
| ARMA redirect | Redirects to Autoregression and moving averages. | Dependence along a series |
| Autoregression and moving averages section | Dependence along a series | |
| The average, and the person section | Marginal models, and two responses at once |
B
| Term | Meaning | Where |
|---|---|---|
| Baseline-category logit model definition | For nominal categories 0,1,\ldots,K-1, the baseline-category logit model is … | Categorical responses |
| Basu’s theorem theorem | A complete sufficient statistic is independent of every ancillary statistic. | Exponential families and conjugacy |
| Bayes action for two hypotheses proposition | These probabilities do not by themselves define an action. If choosing action a_0 when H_1 is true costs L_{01} and choosing a_1 when H_0 is true costs L_{10}, the posterior expected losses are … | Bayesian hypotheses and model comparison |
| Bayes factor redirect | Redirects to Marginal likelihood and Bayes factor. | Bayesian hypotheses and model comparison |
| Bayes factor for independence in the table section | Bayesian hypotheses and model comparison | |
| Bayes’ rule theorem | For events A and B with \mathbb{P}(B)>0 … | Bayesian foundations |
| Bayes’ rule for densities section | Bayesian foundations | |
| Bayesian credible interval definition | Bayesian analysis introduces a probability distribution for the unknown parameter. Conditional on observed data x, a statement such as … | Bayesian foundations |
| Bernoulli and binomial section | Random variables and distributions | |
| Bernoulli distribution definition | X has the Bernoulli distribution with success probability p\in[0,1], written X\sim\operatorname{Bern}(p), when \mathbb{P}(X=1)=p and \mathbb{P}(X=0)=1-p. Then \mathbb{E} X=p and \operatorname{Var}(X)=p(1-p). | Random variables and distributions |
| A Bernoulli response section | Random slopes, and a binary mixed model | |
| Beta distribution definition · distribution | For \alpha,\beta>0, the Beta density is … | The Beta–Binomial model |
| The Beta distribution is the two-category Dirichlet proposition | When K=2, write p_1=p and p_2=1-p. Restricting the Dirichlet density to the one-dimensional simplex gives … | Categorical data and the Dirichlet distribution |
| The Beta prior section | The Beta–Binomial model | |
| Beta–Binomial conjugacy proposition | Therefore … | The Beta–Binomial model |
| Beta–Binomial posterior prediction proposition | If \theta\mid x\sim\operatorname{Beta}(a,b) and, conditionally on \theta, \widetilde X\sim\operatorname{Bin}(m,\theta), then … | The Beta–Binomial model |
| Beta–Binomial prediction section | The Beta–Binomial model | |
| Between and within section | Analysis of variance | |
| Bids for a bankrupt firm section | Underdispersion, positive responses, and quasi-likelihood | |
| Binary responses section | Introduction to generalised linear models | |
| Binomial section | Estimation | |
| Binomial counts section | Standard errors, and models for counts | |
| Binomial distribution definition · distribution | X has the binomial distribution with n trials and success probability p, written X\sim\operatorname{Bin}(n,p), when … | Random variables and distributions |
| The binomial distribution section | Introduction to generalised linear models | |
| Binomial test definition | For X\sim\operatorname{Bin}(n,p) and H_0:p=p_0, the test statistic is X, with null distribution \operatorname{Bin}(n,p_0). | Hypothesis tests |
| Binomial tests section | Hypothesis tests | |
| Birth weight section | Linear model examples | |
| Bivariate exponential example section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| A bivariate normal sample section | Linear regression | |
| Blinding and placebo definition | In a blind experiment the units do not know which treatment they received. In a double-blind experiment the people who give the treatments and measure the responses do not know either. | Studies, data, and evidence |
| Blocks section | Analysis of variance | |
| Bonferroni correction definition | To make m comparisons with a familywise error rate of at most \alpha, run each at level \alpha/m. Equivalently, multiply each p-value by m, capping at 1. | Analysis of variance examples |
| Both normal parameters unknown section | Conjugate analysis for normal models | |
| Bradley–Terry section | Paired comparisons, and how far a fit is from the data | |
| Bradley–Terry question | Paired comparisons, and how far a fit is from the data |
C
| Term | Meaning | Where |
|---|---|---|
| Canonical link definition | The link g is canonical when the linear predictor is the natural parameter of the family: g(\mu_i)=b(\theta_i), in the notation of equation (6). | Introduction to generalised linear models |
| Categorical and multinomial models definition | For two categories, a Bernoulli trial has probabilities (p,1-p) and an aggregated count has a binomial distribution. For K categories, let … | Categorical data and the Dirichlet distribution |
| Categorical response types question | Categorical responses | |
| Central limit theorem theorem | If the X_i are independent and identically distributed with mean \mu and variance \sigma^2\in(0,\infty), then for every x … | Functions of random variables and limit theorems |
| Chance or an effect? section | Hypothesis tests | |
| Checking the assumptions section | Regression examples | |
| Checking the final model section | Linear model examples | |
| Checking the model section | Analysis of variance examples | |
| Chi-squared distribution definition | If Z_1,\ldots,Z_m are independent standard normal variables, then V=Z_1^2+\cdots+Z_m^2 has the chi-squared distribution with m degrees of freedom, written \chi^2_m. | Statistical inference |
| The chi-squared test for a variance section | Hypothesis tests | |
| Chi-squared test of independence definition | For an r\times c contingency table with fitted counts E_{ij}=R_iC_j/n, the statistic … | Chi-squared tests |
| The cholesterol fit section | Introduction to generalised linear models | |
| Cholesterol, with standard errors section | Standard errors, and models for counts | |
| Classical calculations as a reference section | Bayesian linear regression | |
| Clotting times section | Introduction to generalised linear models | |
| Clotting times, three ways section | Underdispersion, positive responses, and quasi-likelihood | |
| Coefficient of determination definition | The coefficient of determination of a fitted line is … | Linear regression |
| Coin tossing and the likelihood section | The Beta–Binomial model | |
| Comparing estimators section | Estimation | |
| Complete pooling, no pooling, and partial pooling section | Hierarchical Bayes | |
| Complete statistic definition | A statistic T is complete for a family if … | Exponential families and conjugacy |
| Conditional coefficient posterior proposition | Bayesian linear regression | |
| Conditional distribution definition | If X and Y are both discrete or both continuous, and f_X(x)>0, the conditional mass function or density of Y given X=x is … | Joint distributions |
| Conditional distributions section | Joint distributions | |
| Conditional posterior of the coefficients section | Bayesian linear regression | |
| Conditional probability definition | If \mathbb{P}(B)>0, the conditional probability of A given B is … | Random experiments and probability |
| Conditional versus marginal dependence question | Marginal models, and two responses at once | |
| Conditional versus marginal effects question | Marginal models, and two responses at once | |
| Confidence interval definition | Let L and U be statistics. The random interval (L,U) is a confidence interval for \theta with confidence 1-\alpha when … | Confidence intervals |
