Free CAS MAS-II (Modern Actuarial Statistics II) Formula Sheet (2026)

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4 Topics
2026 Syllabus
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All MAS-II Formulas

Introduction to Credibility 27 items
Limited fluctuation full credibility standard (claim count)
nF=(z(1+P)/2k)2n_F = \left( \dfrac{z_{(1+P)/2}}{k} \right)^2
where P = probability the estimate is within k of the expected value (Poisson claim counts). Example: P = 0.90, k = 0.05 gives nF=(1.645/0.05)2=1082n_F = (1.645/0.05)^2 = 1082.
Full credibility for pure premium
nFPP=nFN×(1+CVX2)n_F^{PP} = n_F^{N} \times (1 + CV_X^2)
where nFNn_F^N is the full credibility standard for claim count, X is severity per claim, CVX=σX/μXCV_X = \sigma_X/\mu_X is severity coefficient of variation. Pure premium adds severity-side variance to the claim-count standard.
Full credibility for aggregate losses (compound Poisson)
Same as pure premium because aggregate losses = N ×\times X under compound Poisson with independence: nFAgg=nFN(1+CVX2)n_F^{Agg} = n_F^N (1 + CV_X^2).
The extra variance over count-only reflects severity variability. If severity is degenerate, formula reduces to the claim-count standard.
Partial credibility (limited fluctuation)
Z=nnFZ = \sqrt{\dfrac{n}{n_F}}
Capped at 1 (full credibility). Credibility-weighted estimate: X^=ZXˉ+(1−Z)M\hat{X} = Z \bar{X} + (1 - Z)M where M is the complement (e.g., manual rate, prior mean). Z grows as n\sqrt{n}, not linearly.
Bühlmann credibility (greatest accuracy)
Z=nn+kZ = \dfrac{n}{n + k} where k=EPVVHMk = \dfrac{\text{EPV}}{\text{VHM}}
EPV = Expected Process Variance, VHM = Variance of Hypothetical Means.
Credibility-weighted estimate: PC=ZXˉ+(1−Z)μP_C = Z \bar{X} + (1 - Z)\mu.
No upper-cap distortion (unlike LFC). Linear least-squares Bayes.
Expected process variance (EPV)
EPV=E[Var(X∣Θ)]\text{EPV} = E[\text{Var}(X | \Theta)]
Average of conditional variances across risk types. Represents within-risk variation (pure noise that no amount of data can resolve). Computed as expected value of the variance for each parameter θ\theta weighted by the prior on Θ\Theta.
Variance of hypothetical means (VHM)
VHM=Var(E[X∣Θ])\text{VHM} = \text{Var}(E[X | \Theta])
Variance of conditional means across risk types. Represents between-risk variation (the signal we want to estimate). Total variance: Var(X)=EPV+VHM\text{Var}(X) = \text{EPV} + \text{VHM} (law of total variance).
Bühlmann-Straub credibility (varying exposures)
For risks with exposure mijm_{ij} (year j of risk i): Zi=mimi+kZ_i = \dfrac{m_i}{m_i + k} where mi=∑jmijm_i = \sum_j m_{ij}, k=vak = \dfrac{v}{a}, v = expected process variance per unit exposure, a = variance of hypothetical means. Allows different observation periods per risk.
Bayesian credibility (conjugate prior families)
Poisson-Gamma: count data X | λ\lambda is Poisson, λ\lambda is Gamma(α,β)(\alpha, \beta). Posterior is Gamma(α+∑Xi,β+n)(\alpha + \sum X_i, \beta + n). Posterior mean: α+∑Xiβ+n\dfrac{\alpha + \sum X_i}{\beta + n}. Equivalent Bühlmann form: Z=n/(n+β)Z = n/(n + \beta).
Bayesian credibility (Normal-Normal)
X | μ\mu ~ N(μ,σ2)(\mu, \sigma^2), μ\mu ~ N(μ0,τ2)(\mu_0, \tau^2). Posterior is Normal with mean σ2/n⋅μ0+τ2Xˉσ2/n+τ2\dfrac{\sigma^2/n \cdot \mu_0 + \tau^2 \bar{X}}{\sigma^2/n + \tau^2}. Bühlmann form: Z=nn+σ2/τ2Z = \dfrac{n}{n + \sigma^2/\tau^2}.
Empirical Bayes (non-parametric) credibility
Estimate EPV and VHM from data:
EPV^=1r∑isi2\widehat{\text{EPV}} = \dfrac{1}{r} \sum_i s_i^2 (avg within-risk sample variance)
VHM^=sXˉ2−EPV^/n\widehat{\text{VHM}} = s_{\bar{X}}^2 - \widehat{\text{EPV}}/n (between-risk variance minus the within-risk contribution).
Empirical Bayes (semi-parametric) credibility
Assume process distribution is parametric (e.g., Poisson) and use the parametric variance form. For Poisson, Var(X∣Θ)=Θ\text{Var}(X | \Theta) = \Theta, so EPV^=Xˉ\widehat{\text{EPV}} = \bar{X} (the grand mean). VHM estimated from cross-risk variation in observed rates. Less data-hungry than full non-parametric.
Linear least-squares (Bühlmann) credibility derivation
Choose Z to minimize E[(ZXˉ+(1−Z)μ−E[X∣Θ])2]E[(Z \bar{X} + (1 - Z)\mu - E[X | \Theta])^2]. Solution: Z=aa+v/nZ = \dfrac{a}{a + v/n} where a = VHM, v = expected process variance per observation, n = sample size. Same form as Bühlmann.
Credibility complement
When data are not fully credible (Z < 1), the complement (1 - Z) weight is given to a benchmark M:
PC=ZXˉ+(1−Z)MP_C = Z \bar{X} + (1 - Z) M.
Common complements: overall manual rate, larger-class rate, prior-year experience adjusted for trend, present rates underlying current rates.
Loss-ratio credibility weighting
Indicated rate change I=Z⋅Actual LRTarget LR+(1−Z)⋅Trended Current Rate ChangeI = Z \cdot \dfrac{\text{Actual LR}}{\text{Target LR}} + (1 - Z) \cdot \text{Trended Current Rate Change}.
Used in rate filings where indication blends company experience with a manual complement. Z derived per Bühlmann from EPV and VHM estimates.
Bühlmann credibility for severity
When estimating expected severity X (not pure premium), Bühlmann form uses claim count as exposure: Z=NN+kZ = \dfrac{N}{N + k} where N = observed claim count and k reflects EPV and VHM in the severity dimension. Pure-premium credibility combines count and severity.