| Confounding variable definition | A confounding variable affects both the explanatory variable and the response, so that an association between those two can appear, disappear or reverse when the confounder is taken into account. | Studies, data, and evidence |
| Conjugacy question | The Beta–Binomial model | |
| Conjugacy is optional remark | Conjugacy is convenient, not compulsory. A prior concentrated around 1/2 could be obtained by truncating a Laplace density … | The Beta–Binomial model |
| Conjugate priors from the exponential-family form section | Exponential families and conjugacy | |
| Conjugate update with every constant shown section | The Beta–Binomial model | |
| Consistent estimator definition | A sequence of estimators T_n, one for each sample size n, is consistent for \theta when \mathbb{P}_\theta(\|T_n-\theta\|>\varepsilon)\to0 as n\to\infty, for every \varepsilon>0 and every \theta. | Estimation |
| Contingency tables section | Chi-squared tests | |
| Continuous distribution definition | X is continuous when its distribution function can be written F(x)=\int_{-\infty}^{x} f(u)\,\,\mathrm{d} u for a non-negative function f with \int_{-\infty}^{\infty} f=1. The function f is a probability density. | Random variables and distributions |
| Contrast definition | A contrast is a linear combination C=\sum_j c_j\mu_j of the treatment means whose coefficients sum to zero. It is estimated by \widehat C=\sum_j c_j\overline Y_{\cdot j}, which is unbiased with variance \sigma^2\sum_j c_j^2/n_j. | Analysis of variance |
| Contrasts section | Analysis of variance | |
| Counting section | Random experiments and probability | |
| Counts section | Standard errors, and models for counts | |
| Counts in categories section | Chi-squared tests | |
| Covariance definition | The covariance of X and Y is \operatorname{Cov}(X,Y)=\mathbb{E}\bigl((X-\mathbb{E} X)(Y-\mathbb{E} Y)\bigr), when the expectation exists. Equivalently, \operatorname{Cov}(X,Y)=\mathbb{E}(XY)-(\mathbb{E} X)(\mathbb{E} Y). | Joint distributions |
| Coverage section | Confidence intervals | |
| Credible-region constructions definition | Three interval constructions are worth distinguishing. An equal-tailed 95\% credible interval (l,u) satisfies … | Bayesian foundations |
| Critical region definition | A critical region C is a set of values of the test statistic. The test rejects H_0 when T\in C. A right-sided region is [c,\infty), a left-sided region is (-\infty,c], and a two-sided region is (-\infty,c_1]\cup[c_2,\infty). | Hypothesis tests |
| Cubic splines and knots question | Smooth means | |
| Cumulative distribution function definition | The cumulative distribution function of X is F(x)=\mathbb{P}(X\le x), for x\in\mathbb{R}. | Random variables and distributions |
| Curved relationships section | Linear regression |
D
| Term | Meaning | Where |
|---|---|---|
| Data frames section | Using R | |
| Deriving the acceptance probability section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Designed experiment definition | In a designed experiment some columns are controlled. Treatments are assigned to units, often at random, and those assignments stay the same if the measurements are repeated. The response column changes; the treatment column does not. | Studies, data, and evidence |
| Detailed balance implies stationarity theorem | If a transition kernel q satisfies equation (3) for a probability distribution \pi, then \pi satisfies global balance and is a stationary distribution of the chain. | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Detailed-balance proof without skipped cases section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Deviance section | Paired comparisons, and how far a fit is from the data | |
| Deviance, analysis of redirect | Redirects to Analysis of deviance. | Paired comparisons, and how far a fit is from the data |
| A difference of means section | Confidence intervals | |
| Direct posterior simulation proposition | 1. draw … | Bayesian linear regression |
| Direct simulation and genuine Gibbs sampling section | Bayesian linear regression | |
| Dirichlet distribution definition · distribution | The conjugate prior for {\boldsymbol{p}} is the Dirichlet distribution. Write … | Categorical data and the Dirichlet distribution |
| Dirichlet moments proposition | Categorical data and the Dirichlet distribution | |
| The Dirichlet prior section | Categorical data and the Dirichlet distribution | |
| Dirichlet–Multinomial conjugacy theorem | After division by this evidence … | Categorical data and the Dirichlet distribution |
| Discrete distribution definition | X is discrete when it takes values in a countable set S. Its probability mass function is f(x)=\mathbb{P}(X=x) for x\in S, and \mathbb{P}(X\in B)=\sum_{x\in B}f(x). | Random variables and distributions |
| Disease and cholesterol section | Introduction to generalised linear models | |
| Dispersion question | Underdispersion, positive responses, and quasi-likelihood | |
| Draws from the usual families section | Simulation | |
| Dummy variable coding definition | A categorical explanatory variable with d levels enters a linear model through d-1 indicator columns, one for each level except a reference level. | Linear model examples |
E
| Term | Meaning | Where |
|---|---|---|
| The eight steps section | Hypothesis tests | |
| Empirical Bayes definition | An empirical Bayes analysis estimates prior or population-level hyperparameters from data and then treats those estimates as fixed in a lower-level Bayesian update. | Hierarchical Bayes |
| Empirical hyperparameters by moment matching section | Hierarchical Bayes | |
| Equally likely outcomes theorem | If \Omega is finite and every outcome has the same probability, then for every event A … | Random experiments and probability |
| Error and residual definition | The true error is … | Bayesian linear regression |
| Errors and power section | Hypothesis tests | |
| Errors are not residuals section | Bayesian linear regression | |
| Estimated parameters section | Chi-squared tests | |
| Estimates and estimators section | Estimation | |
| Estimator definition | An estimator of \theta is a statistic T=T(X_1,\ldots,X_n) whose value is used as a guess at \theta. Its observed value t=T(x_1,\ldots,x_n) is the estimate. | Estimation |
| Event definition | An event is a subset of the sample space. The event occurs when the outcome lies in that subset. | Random experiments and probability |
| Evidence and model odds question | Bayesian hypotheses and model comparison | |
| Exchangeability definition | The parameters \theta_1,\ldots,\theta_J are exchangeable if their joint density is invariant to every permutation \pi … | Hierarchical Bayes |
| Expectation definition | If X is discrete with mass function f, its expectation is \mathbb{E} X=\sum_x x\,f(x), when the sum converges absolutely. | Random variables and distributions |
| Expectation and variance section | Random variables and distributions | |
| Expectation, covariance and matrices section | Joint distributions | |
| Expectation of a function theorem | If g is a real function and X is discrete with mass function f, then \mathbb{E} g(X)=\sum_x g(x)\,f(x). | Random variables and distributions |
| Expectation vector and covariance matrix definition | For a random column vector \mathbf{X}=(X_1,\ldots,X_n)^\top the expectation vector is \boldsymbol\mu=\mathbb{E}\mathbf{X}, taken coordinate-wise, and the covariance matrix \Sigma has entries \Sigma_{ij}=\operatorname{Cov}(X_i,X_j). | Joint distributions |
| Exponential section | Random variables and distributions | |
| Exponential distribution definition · distribution | X has the exponential distribution with rate \lambda>0, written X\sim\operatorname{Exp}(\lambda), when its density is \lambda e^{-\lambda x} for x>0. | Random variables and distributions |