Conjugate priors summary
Likelihood / Conjugate Prior / Posterior:
Poisson / Gamma / Gamma
Binomial / Beta / Beta
Normal (known σ\sigma) / Normal / Normal
Normal (unknown σ\sigma) / Normal-Inverse-Gamma / Normal-Inverse-Gamma
Exponential / Gamma / Gamma
Conjugacy gives closed-form posteriors; useful for sequential updating.
Credibility-weighted reserving (Cape Cod)
ELR (expected loss ratio) Cape Cod: ELR^=∑Ci∑(Pi⋅fi)\widehat{\text{ELR}} = \dfrac{\sum C_i}{\sum (P_i \cdot f_i)} where CiC_i = paid losses, PiP_i = premium, fif_i = expected reporting percent.
Applied to ultimate losses: Ulti=Ci+(1−fi)⋅Pi⋅ELR^\text{Ult}_i = C_i + (1 - f_i) \cdot P_i \cdot \widehat{\text{ELR}}.
Effect of trend and benefit changes on credibility
When experience period has different trend or benefit levels than the projection period, develop and trend experience to current level BEFORE applying credibility. Credibility weight Z is unchanged; the trended experience replaces Xˉ\bar{X}.
Mahler credibility (changing risk parameters)
When risk parameter evolves over time (loss development, distribution drift), credibility on older data declines. Mahler's correlation: ρ1,t2\rho_{1, t}^2 is the squared correlation between year 1 and year t hypothetical means. Older years receive lower weight in the credibility-weighted estimate.
Exposure-weighted risk average in Buhlmann-Straub
Xˉi=∑jmij Xijmi\bar{X}_i = \frac{\sum_j m_{ij}\,X_{ij}}{m_i} — m_{ij} = exposure for risk i period j, X_{ij} = loss per unit exposure, m_i = total exposure
Bayesian predictive premium
PB=∫μ(θ) π(θ∣x) dθP_B = \int \mu(\theta)\,\pi(\theta\mid\mathbf{x})\,d\theta — μ(θ) = hypothetical mean, π(θ|x) = posterior density of Θ, x = observed data
Classical limited fluctuation reliability criterion
Pr⁡(∣Xˉ−μ∣≤kμ)≥1−α\Pr(|\bar{X} - \mu| \le k\mu) \ge 1 - \alpha — X̄ = sample mean, μ = true mean, k = tolerance fraction, α = error probability
Poisson-Gamma posterior mean
α+∑xiβ+n\dfrac{\alpha + \sum x_i}{\beta + n} — α, β = Gamma prior hyperparameters, Σx_i = sum of observed claim counts, n = number of exposure periods
Bayesian predictive mean
PB=∫μ(θ) π(θ∣x) dθP_B = \int \mu(\theta)\,\pi(\theta\mid\mathbf{x})\,d\theta — μ(θ) = hypothetical mean given θ, π(θ|x) = posterior density of θ given data x
Exposure-weighted risk average (Buhlmann-Straub)
Xˉi=∑jmijXijmi\bar{X}_i = \frac{\sum_j m_{ij} X_{ij}}{m_i} — m_ij = exposure in period j, X_ij = loss per exposure, m_i = total exposure for risk i
Credibility-weighted premium
Pc=ZXˉ+(1−Z)μP_c = Z\bar{X} + (1-Z)\mu — Z = credibility weight, X̄ = observed risk mean, μ = collective/prior/manual mean
Linear Mixed Models 22 items
Linear mixed model general form
Y=Xβ+Zb+ε\mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \mathbf{Z} \mathbf{b} + \boldsymbol{\varepsilon}
β\boldsymbol{\beta} = fixed-effect coefficients, b\mathbf{b} = random effects (with b∼N(0,G)\mathbf{b} \sim N(\mathbf{0}, \mathbf{G})), ε∼N(0,R)\boldsymbol{\varepsilon} \sim N(\mathbf{0}, \mathbf{R}).
Fixed effects vs random effects
Fixed: coefficients for ALL levels of interest (e.g., treatment vs control). Estimated directly.
Random: assume levels sampled from a population; estimate VARIANCE of the level effects rather than each effect.
Random intercept model
Yij=(β0+b0i)+β1xij+εijY_{ij} = (\beta_0 + b_{0i}) + \beta_1 x_{ij} + \varepsilon_{ij} where b0i∼N(0,σb2)b_{0i} \sim N(0, \sigma_b^2). Each group i gets its own intercept shifted by b0ib_{0i} from the grand intercept. Common slope β1\beta_1 across groups.
Random slope model
Yij=β0+(β1+b1i)xij+εijY_{ij} = \beta_0 + (\beta_1 + b_{1i}) x_{ij} + \varepsilon_{ij} where b1i∼N(0,σb12)b_{1i} \sim N(0, \sigma_{b1}^2). Each group has its own slope. Random intercepts and slopes can co-occur with a correlation parameter ρ\rho between them.
Intraclass correlation coefficient (ICC)
ICC=σb2σb2+σ2\text{ICC} = \dfrac{\sigma_b^2}{\sigma_b^2 + \sigma^2}
Proportion of total variance attributable to between-group differences. ICC close to 0 = groups largely homogeneous. ICC close to 1 = strong clustering. High ICC justifies the random effect.
Best linear unbiased predictor (BLUP)
Predicted random effect: b^=GZ′(ZGZ′+R)−1(Y−Xβ^)\hat{\mathbf{b}} = \mathbf{G} \mathbf{Z}' (\mathbf{Z} \mathbf{G} \mathbf{Z}' + \mathbf{R})^{-1} (\mathbf{Y} - \mathbf{X} \hat{\boldsymbol{\beta}}).
Shrinkage estimator: BLUP shrinks group means toward the grand mean. Equivalent to empirical Bayes / credibility weighting.
REML vs ML estimation
ML: maximizes full likelihood. Variance components biased downward in small samples (does not account for fixed-effect estimation).
REML (restricted ML): maximizes likelihood of contrasts orthogonal to fixed effects. Less biased variance estimates. Default in mixed-model software for variance-component inference.
Likelihood ratio test for variance components
Test H0:σb2=0H_0: \sigma_b^2 = 0 against Ha:σb2>0H_a: \sigma_b^2 > 0. Test statistic: Λ=−2(ℓ0−ℓa)\Lambda = -2(\ell_0 - \ell_a).
Null distribution is mixture 12χ02+12χ12\frac{1}{2}\chi_0^2 + \frac{1}{2}\chi_1^2 (since σb2=0\sigma_b^2 = 0 is on the boundary). Standard χ12\chi_1^2 p-value is conservative; divide by 2 for correct test.