| Exponential families section | Introduction to generalised linear models | |
| Exponential family definition | A random variable Y whose distribution depends on a parameter \theta belongs to a (one-parameter) exponential family, with dispersion \phi, if its density or mass function can be written … | Introduction to generalised linear models |
| Exponential family question | Exponential families and conjugacy |
F
| Term | Meaning | Where |
|---|---|---|
| F distribution definition | If U is \chi^2_{m_1}, V is \chi^2_{m_2}, and U and V are independent, then F=(U/m_1)/(V/m_2) has the F distribution with (m_1,m_2) degrees of freedom, written F_{m_1,m_2}. | Statistical inference |
| F test for equal variances definition | For two independent normal samples, F=s_1^2/s_2^2 has the F_{n_1-1,n_2-1} distribution when \sigma_1^2=\sigma_2^2. | Hypothesis tests |
| Finding the conjugate family by inspection section | Conjugate analysis for normal models | |
| Finite-state Markov chain definition | Let E be a finite set of states and let X_0,X_1,X_2,\ldots take values in E. The process is a Markov chain when, for every n and every choice of states … | Markov chains |
| A first hierarchy: TG5 cattle studies section | Hierarchical Bayes | |
| Fisher scoring section | Introduction to generalised linear models | |
| Fisher scoring and IRLS question | Introduction to generalised linear models | |
| Fisher–Neyman factorisation theorem | Suppose the family \{p({\boldsymbol{x}}\mid\theta):\theta\in\Omega\} is dominated by a common measure. Then T({\boldsymbol{X}}) is sufficient for \theta if and only if there are nonnegative functions g_\theta and h such that … | Exponential families and conjugacy |
| Fitted values, residuals and explained variation section | Linear regression | |
| Five-number summary and boxplot definition | The five-number summary of a sample is its minimum, first quartile, median, third quartile and maximum. | Descriptive statistics |
| Fixed and random effects question | Random intercepts | |
| Four ways of counting draws proposition | Random experiments and probability | |
| A four-state random walk section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| From a sample to a parameter section | Statistical inference | |
| From a standard error to an interval section | Confidence intervals | |
| From data to an estimate section | Estimation examples | |
| From one tutorial class to many section | Hierarchical Bayes | |
| From random responses to fitted values section | Bayesian linear regression | |
| From reality to a model and back section | Statistical inference | |
| From two categories to many section | Categorical data and the Dirichlet distribution | |
| Full Bayesian treatment section | Hierarchical Bayes | |
| Full conditionals and Gibbs question | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Full conditionals under independent priors proposition | Holding \sigma^2 fixed, the calculation from equation (1) applies verbatim … | Conjugate analysis for normal models |
G
| Term | Meaning | Where |
|---|---|---|
| GAM redirect | Redirects to Sums of smooth functions. | Additive smooths |
| Gamma and chi-squared section | Random variables and distributions | |
| Gamma and inverse-Gamma kernels question | Conjugate analysis for normal models | |
| Gamma distribution definition · distribution | X has the gamma distribution with shape \alpha>0 and rate \lambda>0, written X\sim\operatorname{Gamma}(\alpha,\lambda), when its density is … | Random variables and distributions |
| GAMs and effective degrees of freedom question | Additive smooths | |
| Gaussian linear model definition | Let {\boldsymbol{x}}_i^\top denote row i of {\color{blue}\boldsymbol{X}}. The Gaussian linear model is … | Bayesian linear regression |
| The Gaussian linear model section | Bayesian linear regression | |
| GEE redirect | Redirects to Marginal model. | Marginal models, and two responses at once |
| Generalised estimating equations question | Marginal models, and two responses at once | |
| Generalised linear model definition | Observations Y_1,\ldots,Y_n follow a generalised linear model when all three of the following are specified. | Introduction to generalised linear models |
| Generic exponential-family conjugate update proposition | Thus the hyperparameters update as … | Exponential families and conjugacy |
| Geometric section | Random variables and distributions | |
| Geometric distribution definition · distribution | X has the geometric distribution with success probability p\in(0,1], written X\sim\operatorname{Geom}(p), when … | Random variables and distributions |
| Gibbs sampling section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Gibbs sampling as an always-accepted MH update section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| GLM redirect | Redirects to Generalised linear model. | Introduction to generalised linear models |
| The GLM components question | Introduction to generalised linear models | |
| GLMM redirect | Redirects to A Bernoulli response. | Random slopes, and a binary mixed model |
| Goodness of fit in R section | Chi-squared examples | |
| Grid approximation and posterior simulation section | Hierarchical Bayes | |
| Grid Monte Carlo approximation definition | Let w_{ab} be the unnormalised hyperposterior value at grid cell (a,b), including the Jacobian \alpha\beta. The discrete approximation is … | Hierarchical Bayes |
H
| Term | Meaning | Where |
|---|---|---|
| Hazard and the Weibull law section | Reliability | |
| Hazard rate definition | If T has density f and survival function R, the hazard rate is h(t)=f(t)/R(t) for t with R(t)>0. | Reliability |
| Hierarchical Bayesian model definition | Hierarchical Bayes is ordinary Bayes for data that arrive in related groups. Each group is allowed to have its own parameter, but those group parameters are modelled as draws from a common population distribution. | Hierarchical Bayes |
| How large a sample? section | Confidence intervals | |
| HPD interval redirect | Redirects to Credible-region constructions. | Bayesian foundations |
| A hundred minutes of traffic section | Forecasting | |
| Hypergeometric section | Random variables and distributions | |
| Hypergeometric distribution definition | A hypergeometric count draws n items without replacement from a population of size N that contains r marked items. If X is the number of marked items in the draw, then … | Random variables and distributions |
| Hyperparameter definition | For a fully Bayesian analysis, the population-level quantities are also unknown and receive priors, for example … | Hierarchical Bayes |
| Hyperprior and transformations section | Hierarchical Bayes | |
| A hypothesis region is not a separate model remark | Two constructions must not be confused … | Bayesian hypotheses and model comparison |
I
| Term | Meaning | Where |
|---|---|---|
| Identification and residual checks question | Forecasting | |
| Independence section | Random experiments and probability | |
| Independence in R section | Chi-squared examples | |
| Independence model section | Bayesian hypotheses and model comparison | |
| Independent events definition | Events A_1,A_2,\ldots are mutually independent when, for every finite subcollection … | Random experiments and probability |
| Independent random variables definition | Discrete random variables X_1,\ldots,X_n are independent when the joint mass function factors for every point … | Joint distributions |
| Informative priors and regularisation section | Bayesian linear regression | |
| The ingredients section | Hypothesis tests | |
| Initialisation, warm-up, and dependence section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Integrating out the study rates section | Hierarchical Bayes | |
| Interaction definition | Two factors interact when the effect of one depends on the level of the other: the difference in mean response between two levels of the first factor is not the same at every level of the second. | Analysis of variance examples |
| Interaction section | Linear model examples | |
| Interior Dirichlet mode proposition | Consequently … | Categorical data and the Dirichlet distribution |
| Interpreting a logistic regression section | Introduction to generalised linear models | |
| Interquartile range definition | The interquartile range is the difference between the third and first quartiles. It is the width of the central half of the sample. | Descriptive statistics |
| Invariance of maximum likelihood proposition | If \widehat\theta is the maximum likelihood estimator of \theta and g is a function, then the maximum likelihood estimator of g(\theta) is g(\widehat\theta). | Estimation |
| Inverse transform section | Simulation | |
| Inverse-Gamma distribution definition · distribution | In the shape–rate convention … | Conjugate analysis for normal models |
| Inverse-transform method definition | Let F be a distribution function and define F^{-1}(u)=\inf\{x:F(x)\ge u\} for u\in(0,1). If U\sim\operatorname{U}[0,1], then X=F^{-1}(U) has distribution function F. | Simulation |
| Invertibility question | Dependence along a series |
J
| Term | Meaning | Where |
|---|---|---|
| Joint distributions and independence section | Joint distributions | |
| Joint probability density definition | Continuous random variables X and Y have joint density f when, for every reasonable region B … | Joint distributions |
| Joint probability mass function definition | Discrete random variables X and Y have joint probability mass function f_{X,Y}(x,y)=\mathbb{P}(X=x,Y=y). The marginal mass functions are the row and column sums … | Joint distributions |
| Jointly normal variables section | Functions of random variables and limit theorems |
K
| Term | Meaning | Where |
|---|---|---|
| Known cell probabilities section | Chi-squared tests | |
| Kyphosis and age section | Smooth means |
L
| Term | Meaning | Where |
|---|---|---|
| Large samples section | Estimation examples | |
| Law of large numbers theorem | If the mean \mu exists, the strong law says … | Functions of random variables and limit theorems |
| Law of total probability theorem | If B_1,\ldots,B_n are disjoint and their union is \Omega, then for every event A … | Random experiments and probability |
| Leaf blotch section | Underdispersion, positive responses, and quasi-likelihood | |
| Least squares section | Linear regression | |
| Least-squares decomposition theorem | Bayesian linear regression | |
| Least-squares estimators are unbiased proposition | If \mathbb{E} Y_i=\beta_0+\beta_1x_i for every i, then \mathbb{E}\widehat\beta_1=\beta_1 and \mathbb{E}\widehat\beta_0=\beta_0. No assumption about the error distribution is needed. | Linear regression |
| Least-squares line definition | For points (x_1,y_1),\ldots,(x_n,y_n) with the x_i not all equal, the least-squares line y=b_0+b_1x minimises … | Linear regression |
| Lidar section | Smooth means | |
| Lifetimes section | Reliability | |
| Likelihood and reference prior section | Bayesian linear regression | |
| Likelihood function definition | Let X_1,\ldots,X_n have joint mass or density function f(x_1,\ldots,x_n;\theta), and let x_1,\ldots,x_n be the observed values. The likelihood function is … | Estimation |
| Likelihood, prior and posterior question | Bayesian foundations | |
| Limit theorems section | Functions of random variables and limit theorems | |
| Limiting distribution theorem | Suppose P is the transition matrix of an irreducible aperiodic chain on a finite set. Then there is a unique probability row vector \boldsymbol\pi with \boldsymbol\pi=\boldsymbol\pi P, and for every start i and every state j … | Markov chains |
| The limiting distribution section | Markov chains | |
| Linear congruential generator definition | A linear congruential generator produces integers by the recursion … | Simulation |
| Links question | Introduction to generalised linear models | |
| Logit, probit and odds question | Introduction to generalised linear models | |
| A log-link coefficient question | Standard errors, and models for counts | |
| Log-partition moment identities proposition | Whenever c(\theta) can be used as a local parameter and differentiation under the normalising integral is valid … | Exponential families and conjugacy |
M
| Term | Meaning | Where |
|---|---|---|
| MAP and squared-error loss question | Bayesian foundations | |
| Margin of error definition | For a random sample with mean \mu and finite variance, the approximate confidence interval for \mu with confidence 1-\alpha is \overline X\pm z_{1-\alpha/2}\,s/\sqrt n. | Estimation examples |
| Marginal hyperposterior proposition | The binomial coefficient does not depend on (\alpha,\beta), so the hyperposterior kernel is … | Hierarchical Bayes |
| Marginal likelihood and Bayes factor definition | The marginal likelihood in model M_j is … | Bayesian hypotheses and model comparison |
| Marginal model definition | A marginal model for clustered responses has three parts. | Marginal models, and two responses at once |
| Marginal posterior of the coefficients section | Bayesian linear regression | |
| Marginal posterior of the variance section | Bayesian linear regression | |
| Marginal Student posterior corollary | Bayesian linear regression | |
| Marginal variance posterior proposition | Bayesian linear regression | |
| Markov chain definition | A stochastic process X_0,X_1,\ldots is a Markov chain if … | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Markov chains and equilibrium section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| The Markov property section | Markov chains | |
| Markov property and invariance question | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Maximum likelihood definition | For a random sample with density or mass function f(x;\theta), the likelihood is … | Estimation |
| Maximum likelihood and Fisher scoring section | Introduction to generalised linear models | |
| MCMC redirect | Redirects to Markov chain. | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Mean and variance section | Introduction to generalised linear models | |
| Mean and variance of an exponential family theorem | If Y has density or mass function equation (7), and if differentiation in \theta may be passed under the integral or the sum, then … | Introduction to generalised linear models |
| The mean response and a new response section | Linear regression | |
| Mean squared error definition | The mean squared error of an estimator T of \theta is \operatorname{MSE}(T)=\mathbb{E}(T-\theta)^2. It splits as … | Estimation |
| Memoryless property theorem | A positive random variable X is memoryless when \mathbb{P}(X>s+t\mid X>s)=\mathbb{P}(X>t) for all s,t>0. The exponential distributions are the continuous memoryless laws, and the geometric distributions are the discrete ones. | Random variables and distributions |
| Method of moments definition | Suppose the first k population moments exist and determine \theta=(\theta_1,\ldots,\theta_k). The method of moments matches them to the corresponding sample moments … | Estimation |