LMM vs GEE (marginal models)
LMM: subject-specific (conditional) interpretation of β\beta. Models the random effects explicitly.
GEE (generalized estimating equations): population-average (marginal) interpretation. Specifies only the mean and working correlation structure. Robust standard errors.
Variance component interpretation in actuarial context
Random territory effect: σb2\sigma_b^2 measures cross-territory loss variation NOT explained by fixed-effect covariates. Higher σb2\sigma_b^2 = more residual heterogeneity at the territory level = more value to including territory random effects (or a credibility step) in the rating plan.
BLUP shrinkage (credibility) factor for group i
Zi=niσu2niσu2+σε2Z_i = \frac{n_i \sigma_u^2}{n_i \sigma_u^2 + \sigma_\varepsilon^2} — n_i = obs in group i, σu2\sigma_u^2 = random-effect variance, σε2\sigma_\varepsilon^2 = residual variance
Boundary-corrected p-value for variance-component LRT
p=12P(χ12≥LRT)p = \tfrac{1}{2} P(\chi^2_1 \ge \text{LRT}) — LRT = likelihood ratio statistic, p = tail probability under 50:50 χ02/χ12\chi^2_0/\chi^2_1 mixture
Marginal distribution of LMM response
y∼N(Xβ,  ZGZ⊤+R)y \sim N(X\beta,\;ZGZ^{\top} + R) — X = fixed-effect design, β = fixed effects, Z = random-effect design, G = Var(u), R = Var(ε)
Marginal covariance matrix V of LMM
V=ZGZ⊤+RV = ZGZ^{\top} + R — Z = random-effect design, G = random-effect covariance, R = residual covariance; V is what REML and ML actually fit
Conditional variance of LMM response given random effects
Var⁡(y∣u)=R\operatorname{Var}(y \mid u) = R — R = residual covariance matrix; conditional on u the LMM reduces to standard linear regression with covariance R
Conditional mean of LMM response given random effects
E[y∣u]=Xβ+ZuE[y \mid u] = X\beta + Zu — X = fixed-effect design, β = fixed effects, Z = random-effect design, u = realized random-effect vector
BLUP shrinkage factor for hierarchical group j
λj=τ2/(τ2+σ2/nj)\lambda_j = \tau^2 / (\tau^2 + \sigma^2/n_j) — τ² = between-group variance, σ² = within-group variance, n_j = group j sample size
Hierarchical partially pooled group rate
β^0+u^j=β^0+λj(yˉj−β^0)\hat{\beta}_0 + \hat{u}_j = \hat{\beta}_0 + \lambda_j(\bar{y}_j - \hat{\beta}_0) — β₀ = grand intercept, λ_j = shrinkage factor, ȳ_j = group j sample mean
Boundary null distribution for variance-component LRT
LRT∼12χ02+12χ12\text{LRT} \sim \tfrac{1}{2}\chi^2_0 + \tfrac{1}{2}\chi^2_1 under H0:τ2=0H_0: \tau^2 = 0 — mixture because zero sits on parameter-space boundary
Marginal residuals in linear mixed model
rM=y−Xβ^r^M = y - X\hat{\beta} — y = response vector, X = fixed-effect design matrix, β̂ = estimated fixed-effect coefficients
Likelihood ratio test statistic for nested LMMs
LR=−2(ℓR−ℓF)LR = -2(\ell_R - \ell_F) — ℓ_R = maximized log-likelihood of reduced model, ℓ_F = maximized log-likelihood of full (nesting) model
Conditional residuals in linear mixed model
rC=y−Xβ^−Zb^r^C = y - X\hat{\beta} - Z\hat{b} — Z = random-effect design matrix, b̂ = predicted random effects (BLUPs), X, β̂ as in marginal residuals
Statistical Learning 74 items
Bias-variance decomposition
E[(y−f^(x))2]=(Bias[f^(x)])2+Var[f^(x)]+σε2E[(y - \hat{f}(x))^2] = (\text{Bias}[\hat{f}(x)])^2 + \text{Var}[\hat{f}(x)] + \sigma_\varepsilon^2
Irreducible error σε2\sigma_\varepsilon^2 cannot be reduced. Complex models lower bias but raise variance; simpler models do the opposite. Optimal complexity minimizes total MSE.
Training, validation, and test sets
Training: fit model parameters.
Validation: tune hyperparameters (e.g., λ\lambda in ridge, depth in trees). Select the model.
Test: estimate generalization error of the FINAL model on truly held-out data.
Common splits: 60/20/20 or 70/15/15. Never tune hyperparameters on test data (causes optimistic estimates).
k-fold cross-validation
Split data into k roughly-equal folds. For each fold i: train on the other k-1 folds, evaluate on fold i. CV error =1k∑i=1kMSEi= \dfrac{1}{k} \sum_{i=1}^k \text{MSE}_i. Typical k = 5 or 10.
LOOCV: k = n. Low bias but high variance and computationally expensive (n model fits).
Bootstrap
Sample with replacement from the original n-row dataset to create B bootstrap samples (each of size n; ~63% of unique original rows). Compute statistic on each. Bootstrap standard error: SE^∗=1B−1∑(θ^b∗−θ^ˉ∗)2\widehat{\text{SE}}^* = \sqrt{\dfrac{1}{B-1} \sum (\hat\theta_b^* - \bar{\hat\theta}^*)^2}.
OLS estimator (matrix form)
β^=(X′X)−1X′y\hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1} \mathbf{X}'\mathbf{y}
Var(β^)=σ2(X′X)−1\text{Var}(\hat{\boldsymbol{\beta}}) = \sigma^2 (\mathbf{X}'\mathbf{X})^{-1}.
Under Gauss-Markov (homoskedastic, uncorrelated errors), OLS is BLUE.
Multiple R-squared and adjusted R-squared
R2=1−SSresSStotR^2 = 1 - \dfrac{SS_{res}}{SS_{tot}}. Never decreases when predictors are added.
Radj2=1−SSres/(n−p−1)SStot/(n−1)R^2_{adj} = 1 - \dfrac{SS_{res}/(n - p - 1)}{SS_{tot}/(n - 1)}. Penalizes for extra predictors p. Adjusted R-squared can decrease when irrelevant variables are added.
AIC and BIC
AIC=2k−2ln⁡L\text{AIC} = 2k - 2\ln L (k = parameters, L = max likelihood).
BIC=kln⁡n−2ln⁡L\text{BIC} = k \ln n - 2\ln L.
Lower is better. BIC has stiffer penalty than AIC for large n, so BIC favors smaller models. AIC targets predictive accuracy; BIC targets the true model under the assumption it is in the candidate set.
Ridge regression (L2 penalty)
β^ridge=arg⁡min⁡∑(yi−xi′β)2+λ∑βj2\hat{\boldsymbol{\beta}}_{\text{ridge}} = \arg\min \sum (y_i - \mathbf{x}_i' \boldsymbol{\beta})^2 + \lambda \sum \beta_j^2
Closed form: β^ridge=(X′X+λI)−1X′y\hat{\boldsymbol{\beta}}_{\text{ridge}} = (\mathbf{X}'\mathbf{X} + \lambda \mathbf{I})^{-1}\mathbf{X}'\mathbf{y}. Shrinks coefficients toward zero (never to zero).
Lasso regression (L1 penalty)
β^lasso=arg⁡min⁡∑(yi−xi′β)2+λ∑∣βj∣\hat{\boldsymbol{\beta}}_{\text{lasso}} = \arg\min \sum (y_i - \mathbf{x}_i' \boldsymbol{\beta})^2 + \lambda \sum |\beta_j|
L1 penalty produces exact zeros: performs variable selection. No closed form (requires coordinate descent or LARS). Most useful when many predictors are irrelevant.
Elastic net
β^=arg⁡min⁡∑(yi−xi′β)2+λ1∑∣βj∣+λ2∑βj2\hat{\boldsymbol{\beta}} = \arg\min \sum (y_i - \mathbf{x}_i' \boldsymbol{\beta})^2 + \lambda_1 \sum |\beta_j| + \lambda_2 \sum \beta_j^2
Combines lasso variable selection with ridge stability. Useful when predictors are correlated (lasso would arbitrarily select one; elastic net keeps both with shrinkage).
Variable selection: forward, backward, stepwise
Forward: start empty; add the predictor that most improves fit (lowest p-value, highest R2R^2, lowest AIC).
Backward: start full; remove the least helpful predictor each step.
Stepwise: alternate forward and backward, allowing previously-added vars to be removed if a better predictor enters.
Logistic regression
Pr⁡(Y=1∣x)=11+e−x′β\Pr(Y = 1 | \mathbf{x}) = \dfrac{1}{1 + e^{-\mathbf{x}' \boldsymbol{\beta}}}
Log-odds (logit): ln⁡p1−p=x′β\ln \dfrac{p}{1 - p} = \mathbf{x}' \boldsymbol{\beta}.
Fit by maximum likelihood. eβje^{\beta_j} is the odds ratio per unit change in xjx_j (others fixed).
Linear discriminant analysis (LDA)
Assumes each class is Gaussian with a SHARED covariance matrix. Decision boundary is linear.
δk(x)=x′Σ−1μk−12μk′Σ−1μk+ln⁡πk\delta_k(\mathbf{x}) = \mathbf{x}' \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu}_k - \frac{1}{2} \boldsymbol{\mu}_k' \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu}_k + \ln \pi_k. Classify to k with highest δk\delta_k.
Quadratic discriminant analysis (QDA)
Each class has its OWN covariance matrix Σk\boldsymbol{\Sigma}_k. Decision boundary is quadratic.
More flexible than LDA but estimates more parameters; preferred when sample size is large and class covariances clearly differ. More variance, less bias.
k-Nearest Neighbors (KNN)
Classify a new point by majority vote among its k nearest training points (using Euclidean or another distance metric).
Low k: low bias, high variance (overfits).
High k: high bias, low variance.
No model fit. Sensitive to feature scaling (always standardize). Curse of dimensionality in high p.
Decision tree splitting criteria
Regression: minimize ∑R(yi−yˉR)2\sum_R (y_i - \bar{y}_R)^2 within each region.
Classification: Gini =∑kpk(1−pk)= \sum_k p_k(1 - p_k); Entropy =−∑kpkln⁡pk= -\sum_k p_k \ln p_k; Classification error =1−max⁡kpk= 1 - \max_k p_k.
Gini and entropy are smooth; classification error is less sensitive. Trees built top-down by greedy recursive splitting.
Bagging (bootstrap aggregating)
Build B trees on bootstrap samples; average predictions (regression) or majority vote (classification).
Reduces variance versus a single tree. Each tree uses all p predictors. Out-of-bag (OOB) prediction = average of predictions from trees not built on a given observation; approximates CV error free of charge.
Random forest
Bagging plus random subset of predictors considered at each split (typical m = p\sqrt{p} for classification, m = p/3 for regression). Decorrelates trees, further reducing variance. OOB error is the standard out-of-sample estimate. Variable importance via mean decrease in Gini or permutation importance.
Boosting (AdaBoost and gradient boosting)
AdaBoost: sequentially fit weak learners (often stumps) to reweighted data (misclassified points up-weighted). Final classifier = weighted vote.
Gradient boosting: fit each new tree to the residuals (or negative gradient of loss) of the current ensemble.
Confusion matrix and classification metrics
TP / FP / FN / TN cells.
Accuracy = (TP + TN) / N
Sensitivity (recall, TPR) = TP / (TP + FN)
Specificity (TNR) = TN / (TN + FP)
Precision (PPV) = TP / (TP + FP)
F1 = harmonic mean of precision and recall = 2⋅P⋅RP+R\dfrac{2 \cdot P \cdot R}{P + R}.
ROC curve and AUC
ROC plots sensitivity (TPR) vs 1 −- specificity (FPR) across classification thresholds. Diagonal = random classifier.
AUC = area under ROC = probability that the model ranks a random positive higher than a random negative. AUC = 1 perfect; AUC = 0.5 random; AUC > 0.8 generally good.
Principal Components Analysis (PCA)
Find orthogonal directions of maximum variance in centered data X\mathbf{X}. Loadings = eigenvectors of X′X/(n−1)\mathbf{X}'\mathbf{X}/(n-1). Variance explained by PC i = eigenvalue λi/∑jλj\lambda_i / \sum_j \lambda_j. Scores: project data onto loadings.
K-means clustering
Partition n points into K clusters minimizing within-cluster sum of squares ∑k=1K∑i∈Ck∥xi−μk∥2\sum_{k=1}^K \sum_{i \in C_k} \|\mathbf{x}_i - \boldsymbol{\mu}_k\|^2. Algorithm: assign each point to nearest centroid; update centroids; repeat until convergence. Local minima possible (run with multiple random starts).
Hierarchical clustering linkage methods
Single linkage: distance between closest points across clusters (chains).
Complete linkage: distance between farthest points (compact, similar-size clusters).