| Metropolis–Hastings section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Metropolis–Hastings cancellations question | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Metropolis–Hastings invariance theorem | The Metropolis–Hastings transition defined by equation (5) satisfies detailed balance with respect to \pi. Consequently, \pi is stationary. | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Metropolis–Hastings update definition | Let \pi(x) be a target density known only up to proportionality. Given X_t=x … | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Mine injuries section | Standard errors, and models for counts | |
| Minimal sufficiency definition | Sufficiency alone does not guarantee the greatest possible compression: the entire sample is trivially sufficient. A statistic T is minimal sufficient if it is a function of every other sufficient statistic. | Exponential families and conjugacy |
| Minimal sufficiency, completeness, and ancillarity section | Exponential families and conjugacy | |
| A model for the sample section | Studies, data, and evidence | |
| Moment generating function definition | The moment generating function of X is M(s)=\mathbb{E} e^{sX}, for those s at which the expectation exists. | Random variables and distributions |
| Moments and the mode section | Categorical data and the Dirichlet distribution | |
| Moments of the sample mean and variance proposition | If X_1,\ldots,X_n are a random sample from a distribution with mean \mu and variance \sigma^2, then … | Statistical inference |
| Monte Carlo estimation question | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Multinomial central limit theorem proposition | Let {\boldsymbol{x}}\sim\operatorname{Multinomial}(n,{\boldsymbol{p}}) and \widehat{\boldsymbol{p}}={\boldsymbol{x}}/n. For fixed interior {\boldsymbol{p}}, define the covariance matrix … | Categorical data and the Dirichlet distribution |
| Multinomial distribution distribution | Common distributions | |
| Multiple comparisons section | Analysis of variance | |
| Multiple linear regression definition | The multiple linear regression model for a response Y and explanatory variables x_1,\ldots,x_p is … | Regression examples |
| Multivariate normal distribution definition | A random vector \mathbf{Z} in \mathbb{R}^n is multivariate normal with mean \boldsymbol\mu and covariance matrix \Sigma, when \Sigma is positive definite and the density is … | Functions of random variables and limit theorems |
N
| Term | Meaning | Where |
|---|---|---|
| Named distributions section | Random variables and distributions | |
| Natural exponential family definition | In the a–b–c–T notation used here, the natural or canonical form of an exponential-family density is … | Exponential families and conjugacy |
| Negative-binomial variation question | Standard errors, and models for counts | |
| Neyman–Pearson approach definition | Fix a significance level \alpha. Among critical regions whose type I error is at most \alpha, choose one whose type II error is as small as possible. | Hypothesis tests |
| Nominal categories section | Categorical responses | |
| Nonconjugate priors section | The Beta–Binomial model | |
| Normal section | Random variables and distributions | |
| Normal section | Estimation | |
| Normal distribution definition · distribution | X has the normal distribution with mean \mu and variance \sigma^2>0, written X\sim\operatorname{N}(\mu,\sigma^2), when its density is … | Random variables and distributions |
| The normal distribution section | Introduction to generalised linear models | |
| Normal hierarchical shrinkage proposition | The final term is constant in \theta_j. The remaining kernel is normal, so … | Hierarchical Bayes |
| Normal linear model definition | A normal linear model for a response vector \mathbf{Y}\in\mathbb{R}^n says … | Functions of random variables and limit theorems |
| The normal linear model section | Functions of random variables and limit theorems | |
| The normal linear model section | Linear regression | |
| Normal mean and variance example section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Normal quantile plot definition | A normal quantile plot, or Q–Q plot, plots the sorted data against the quantiles of the standard normal distribution at the same fractions. | Descriptive statistics |
| Normal sample theorem theorem | If X_1,\ldots,X_n are a random sample from \operatorname{N}(\mu,\sigma^2), then … | Statistical inference |
| Normal–inverse-Gamma conjugacy theorem | Bayesian linear regression | |
| Normal–inverse-Gamma prior definition | Full joint conjugacy is obtained from the scaled prior … | Bayesian linear regression |
| Normal–inverse-Gamma variance update proposition | Hence … | Conjugate analysis for normal models |
| Normal–Normal conjugacy proposition | Conjugate analysis for normal models | |
| Normal–Normal model definition | Assume … | Conjugate analysis for normal models |
| Null and alternative hypotheses definition | The null hypothesis H_0 and the alternative hypothesis H_1 are two disjoint statements about the parameter, H_0:\theta\in\Theta_0 and H_1:\theta\in\Theta_1. H_0 is the statement being tested, often a single value \theta_0. | Hypothesis tests |
| Numerical example: five future tosses section | The Beta–Binomial model |
O
| Term | Meaning | Where |
|---|---|---|
| Observational studies and experiments section | Studies, data, and evidence | |
| Observational study definition | In an observational study the investigator does not assign the conditions being compared. A survey is the usual case. | Studies, data, and evidence |
| OLS redirect | Redirects to Ordinary least squares. | Bayesian linear regression |
| One categorical and one quantitative variable section | Descriptive statistics | |
| One categorical variable, and two section | Descriptive statistics | |
| One explanatory variable section | Regression examples | |
| One factor section | Analysis of variance | |
| One factor section | Analysis of variance examples | |
| One model, many designs section | Linear model examples | |
| One normal sample section | Confidence intervals | |
| One proportion section | Hypothesis tests | |
| One proportion: exact or approximate section | Testing examples | |
| One quantitative variable section | Descriptive statistics | |
| One variable, and a sum section | Functions of random variables and limit theorems | |
| One-sample t pivot theorem | If X_1,\ldots,X_n are independent and \operatorname{N}(\mu,\sigma^2), then … | Confidence intervals |
| One-sample t test definition | For a normal random sample, the one-sample t statistic for H_0:\mu=\mu_0 is … | Hypothesis tests |
| The one-sample t test section | Hypothesis tests | |
| One-way analysis of variance definition | In a completely randomised one-factor design, responses at level j=1,\ldots,k follow … | Analysis of variance |
| Ordinal data, built from binary splits section | Categorical responses | |
| Ordinary least squares proposition | Assume that \operatorname{rank}({\color{blue}\boldsymbol{X}})=p. Then {\color{blue}\boldsymbol{X}}^\top{\color{blue}\boldsymbol{X}} is invertible and … | Bayesian linear regression |
| Outcomes and events section | Random experiments and probability |
P
| Term | Meaning | Where |
|---|---|---|
| Paired samples section | Hypothesis tests | |
| Paired samples section | Testing examples | |
| Pearson chi-squared approximation theorem | If p_i=\pi_i for every i, the Pearson statistic T has approximately the \chi^2_{k-1} distribution for large n. | Chi-squared tests |
| Pearson statistic definition | Let (X_1,\ldots,X_k) be multinomial with n trials and cell probabilities p_1,\ldots,p_k. For reference probabilities \pi_1,\ldots,\pi_k the Pearson statistic is … | Chi-squared tests |
| Pearson test with estimated parameters theorem | Let (X_1,\ldots,X_k) be multinomial with cell probabilities p_i(\theta), where \theta has r free coordinates. | Chi-squared tests |
| Pilot adaptation and a fixed proposal section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Pitman–Koopman–Darmois, informal form theorem | For regular iid models with parameter-independent support, the theorem says, roughly, that the existence for every n of a sufficient statistic whose dimension stays fixed essentially characterises exponential families. | Exponential families and conjugacy |
| Pivot definition | A pivot is a function h({\boldsymbol{X}};\theta) of the data and the parameter whose distribution does not depend on any unknown parameter. | Confidence intervals |
| Plots and model fits section | Using R | |
| Poisson section | Random variables and distributions | |
| Poisson distribution definition · distribution | X has the Poisson distribution with mean \lambda>0, written X\sim\operatorname{Poisson}(\lambda), when … | Random variables and distributions |
| The Poisson distribution section | Introduction to generalised linear models | |
| Poisson example and Gamma conventions section | Exponential families and conjugacy | |
| Poisson imperfect-fruit example section | Bayesian hypotheses and model comparison | |
| Poisson limit of the binomial theorem | If X_n\sim\operatorname{Bin}(n,p_n) with np_n\to\lambda as n\to\infty, then \mathbb{P}(X_n=x)\to e^{-\lambda}\lambda^x/x! for every x=0,1,2,\ldots … | Random variables and distributions |
| Pooled variance definition | For two independent samples assumed to share a variance \sigma^2, the pooled sample variance is … | Confidence intervals |
| Pooling question | Hierarchical Bayes | |
| Positive continuous responses section | Underdispersion, positive responses, and quasi-likelihood | |
| Positive continuous responses question | Underdispersion, positive responses, and quasi-likelihood | |
| Posterior hypothesis probability definition | Suppose a parameter space \Omega is partitioned into regions \Omega_0 and \Omega_1. A test that instead reports a long-run p-value under a null sampling model is a hypothesis test. | Bayesian hypotheses and model comparison |
| Posterior odds and Bayes factors section | Bayesian hypotheses and model comparison | |
| Posterior point summaries definition | For a scalar parameter, common point summaries include … | Bayesian foundations |
| Posterior prediction section | Bayesian linear regression | |
| Posterior prediction question | Bayesian foundations | |
| Posterior prediction under the reference prior theorem | Bayesian linear regression | |
| Posterior predictive target definition | Prediction requires a new random quantity; estimating a fitted mean and predicting a new observation are not the same problem. For a new predictor vector \widetilde{\boldsymbol{x}}, suppose … | Bayesian linear regression |
| Posterior summaries section | Bayesian foundations | |
| Posterior summaries from draws proposition | Let \theta^{(1)},\ldots,\theta^{(L)} be posterior draws and let \theta_{(1)}\leq\cdots\leq\theta_{(L)} be their order statistics. | Bayesian foundations |
| Posterior-odds factorisation theorem | For two models M_1 and M_2, Bayes’ rule gives … | Bayesian hypotheses and model comparison |
| Power by simulation section | Testing examples | |
| Power function definition | For a test with critical region C, the power at \theta is \operatorname{Power}(\theta)=\mathbb{P}_\theta(T\in C). On \Theta_1 it equals one minus the type II error at \theta, and on \Theta_0 it is the type I error. | Hypothesis tests |
| Precision weighting remark | Writing … | Conjugate analysis for normal models |
| Prediction by simulation section | Bayesian foundations | |
| Prediction interval definition | For a new response Y\sim\operatorname{N}(\beta_0+\beta_1x,\sigma^2) independent of the sample, the interval … | Linear regression |
| Predictive-variance decomposition proposition | Under the conditional-independence assumption above … | Bayesian foundations |
| Prior and posterior prediction section | Bayesian foundations | |
| Prior and posterior predictive distributions definition | Before data are observed, a new observation \widetilde x is generated by first drawing \theta from its prior and then drawing \widetilde x from the sampling model. Marginalising \theta gives the prior predictive density … | Bayesian foundations |
| Prior predictive and posterior predictive distributions section | Bayesian foundations | |
| Priors and transparency section | Bayesian foundations | |
| Probability section | Random experiments and probability | |
| Probability generating function definition | If N takes values in \{0,1,2,\ldots\}, its probability generating function is G(z)=\mathbb{E} z^N=\sum_{n=0}^{\infty}\mathbb{P}(N=n)\,z^n, for \|z\|\le 1. | Random variables and distributions |
| Probability measure definition | A probability \mathbb{P} assigns a number to each event and satisfies three axioms: \mathbb{P}(A)\ge 0, \mathbb{P}(\Omega)=1, and for any sequence of disjoint events … | Random experiments and probability |
| Probability statements and decisions section | Bayesian hypotheses and model comparison | |
| Process and realisation question | Dependence along a series | |
| Product rule theorem | If \mathbb{P}(A_1\cap\cdots\cap A_{n-1})>0, then … | Random experiments and probability |
| A proper approximation to the flat prior remark | The honest bounded version of the improper kernel p(\mu)\propto1 is … | Conjugate analysis for normal models |
| Proper prior and proper posterior definition | A prior is proper if it is a probability distribution, so its total mass is one. A posterior is proper if the product of likelihood and prior has a finite, nonzero normalising constant. | Bayesian foundations |
| Properties of a probability theorem | For events A and B … | Random experiments and probability |
| A proportion section | Confidence intervals | |
| Proportional odds question | Categorical responses | |
| Proportional-odds model definition | A proportional-odds model for ordered categories 1,\ldots,K and covariates {\boldsymbol{x}} is … | Categorical responses |
| Proportions section | Estimation examples | |
| Pseudo-counts and sequential updating section | The Beta–Binomial model | |
| p-value definition | The p-value of an observed statistic t is the probability, under H_0, that T is at least as extreme as t, in the direction of H_1. For a two-sided normal or t statistic it is twice the one-sided tail probability beyond \|t\|. | Hypothesis tests |
Q
| Term | Meaning | Where |
|---|---|---|
| Quasi-likelihood section | Underdispersion, positive responses, and quasi-likelihood | |
| Quasilikelihood question | Underdispersion, positive responses, and quasi-likelihood | |
| Quasi-Poisson section | Standard errors, and models for counts | |
| Quasi-score and quasi-likelihood definition | If \mathrm{Var}(Y)=\phi V(\mu), the quasi-score in the mean is … | Underdispersion, positive responses, and quasi-likelihood |
R
| Term | Meaning | Where |
|---|---|---|
| A random intercept section | Random intercepts | |
| Random intercepts and slopes question | Random slopes, and a binary mixed model | |
| Random numbers section | Simulation | |
| Random sample definition | X_1,\ldots,X_n are a random sample from a distribution F when they are independent and each has distribution F. The joint density or mass function is then the product of the marginals. | Statistical inference |
| Random variable definition | A random variable is a function X from the sample space \Omega to the real numbers. The event \{X\le x\} is the set of outcomes \omega with X(\omega)\le x. | Random variables and distributions |
| Random variables section | Random variables and distributions | |
| Randomisation and blocking section | Analysis of variance examples | |