Average linkage: mean pairwise distance.
Ward: minimize the increase in within-cluster variance from merging.
Silhouette score
For each point: si=bi−aimax⁡(ai,bi)s_i = \dfrac{b_i - a_i}{\max(a_i, b_i)} where aia_i = mean distance to its own cluster, bib_i = mean distance to nearest other cluster. Range −1-1 to +1+1. Closer to 1 = well-clustered; close to 0 = boundary point; negative = likely misclustered.
Naive Bayes classifier
Assumes predictors are independent given the class: P(Y=k∣x)∝P(Y=k)∏jP(xj∣Y=k)P(Y = k | \mathbf{x}) \propto P(Y = k) \prod_j P(x_j | Y = k). Despite the unrealistic independence assumption, performs well for text classification and other high-dimensional discrete problems.
Generalized linear models (GLM) actuarial workhorse
Three components: random component (exponential family distribution), systematic component (η=x′β\eta = \mathbf{x}' \boldsymbol{\beta}), and link function g such that g(μ)=ηg(\mu) = \eta. Common: Poisson with log link for claim counts; Gamma with log link for severity; Tweedie for pure premium.
GLM deviance and goodness-of-fit
Deviance D=−2(ℓfitted−ℓsaturated)D = -2 (\ell_{\text{fitted}} - \ell_{\text{saturated}}). For Gaussian, D = residual sum of squares. For Poisson, D=2∑[yiln⁡(yi/μ^i)−(yi−μ^i)]D = 2 \sum [y_i \ln(y_i / \hat{\mu}_i) - (y_i - \hat{\mu}_i)]. Asymptotically χn−p2\chi^2_{n - p}. Likelihood ratio test: ΔD∼χΔp2\Delta D \sim \chi^2_{\Delta p} for nested model comparison.
Splines and polynomial regression
Polynomial: include x,x2,…,xdx, x^2, \ldots, x^d as predictors. Global; sensitive to boundary points.
Regression splines: piecewise polynomials joined at knots with continuity constraints. Cubic splines (degree 3) are common.
Natural splines: cubic in interior, linear past boundary knots (less variance at edges).
Generalized additive models (GAM)
g(μ)=β0+∑jfj(xj)g(\mu) = \beta_0 + \sum_j f_j(x_j) where each fjf_j is a smooth function (spline) of one predictor. Captures nonlinearity without interactions. Interpretable: plot each fjf_j to see its effect. Fit by backfitting or penalized likelihood. Used in actuarial GLM extensions.
Gini index from Lorenz curve
G=2∫01(x−L(x)) dxG = 2\int_0^1 (x - L(x))\,dx — G = Gini index, x = cumulative exposure share, L(x) = cumulative actual loss share at x
Root mean squared error (RMSE)
RMSE=1n∑(yi−y^i)2\text{RMSE} = \sqrt{\tfrac{1}{n}\sum (y_i - \hat{y}_i)^2} — n = hold-out size, y_i = actual, ŷ_i = predicted for policy i
Double lift chart sort ratio
Ri=y^iB/y^iAR_i = \hat{y}^B_i / \hat{y}^A_i — R = sort key, ŷ^B = challenger prediction for policy i, ŷ^A = incumbent prediction for policy i
GLM deviance against saturated model
D=2∑(ℓ(yi;yi)−ℓ(yi;y^i))D = 2\sum (\ell(y_i; y_i) - \ell(y_i; \hat{y}_i)) — D = deviance, ℓ(y;y) = saturated log-likelihood, ℓ(y;ŷ) = fitted log-likelihood
Gradient boosting pseudo-residual
rim=−∂L(yi,f(xi))∂f(xi)∣f=Fm−1r_{im} = -\left.\frac{\partial L(y_i, f(x_i))}{\partial f(x_i)}\right|_{f=F_{m-1}} — L = loss, y_i = target, F_{m-1} = current ensemble, r_{im} = stage-m pseudo-residual
Variance of bagged ensemble prediction
Var(fˉ)=ρσ2+1−ρBσ2\text{Var}(\bar{f}) = \rho\sigma^{2} + \frac{1-\rho}{B}\sigma^{2} — ρ = pairwise tree correlation, σ² = individual tree variance, B = number of trees
AdaBoost stage weight alpha
αm=log⁡ ⁣((1−errm)/errm)\alpha_m = \log\!\big((1-\text{err}_m)/\text{err}_m\big) — err_m = weighted classification error at stage m, α_m = log-odds weight assigned to stump m
Bagged regression ensemble prediction
f^bag(x)=1B∑b=1Bf^∗b(x)\hat{f}_{\text{bag}}(x) = \frac{1}{B}\sum_{b=1}^{B}\hat{f}^{*b}(x) — B = number of bootstrap trees, f^∗b\hat{f}^{*b} = b-th tree fit on bootstrap sample, x = input vector
Gini coefficient from AUROC (binary classifier)
Gini=2⋅AUROC−1\text{Gini} = 2 \cdot \text{AUROC} - 1 — AUROC = area under ROC curve; Gini ranges from 0 (random) to 1 (perfect).
AUROC as Mann-Whitney concordant-pair estimator
AUROC^=#{p^i>p^j}+0.5 #{p^i=p^j}n1n0\widehat{\text{AUROC}} = \dfrac{\#\{\hat{p}_i > \hat{p}_j\} + 0.5\,\#\{\hat{p}_i = \hat{p}_j\}}{n_1 n_0} — i indexes positives, j negatives; n_1, n_0 counts; ties get 0.5 credit.
Lorenz-based Gini coefficient
Gini=2∫01(x−L(x)) dx\text{Gini} = 2 \int_{0}^{1} (x - L(x))\,dx — x = cumulative exposure share, L(x) = cumulative loss share at depth x.
Lift in a scored bin
Liftb=positivesb/nbpositivestotal/N\text{Lift}_b = \dfrac{\text{positives}_b / n_b}{\text{positives}_{\text{total}} / N} — n_b = records in bin b, N = total records; positives counted in numerator bin and denominator overall.