| Randomised block design definition | A randomised block design measures every one of t treatments once in each of b blocks. The response for block i and treatment j is … | Analysis of variance |
| Randomised controlled trial definition | In a randomised controlled trial each experimental unit receives one of several treatments, chosen at random with equal probability, and one of the treatments is a control, such as the standard treatment or a placebo. | Studies, data, and evidence |
| Randomness as a model section | Random experiments and probability | |
| Random-walk and independent proposals section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Rat tumours: why a hierarchy? section | Hierarchical Bayes | |
| A ratio of variances section | Confidence intervals | |
| Reading a log-linear coefficient section | Standard errors, and models for counts | |
| Reading the addiction model section | Paired comparisons, and how far a fit is from the data | |
| Recognising the posterior distributions section | Bayesian linear regression | |
| The regression effect section | Linear regression | |
| Reliability function definition | Under independent component states, the reliability function is r(\mathbf{p})=\mathbb{P}(\varphi(\mathbf{X})=1), a function of the vector \mathbf{p}=(p_1,\ldots,p_n). | Reliability |
| The reliability function section | Reliability | |
| The residual sum-of-squares decomposition section | Bayesian linear regression | |
| Residuals are not section | Paired comparisons, and how far a fit is from the data | |
| A roughness penalty section | Smooth means | |
| Roughness penalty question | Smooth means | |
| R’s distribution interface remark | For a common distribution, R prefixes the distribution name by d for density, p for cumulative probability, q for quantile, and r for random generation. |
Bayesian foundations |
| A running example: milk cartons section | Statistical inference | |
| Running the Gibbs sampler section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Runs in the past week section | Introduction to generalised linear models |
S
| Term | Meaning | Where |
|---|---|---|
| Sample correlation coefficient definition | For pairs (x_1,y_1),\ldots,(x_n,y_n), the sample correlation coefficient is … | Estimation |
| Sample quantile definition | A sample p-quantile, for 0<p<1, is a value q_p such that a fraction p of the data is at most q_p. The 0.25, 0.5 and 0.75 quantiles are the first quartile, the median and the third quartile … | Descriptive statistics |
| Sample space definition | The sample space \Omega of a random experiment is the set of all outcomes that experiment can produce. | Random experiments and probability |
| Sample variance and standard deviation definition | The sample variance is s^2=\frac1{n-1}\sum_{i=1}^n(x_i-\overline x)^2, and the sample standard deviation is s=\sqrt{s^2}, in the units of the data. | Descriptive statistics |
| The sampling distribution of a normal sample section | Statistical inference | |
| Sampling distribution of the OLS estimator proposition | Bayesian linear regression | |
| Sampling distribution of the regression estimators theorem | Under simple linear regression … | Linear regression |
| Sampling from a population section | Studies, data, and evidence | |
| Sandwich robustness question | Marginal models, and two responses at once | |
| Saturated model and deviance question | Paired comparisons, and how far a fit is from the data | |
| Score and information question | Introduction to generalised linear models | |
| Seeing the approximation section | Chi-squared examples | |
| Sequential coherence proposition | Sequential updating is exact. | The Beta–Binomial model |
| Several explanatory variables section | Regression examples | |
| Sign test definition | For independent pairs (X_i,Y_i), i=1,\ldots,n, let N be the number of pairs with Y_i>X_i, ties discarded. If H_0:\mathbb{P}(Y_i>X_i)=1/2 holds, N\sim\operatorname{Bin}(n,1/2), and H_0 is tested with the binomial test. | Hypothesis tests |
| The sign test section | Hypothesis tests | |
| Significance is not adequacy question | Paired comparisons, and how far a fit is from the data | |
| Simple linear regression definition | The simple linear regression model says … | Linear regression |
| Simple random sample definition | A simple random sample of size n from a finite population of size N is drawn without replacement so that every subset of n members is equally likely to be the sample. | Studies, data, and evidence |
| Simulation section | Markov chains | |
| Sleep deprivation section | Random slopes, and a binary mixed model | |
| Slope t pivot theorem | Under simple linear regression … | Linear regression |
| A slope that belongs to the block section | Random slopes, and a binary mixed model | |
| Small normal samples section | Estimation examples | |
| Standard errors in a GLM section | Standard errors, and models for counts | |
| Stationarity section | Dependence along a series | |
| Stationarity and detailed balance definition | A distribution \pi is stationary if a state distributed as \pi remains distributed as \pi after one transition. The global balance equation is … | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Statistic definition | A statistic is a function of the sample that does not depend on the unknown parameter. Its sampling distribution is the distribution of that function when the sample is still random. | Statistical inference |
| Statistic and estimator definition | Write the complete sample as {\boldsymbol{X}}=(X_1,\ldots,X_n) and its realised value as {\boldsymbol{x}}=(x_1,\ldots,x_n). A statistic is any function … | Exponential families and conjugacy |
| Statistics and inferential reduction section | Exponential families and conjugacy | |
| Stratified and cluster sampling definition | In stratified sampling the population is split into subgroups called strata, and a simple random sample is drawn from each, usually of size nN_s/N for a stratum of size N_s. | Studies, data, and evidence |
| Structure section | Reliability | |
| Structure function definition | The structure function \varphi:\{0,1\}^n\to\{0,1\} equals 1 when the system works and 0 when it has failed. A system is monotone when \mathbf{x}\le\mathbf{y} implies \varphi(\mathbf{x})\le\varphi(\mathbf{y}) … | Reliability |
| Student t distribution definition | If Z is standard normal, V is \chi^2_m, and Z and V are independent, then T=Z/\sqrt{V/m} has the Student t distribution with m degrees of freedom, written t_m. | Statistical inference |
| Studentised coefficient pivot corollary | Bayesian linear regression | |
| Study-rate full conditional proposition | Normalising gives … | Hierarchical Bayes |
| Sufficiency question | Exponential families and conjugacy | |
| Sufficiency redirect | Redirects to Sufficient statistic. | Exponential families and conjugacy |
| Sufficiency and the Fisher–Neyman theorem section | Exponential families and conjugacy | |
| Sufficient statistic definition | A statistic T({\boldsymbol{X}}) is sufficient for \theta if the conditional distribution of {\boldsymbol{X}} given T({\boldsymbol{X}})=t does not depend on \theta. | Exponential families and conjugacy |
| Summary statistics section | Descriptive statistics | |
| Sums of smooth functions section | Additive smooths | |
| Survival function definition | The survival function of a lifetime T with distribution function F is R(t)=\mathbb{P}(T>t)=1-F(t) for t\ge 0. | Reliability |
| Systematic-scan Gibbs sampler definition | Suppose the target is a joint density \pi(x_1,\ldots,x_p) that is difficult to sample from directly, while every full conditional is available. One Gibbs sweep updates coordinates in order … | Markov chains, Gibbs sampling, and Metropolis–Hastings |
T
| Term | Meaning | Where |
|---|---|---|