KNN regression prediction (unweighted mean)
f^(x0)=1K∑i∈NK(x0)yi\hat{f}(x_0) = \frac{1}{K}\sum_{i \in \mathcal{N}_K(x_0)} y_i — K = neighbor count, NK(x0)\mathcal{N}_K(x_0) = K nearest training points to query x0x_0, yiy_i = neighbor response
Inverse-distance weighted KNN regression prediction
f^(x0)=∑i∈NKwiyi∑i∈NKwi,  wi=1d(x0,xi)\hat{f}(x_0) = \frac{\sum_{i \in \mathcal{N}_K} w_i y_i}{\sum_{i \in \mathcal{N}_K} w_i}, \; w_i = \frac{1}{d(x_0, x_i)} — wiw_i = neighbor weight, d(x0,xi)d(x_0,x_i) = distance from query to neighbor i, yiy_i = response
Euclidean distance between feature vectors
d(x,y)=∑j(xj−yj)2d(x,y) = \sqrt{\sum_j (x_j - y_j)^2} — xj,yjx_j, y_j = j-th standardized feature of points x and y, sum runs over all p features
KNN classifier posterior probability estimate
P^(Y=j∣X=x0)=1K∑i∈NK(x0)1{yi=j}\hat{P}(Y=j \mid X=x_0) = \frac{1}{K}\sum_{i \in \mathcal{N}_K(x_0)} \mathbf{1}\{y_i=j\} — K = neighbor count, NK(x0)\mathcal{N}_K(x_0) = K nearest training points to query x0x_0, yiy_i = class label
Proportion of variance explained by a principal component
PVEm=λm/∑j=1pλj\mathrm{PVE}_m = \lambda_m / \sum_{j=1}^{p}\lambda_j — λm\lambda_m = mth eigenvalue of sample covariance (or correlation) matrix, p = number of variables
First loading vector as variance maximizer
ϕ1=arg⁡max⁡∥ϕ∥=11n∑i=1n(ϕ⊤x~i)2\phi_1 = \arg\max_{\|\phi\|=1}\frac{1}{n}\sum_{i=1}^{n}(\phi^\top \tilde{x}_i)^2 — ϕ\phi = unit vector, x~i\tilde{x}_i = centered observation, n = sample size
Principal component score
zim=ϕm⊤x~i=∑j=1pϕjmx~ijz_{im} = \phi_m^\top \tilde{x}_i = \sum_{j=1}^{p}\phi_{jm}\tilde{x}_{ij} — ϕm\phi_m = mth unit loading vector, x~i\tilde{x}_i = centered observation, p = number of variables
Total variance identity in PCA
∑m=1pλm=tr(S)=∑j=1psj2\sum_{m=1}^{p}\lambda_m = \mathrm{tr}(S) = \sum_{j=1}^{p}s_j^{2} — λm\lambda_m = mth eigenvalue, S = sample covariance matrix, sj2s_j^2 = sample variance of variable j
Proportion of variance explained by PC j
PVEj=λj/∑k=1pλk\text{PVE}_j = \lambda_j / \sum_{k=1}^{p} \lambda_k — λj\lambda_j = eigenvalue of PC j, p = number of variables
K-means total within-cluster sum of squares objective
W=∑k=1K∑i∈Ck∥xi−xˉk∥2W = \sum_{k=1}^{K}\sum_{i \in C_k} \lVert x_i - \bar{x}_k \rVert^{2} — K = number of clusters, C_k = cluster k, x_i = observation, \bar{x}_k = cluster k centroid
Centroid linkage distance between two clusters
d(A,B)=∥xˉA−xˉB∥d(A,B) = \lVert \bar{x}_A - \bar{x}_B \rVert — A, B = clusters, \bar{x}_A, \bar{x}_B = centroids of A and B, ∥⋅∥\lVert \cdot \rVert = Euclidean norm
Eigenvalue from prcomp sdev output
λj=sdevj2\lambda_j = \text{sdev}_j^2 — λj\lambda_j = variance captured by PC j, sdevj\text{sdev}_j = standard deviation of PC j from prcomp
Single linkage distance between two clusters
d(A,B)=min⁡i∈A, j∈Bd(xi,xj)d(A,B) = \min_{i \in A,\, j \in B} d(x_i, x_j) — A, B = clusters, x_i, x_j = points in A and B, d = pairwise distance metric
Unit-length constraint on PCA loadings
∑i=1pϕij2=1\sum_{i=1}^{p} \phi_{ij}^{2} = 1 — ϕij\phi_{ij} = loading of variable i on PC j, p = number of variables
PC score for a standardized observation
zij=∑k=1pϕkj(xik−xˉk)/skz_{ij} = \sum_{k=1}^{p} \phi_{kj} (x_{ik} - \bar{x}_k)/s_k — ϕkj\phi_{kj} = loading, xikx_{ik} = raw value, xˉk\bar{x}_k = mean, sks_k = SD
Complete linkage distance between two clusters
d(A,B)=max⁡i∈A, j∈Bd(xi,xj)d(A,B) = \max_{i \in A,\, j \in B} d(x_i, x_j) — A, B = clusters, x_i, x_j = points in A and B, d = pairwise distance metric
Sigmoid activation function
g(z)=11+e−zg(z) = \frac{1}{1 + e^{-z}} — z = pre-activation, g(z) = predicted probability of class 1, range (0,1)
Permutation importance of feature j
Imp(j)=L(y,f^(x(πj)))−L(y,f^(x))\text{Imp}(j) = L(y, \hat{f}(x^{(\pi_j)})) - L(y, \hat{f}(x)) — L = validation loss, π_j = permutation of column j, larger gap = more important variable
Partial dependence of prediction on variable j
PDj(xj)=1n∑i=1nf^(xj,x−j(i))\text{PD}_j(x_j) = \frac{1}{n}\sum_{i=1}^{n} \hat{f}(x_j, x_{-j}^{(i)}) — n = observations, x_{-j}^{(i)} = other features at observed values, average marginal effect
Feedforward neural network forward pass
y^=gL(WLgL−1(⋯g1(W1x+b1)⋯ )+bL)\hat{y} = g_L(W_L g_{L-1}(\cdots g_1(W_1 x + b_1)\cdots) + b_L) — x = input, W_k = weight matrix, b_k = bias, g_k = activation at layer k, L = number of layers
Regression tree residual sum of squares
RSS(T)=∑m=1M∑i∈Rm(yi−y^Rm)2\text{RSS}(T) = \sum_{m=1}^{M}\sum_{i\in R_m}(y_i-\hat{y}_{R_m})^2 — RmR_m = terminal region m, y^Rm\hat{y}_{R_m} = regional mean, M = number of leaves
Regression tree training risk
R(T)=∑leaves t∑i∈t(yi−yˉt)2R(T) = \sum_{\text{leaves } t}\sum_{i \in t}(y_i - \bar{y}_t)^2 — y_i = response in leaf t, ȳ_t = mean response in leaf t