| The table section | Descriptive statistics | |
| Tests and intervals section | Hypothesis tests | |
| Tests for a normal mean and variance section | Hypothesis tests | |
| The three components of a GLM section | Introduction to generalised linear models | |
| Three corrections and a problem-solving checklist section | Bayesian linear regression | |
| Three distributions built from normal variables section | Statistical inference | |
| Three easy confusions remark | 1. \widehat{\boldsymbol{\beta}} has an exact normal sampling distribution when \sigma^2 is known. The t distribution applies after studentisation by s. | Bayesian linear regression |
| Three examples the linear model does not fit section | Introduction to generalised linear models | |
| Three links section | Introduction to generalised linear models | |
| Three links, one cholesterol slope section | Paired comparisons, and how far a fit is from the data | |
| Throw out mean plus error section | Introduction to generalised linear models | |
| Tower property theorem | If the expectations exist, then \mathbb{E}\bigl(\mathbb{E}(Y\mid X)\bigr)=\mathbb{E} Y. More generally, \mathbb{E}\bigl(\mathbb{E}(Y\mid X,Z)\mid X\bigr)=\mathbb{E}(Y\mid X). | Joint distributions |
| Transformation rule theorem | Functions of random variables and limit theorems | |
| The transformation rule section | Functions of random variables and limit theorems | |
| Transformations section | Descriptive statistics | |
| Transformations section | Linear model examples | |
| Transforms section | Random variables and distributions | |
| Translation to another common notation remark | Some books write the same density as … | Exponential families and conjugacy |
| Tukey intervals definition | In a one-way design with k groups of equal size r, and n=kr, the probability is 1-\alpha that every one of the \binom k2 differences \mu_i-\mu_j simultaneously satisfies … | Analysis of variance |
| Two factors section | Analysis of variance examples | |
| Two independent samples section | Testing examples | |
| Two kinds of interval question | Bayesian foundations | |
| Two models with a real distribution section | Standard errors, and models for counts | |
| Two normal samples section | Confidence intervals | |
| Two proportions section | Hypothesis tests | |
| Two proportions section | Testing examples | |
| Two pseudo-count conventions remark | For posterior means, (\alpha,\beta) act as prior success and failure counts. For an interior posterior mode, the corresponding counts are (\alpha-1,\beta-1). | The Beta–Binomial model |
| Two quantitative variables section | Descriptive statistics | |
| Two responses, one person section | Marginal models, and two responses at once | |
| Two samples section | Hypothesis tests | |
| Two samples with unequal variances section | Estimation examples | |
| Two stationarity definitions question | Dependence along a series | |
| Two treatments: the paired t test section | Analysis of variance | |
| Two-proportion z test definition | For independent X_1\sim\operatorname{Bin}(n_1,p_1) and X_2\sim\operatorname{Bin}(n_2,p_2), let \widehat p=(X_1+X_2)/(n_1+n_2). The statistic for H_0:p_1=p_2 is … | Hypothesis tests |
| Two-sample t pivot theorem | For two independent normal samples with a common variance … | Confidence intervals |
| Two-sample t test definition | For two independent normal samples with a common variance, the two-sample t statistic for H_0:\mu_1-\mu_2=\delta_0 is … | Hypothesis tests |
| Type I and type II errors definition | A type I error rejects H_0 when H_0 is true. A type II error accepts H_0 when H_1 is true. The size, or significance level, is the probability of a type I error. | Hypothesis tests |
U
| Term | Meaning | Where |
|---|---|---|
| Unbiased estimator definition | An estimator T of \theta is unbiased when \mathbb{E}_\theta T=\theta for every \theta in the parameter space. The bias is \mathbb{E}_\theta T-\theta. | Estimation |
| Uniform section | Random variables and distributions | |
| Uniform distribution definition | X is uniform on the interval [a,b], written X\sim\operatorname{U}[a,b], when its density is 1/(b-a) on that interval and zero elsewhere. Then \mathbb{E} X=(a+b)/2 and \operatorname{Var}(X)=(b-a)^2/12. | Random variables and distributions |
| Uniformity depends on a domain remark | The Jacobian calculation is local; calling the prior “uniform in (u,v)” also requires a stated domain. | Hierarchical Bayes |
| Unit roots and ARIMA question | Forecasting | |
| Unknown mean, known variance section | Conjugate analysis for normal models | |
| Unknown variance, known mean section | Conjugate analysis for normal models | |
| Unrestricted model section | Bayesian hypotheses and model comparison |
V
| Term | Meaning | Where |
|---|---|---|
| Values, vectors, and assignment section | Using R | |
| Variance definition | The variance of X is \operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2, when that expectation exists. Equivalently, \operatorname{Var}(X)=\mathbb{E} X^2-(\mathbb{E} X)^2. The standard deviation is the square root of the variance. | Random variables and distributions |
| A variance section | Confidence intervals | |
| Variance-ratio F pivot theorem | For two independent normal samples … | Confidence intervals |
| Vector-response regression question | Marginal models, and two responses at once |
W
| Term | Meaning | Where |
|---|---|---|
| Wald interval definition | For X\sim\operatorname{Bin}(n,p) with \widehat p=X/n, the Wald interval with approximate confidence 1-\alpha is … | Confidence intervals |
| Warm-up and dependence question | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Warm-up is not a convergence proof remark | The chain should be initialised in a plausible region when one is available. | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Warp breaks section | Marginal models, and two responses at once | |
| Weibull distribution definition | A lifetime T is Weibull with shape \beta>0 and rate \lambda>0 when its density is \lambda\beta\,(\lambda t)^{\beta-1}\exp(-(\lambda t)^\beta) for t>0. | Reliability |
| Welch interval definition | For independent normal samples of sizes n_1 and n_2 with sample variances s_X^2 and s_Y^2, the Welch interval for \mu_X-\mu_Y with approximate confidence 1-\alpha is … | Estimation examples |
| What a regression statement is hiding section | Introduction to generalised linear models | |
| What is fixed, and what is random section | Random intercepts | |
| What is random? section | Bayesian foundations | |
| What the rest of this section does section | Statistical inference | |
| What the software prints section | Introduction to generalised linear models | |
| When the likelihood equation fails section | Estimation | |
| Where the shrinkage weight comes from section | Hierarchical Bayes | |
| Which groups differ? section | Analysis of variance examples | |
| White noise is not necessarily independent question | Dependence along a series | |
| Why a plug-in prediction is incomplete section | Bayesian foundations | |
| Why another sampler is needed section | Markov chains, Gibbs sampling, and Metropolis–Hastings | |
| Why exponential families keep appearing section | Exponential families and conjugacy | |
| Why hierarchical Bayes? section | Hierarchical Bayes | |
| Wold decomposition theorem | If \{X_t\} is weakly stationary with mean zero, then … | Dependence along a series |
| Worked factorisation: a normal mean section | Exponential families and conjugacy | |
| Worked prior-sensitivity calculation section | The Beta–Binomial model | |
| Worked regression example section | Bayesian linear regression | |
| Working correlation question | Marginal models, and two responses at once |