Cost-complexity pruning criterion
Cα(T)=Loss(T)+α∣T∣C_\alpha(T) = \text{Loss}(T) + \alpha|T| — Loss(T) = tree RSS or weighted impurity, ∣T∣|T| = number of terminal nodes, α\alpha = complexity parameter from CV
Gini index node impurity
Gm=∑k=1Kp^mk(1−p^mk)=1−∑kp^mk2G_m = \sum_{k=1}^{K}\hat{p}_{mk}(1-\hat{p}_{mk}) = 1 - \sum_{k}\hat{p}_{mk}^{2} — p^mk\hat{p}_{mk} = proportion of class k at node m, K = number of classes
One-standard-error rule threshold
CV(α)≤CVmin⁡+SE\text{CV}(\alpha) \le \text{CV}_{\min} + \text{SE} — CV(α) = K-fold CV error at penalty α, CV_min = minimum CV error, SE = standard error across folds; pick largest qualifying α
Weakest-link strength at internal node
g(t)=R(t)−R(Tt)∣Tt∣−1g(t) = \frac{R(t) - R(T_t)}{|T_t| - 1} — R(t) = risk if t is a single leaf, R(T_t) = risk of subtree rooted at t, |T_t| = leaves in that subtree
Cross-entropy node impurity
Dm=−∑k=1Kp^mklog⁡p^mkD_m = -\sum_{k=1}^{K}\hat{p}_{mk}\log\hat{p}_{mk} — p^mk\hat{p}_{mk} = class k proportion at node m, K = classes, 0log⁡0=00\log 0 = 0 by convention
Cost-complexity criterion for a subtree
Rα(T)=R(T)+α∣T∣R_\alpha(T) = R(T) + \alpha|T| — R(T) = training risk (RSS or misclassification), α = complexity penalty per leaf, |T| = number of terminal nodes
Complete linkage inter-cluster distance
d(A,B)=max⁡a∈A, b∈Bd(a,b)d(A,B) = \max_{a\in A,\,b\in B} d(a,b) — A, B = clusters, d(a,b) = pairwise distance between points a in A and b in B
Average linkage inter-cluster distance
d(A,B)=1∣A∣∣B∣∑a∈A∑b∈Bd(a,b)d(A,B) = \frac{1}{|A||B|}\sum_{a\in A}\sum_{b\in B} d(a,b) — |A|, |B| = cluster sizes, d(a,b) = pairwise distance between a and b
K-means centroid update
xˉk=1∣Ck∣∑i∈Ckxi\bar{x}_k = \frac{1}{|C_k|}\sum_{i\in C_k} x_i — |C_k| = number of points in cluster k, x_i = data point assigned to cluster k, xˉk\bar{x}_k = updated centroid
Within-cluster sum of squares (WCSS) for K-means
WCSS=∑k=1K∑i∈Ck∥xi−xˉk∥2\text{WCSS} = \sum_{k=1}^{K}\sum_{i\in C_k}\|x_i - \bar{x}_k\|^{2} — K = number of clusters, C_k = cluster k, x_i = point i, xˉk\bar{x}_k = centroid of cluster k
Time Series with Constant Variance 28 items
Stationarity (weak and strict)
Weak (covariance) stationarity: mean and variance constant over time; autocovariance γ(t,t+h)\gamma(t, t + h) depends only on lag h (not on t).
Strict stationarity: full joint distribution invariant under time shifts. Strict implies weak (with finite second moments) but not vice versa.
Autocovariance and autocorrelation functions
γh=Cov(Xt,Xt+h)\gamma_h = \text{Cov}(X_t, X_{t + h})
ρh=γhγ0\rho_h = \dfrac{\gamma_h}{\gamma_0} (autocorrelation function ACF).
ρ0=1\rho_0 = 1. For weakly stationary processes, ρh\rho_h depends only on lag h. Sample ACF: ρ^h=∑t=1n−h(Xt−Xˉ)(Xt+h−Xˉ)∑t=1n(Xt−Xˉ)2\hat\rho_h = \dfrac{\sum_{t=1}^{n-h}(X_t - \bar{X})(X_{t+h} - \bar{X})}{\sum_{t=1}^n (X_t - \bar{X})^2}.
Partial autocorrelation function (PACF)
ϕhh\phi_{hh} = correlation between XtX_t and Xt−hX_{t-h} after removing the linear effects of intermediate lags Xt−1,…,Xt−h+1X_{t-1}, \ldots, X_{t-h+1}. For AR(p) process, PACF cuts off after lag p. For MA(q), PACF tails off. Used with ACF for Box-Jenkins model identification.
White noise and random walk
White noise: εt\varepsilon_t iid mean 0 variance σ2\sigma^2. ρh=0\rho_h = 0 for h≠0h \neq 0. Stationary.
Random walk: Xt=Xt−1+εtX_t = X_{t-1} + \varepsilon_t. Non-stationary; variance grows linearly with t. Differencing once produces white noise. Conditional forecast: E[Xt+h∣Ft]=XtE[X_{t+h} | \mathcal{F}_t] = X_t.
Autoregressive process AR(p)
Xt=c+ϕ1Xt−1+ϕ2Xt−2+⋯+ϕpXt−p+εtX_t = c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_p X_{t-p} + \varepsilon_t
Stationary if all roots of 1−ϕ1z−⋯−ϕpzp=01 - \phi_1 z - \cdots - \phi_p z^p = 0 lie OUTSIDE the unit circle. AR(1): ρh=ϕh\rho_h = \phi^h, so ACF decays geometrically. PACF cuts off at lag p.
Moving average process MA(q)
Xt=μ+εt+θ1εt−1+⋯+θqεt−qX_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q}
Always stationary (finite linear combination of white noise). Invertible if roots of 1+θ1z+⋯+θqzq=01 + \theta_1 z + \cdots + \theta_q z^q = 0 lie OUTSIDE the unit circle. ACF cuts off at lag q; PACF tails off.
ARMA(p, q) and ARIMA(p, d, q)
ARMA: combines AR(p) and MA(q) for stationary series. ACF and PACF both tail off.
ARIMA: ARMA on the d-th difference ΔdXt\Delta^d X_t. Used when the original series is non-stationary but differencing produces stationarity. d = 1 for random walks; d = 2 for series with linear trends in slope.
Yule-Walker equations (AR estimation)
For AR(p): ρh=ϕ1ρh−1+ϕ2ρh−2+⋯+ϕpρh−p\rho_h = \phi_1 \rho_{h-1} + \phi_2 \rho_{h-2} + \cdots + \phi_p \rho_{h-p} for h≥1h \geq 1.
Matrix form: ρ=Rϕ\boldsymbol{\rho} = \mathbf{R} \boldsymbol{\phi}, solve ϕ^=R−1ρ\hat{\boldsymbol{\phi}} = \mathbf{R}^{-1} \boldsymbol{\rho} using sample autocorrelations. Method of moments estimator.
Box-Jenkins methodology
1. Identification: plot the series, inspect ACF and PACF, difference if non-stationary, choose tentative (p, d, q).
2. Estimation: fit ARIMA by MLE or least squares.
3. Diagnostic checking: residuals should look like white noise (Ljung-Box test; ACF of residuals).
4.
Ljung-Box test for residual autocorrelation
Q=n(n+2)∑k=1mρ^k2n−kQ = n(n + 2) \sum_{k=1}^m \dfrac{\hat\rho_k^2}{n - k}
Under H0H_0 of white-noise residuals, Q∼χm−p−q2Q \sim \chi^2_{m - p - q} where p + q is the number of fitted ARMA parameters. Reject if Q exceeds critical value; indicates residual autocorrelation and model misspecification.
ARIMA forecast and forecast variance
One-step forecast: X^n+1∣n\hat{X}_{n+1 | n} is the conditional expectation given history. For AR(1) without drift: X^n+h∣n=ϕhXn\hat{X}_{n+h | n} = \phi^h X_n.
Forecast error variance grows with horizon: for AR(1), Var(eh)=σ2(1+ϕ2+⋯+ϕ2(h−1))\text{Var}(e_h) = \sigma^2(1 + \phi^2 + \cdots + \phi^{2(h-1)}).
Seasonal ARIMA (SARIMA)
ARIMA(p,d,q)(P,D,Q)s\text{ARIMA}(p, d, q)(P, D, Q)_s. Adds seasonal AR, differencing, and MA terms at lag s (e.g., 12 for monthly).
Example: SARIMA(0, 1, 1)(0, 1, 1)12_{12} is the 'airline model' often used as a baseline for monthly series with both trend and seasonality.
AR(1) bounded h-step forecast variance
Var(Y^t+h∣t)=σε2(1−ϕ12h)/(1−ϕ12)\text{Var}(\hat{Y}_{t+h\mid t}) = \sigma_\varepsilon^2 (1 - \phi_1^{2h}) / (1 - \phi_1^2) — σε2\sigma_\varepsilon^2 = shock variance, ϕ1\phi_1 = AR coefficient, h = horizon
Long-run mean of stationary AR(p) process
μ=c/(1−ϕ1−⋯−ϕp)\mu = c / (1 - \phi_1 - \dots - \phi_p) — c = printed intercept, ϕi\phi_i = AR coefficients, μ\mu = unconditional mean (not the intercept)
t-ratio for ARIMA coefficient significance
ti=ϕ^i/SE(ϕ^i)t_i = \hat{\phi}_i / \text{SE}(\hat{\phi}_i) — ϕ^i\hat{\phi}_i = estimated coefficient, SE = standard error; significant when ∣ti∣>1.96|t_i|>1.96
AR(1) h-step ahead mean-corrected forecast
Y^t+h∣t=μ+ϕ1h(Yt−μ)\hat{Y}_{t+h\mid t} = \mu + \phi_1^{h}(Y_t - \mu) — μ\mu = long-run mean, ϕ1\phi_1 = AR(1) coefficient, YtY_t = last observation, h = horizon
Random walk with drift
Yt=δ+Yt−1+εtY_t = \delta + Y_{t-1} + \varepsilon_t — δ = constant drift per period, Y_{t-1} = prior level, ε_t = iid shock with mean 0 and variance σ²
Random walk h-step forecast variance
Var(Y^n+h)=hσ2\text{Var}(\hat{Y}_{n+h}) = h\sigma^2 — h = forecast horizon, σ² = per-period shock variance; SE grows as σh\sigma\sqrt{h}
Deterministic linear trend regression
Yt=β0+β1t+εtY_t = \beta_0 + \beta_1 t + \varepsilon_t — Y_t = series at time t, t = time index, β_1 = fixed per-period drift, ε_t = iid noise with mean 0 and variance σ²
Seasonal regression with dummy variables
Yt=β0+β1t+∑j=1s−1γjDj,t+εtY_t = \beta_0 + \beta_1 t + \sum_{j=1}^{s-1}\gamma_j D_{j,t} + \varepsilon_t — s = period, D_{j,t} = season-j indicator, γ_j = additive seasonal effect vs baseline
Invertibility condition for an MA(q) process
All roots of θ(z)=1+θ1z+⋯+θqzq=0\theta(z) = 1 + \theta_1 z + \dots + \theta_q z^q = 0 satisfy ∣z∣>1|z| > 1 — θ_j = MA coefficients, q = MA order, z = complex root
Stationarity condition for an AR(p) process
All roots of ϕ(z)=1−ϕ1z−⋯−ϕpzp=0\phi(z) = 1 - \phi_1 z - \dots - \phi_p z^p = 0 satisfy ∣z∣>1|z| > 1 — φ_i = AR coefficients, p = AR order, z = complex root
White-noise confidence bands for sample ACF
±1.96/n\pm 1.96/\sqrt{n} — n = sample size; sample autocorrelations inside the band are not significantly different from zero at the 5% level
Airline model for monthly time series
(1−B)(1−B12)Yt=(1+θ1B)(1+Θ1B12)εt(1-B)(1-B^{12})Y_t = (1+\theta_1 B)(1+\Theta_1 B^{12})\varepsilon_t — B = backshift, θ_1 = regular MA, Θ_1 = seasonal MA, ε_t = white noise
AR(1) h-step-ahead forecast
Y^n+h=μ+ϕ1h(Yn−μ)\hat{Y}_{n+h} = \mu + \phi_1^h (Y_n - \mu) — μ = long-run mean, φ₁ = AR(1) coefficient, Y_n = most recent observation, h = horizon
AR(1) long-run mean
μ=c1−ϕ1\mu = \dfrac{c}{1 - \phi_1} — c = intercept, φ₁ = AR(1) coefficient with |φ₁| < 1
MA(1) lag-1 autocorrelation
ρ1=θ11+θ12\rho_1 = \dfrac{\theta_1}{1 + \theta_1^2} — θ₁ = MA(1) coefficient with |θ₁| < 1 for invertibility; ρ_k = 0 for k ≥ 2
AR(1) unconditional variance
γ0=σ21−ϕ12\gamma_0 = \dfrac{\sigma^2}{1 - \phi_1^2} — σ² = innovation variance, φ₁ = AR(1) coefficient with |φ₁| < 1

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