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All MAS-I Formulas

Probability Models 30 items

Gamma waiting time density for the n-th arrival

f_{S_{n}} (s) = \frac{λ ^{n} s ^{n - 1} e ^{- λ s}}{( n - 1 )!}

— λ = rate, n = event index, s = waiting time, mean n/λ, variance n/λ²

Compound Poisson mean and variance

E [S (t)] = λ t E [X], Var (S (t)) = λ t E [X^{2}]

— λ = rate, t = time, X = iid severity, E[X²] = Var(X)+(E[X])²

NHPP mean function over an interval

Λ (a, b) = \int_{a}^{b} λ (s) d s

— λ(s) = time-varying intensity, [a,b] = interval; equals both mean and variance of N(b)−N(a)

Monte Carlo sample size for target half-width

n \approx (1.96 s / h)^{2}

— s = pilot sample SD, h = target half-width, n = required number of draws

Exponential inversion from a uniform

X = - \frac{1}{λ} ln (1 - U)

— U = Uniform(0,1) draw, λ = exponential rate, X = exponential severity draw

Monte Carlo 95% confidence interval half-width

θ^_{n} \pm 1.96 s / n

— \hat\theta_n = sample mean of g(X_i), s = sample SD, n = independent draws

Inversion method draw from a uniform

X = F^{- 1} (U)

— U = Uniform(0,1) draw, F^{-1} = generalized inverse CDF, X = draw from target distribution F

Reversionary annuity to y after x dies

a ˉ_{y ∣ x} = a ˉ_{y} - a ˉ_{x y}

—

a ˉ_{y}

= single-life annuity on y,

a ˉ_{x y}

= joint-life annuity

Joint-life survival under common shock

_{t} p_{x y} =_{t} p_{x}^{*} \cdot_{t} p_{y}^{*} \cdot e^{- λ t}

—

_{t} p^{*}

= private survival,

λ

= shared hazard rate, t = time

Survival function from cumulative hazard

S (t) = exp (- \int_{0}^{t} h (s) d s) = e^{- H (t)}

— H(t) = cumulative hazard, h(s) = hazard rate, S(t) = survival probability

Last-survivor survival probability (inclusion-exclusion)

_{t} p_{x y} =_{t} p_{x} +_{t} p_{y} -_{t} p_{x y}

—

_{t} p_{x}

= prob x survives t,

_{t} p_{x y}

= joint survival

Mean residual life at age t

e (t) = \frac{\int _{t}^{\infty} S ( u ) d u}{S ( t )}

— S = survival function, t = current age, e(t) = expected remaining lifetime given survival to t

Joint-life continuous annuity under constant force and interest

a ˉ_{x y} = \frac{1}{μ _{x} + μ _{y} + δ}

—

μ_{x}, μ_{y}

= constant forces of mortality,

δ

= force of interest

Conditional survival probability (t-p-s)

_{t} p_{s} = \frac{S ( s + t )}{S ( s )}

— S = survival function, s = current age, t = additional years survived

Hazard rate from density and survival

h (t) = \frac{f ( t )}{S ( t )} = - \frac{d}{d t} ln S (t)

— f = density, S = survival function, h = instantaneous failure rate

Constant-force whole life insurance EPV

A ˉ_{x} = \frac{μ}{μ + δ}

— μ = constant force of mortality, δ = constant force of interest

Whole life insurance EPV (discrete)

A_{x} = \sum_{k = 0}^{\infty} v^{k + 1}_{k ∣} q_{x}

— v = 1/(1+i),

_{k ∣} q_{x}

= prob of death in year k+1 for life age x

Insurance-annuity fundamental identity (discrete)

A_{x} = 1 - d a ¨_{x}

— d = i/(1+i) effective discount rate,

a ¨_{x}

= whole life annuity-due EPV,

A_{x}

= whole life insurance EPV

Whole life annuity-due EPV

a ¨_{x} = \sum_{k = 0}^{\infty} v^{k}_{k} p_{x}

— v = 1/(1+i),

_{k} p_{x}

= prob life age x survives k years

Waiting time to the n-th event in a Poisson process

S_{n} = \sum_{i = 1}^{n} T_{i} \sim Gamma (n, λ)

E [S_{n}] = n / λ

— T_i = iid Exponential(λ) gaps, n = event number, λ = rate

Non-homogeneous Poisson process mean function

N (b) - N (a) \sim Poisson (\int_{a}^{b} λ (s) d s)

— λ(s) = intensity function, [a,b] = time interval

Poisson process count probability

P (N (t) = k) = \frac{e ^{- λ t} ( λ t ) ^{k}}{k !}

— λ = rate, t = interval length, k = number of events

Fundamental matrix of an absorbing Markov chain

N = (I - Q)^{- 1}

— I = identity, Q = transient-to-transient block of P, N_{ij} = expected visits to transient state j starting from i

Series system reliability with independent components

R_{s} = \prod_{i = 1}^{n} R_{i}

— R_i = reliability of component i, n = number of components in series

Bridge reliability by conditioning on the center element

R_{bridge} = R_{C} \cdot R_{works} + (1 - R_{C}) \cdot R_{fails}

— R_C = bridge element reliability, R_works = system reliability given C works, R_fails = given C fails

Parallel system reliability with independent components

R_{p} = 1 - \prod_{i = 1}^{n} (1 - R_{i})

— R_i = reliability of component i, 1 - R_i = unreliability, n = number of parallel components

Limited expected value (survival form)

E [X \land u] = \int_{0}^{u} S (x) d x

— X = non-negative loss, u = cap/limit, S(x) = survival function 1−F(x)

Loss elimination ratio for ordinary deductible

LER (d) = E [X \land d] / E [X]

— d = ordinary deductible, X = ground-up loss severity

Expected layer cost between deductible d and limit u

E [min (X, u) - min (X, d)] = E [X \land u] - E [X \land d]

— X = loss, d = attachment, u = exhaustion point

Exponential limited expected value

E [X \land u] = θ (1 - e^{- u / θ})

— θ = exponential mean, u = policy limit/cap

Statistics 35 items

Collective risk model aggregate loss

S = \sum_{i = 1}^{N} X_{i}

— N = random claim count, X_i = iid severities independent of N

Compound distribution variance

Var (S) = E [N] Var (X) + Var (N) E [X]^{2}

— N = claim count, X = severity, S = aggregate loss

Panjer recursion for aggregate loss pmf

f_{S} (s h) = \frac{1}{1 - a f _{X} ( 0 )} \sum_{y = 1}^{s} (a + b y / s) f_{X} (y h) f_{S} ((s - y) h)

— (a,b) = (a,b,0) class parameters, h = grid step

Normal approximation stop-loss premium

E [(S - d)_{+}] = σ [ϕ (z) - z (1 - Φ (z))]

— z = (d - E[S])/σ, σ = SD of S, φ = standard normal pdf, Φ = cdf

MLE of the exponential rate parameter

λ^= n / \sum X_{i} = 1/ X ˉ

— n = sample size, ΣX_i = sufficient statistic, X̄ = sample mean

Exponential family canonical density

f (x; θ) = h (x) c (θ) exp (\sum_{j = 1}^{k} w_{j} (θ) t_{j} (x))

— h, c base functions; w natural params; t sufficient kernels

Fisher-Neyman factorization theorem

f (x_{1}, \dots, x_{n}; θ) = g (T (x), θ) \cdot h (x)

— T = sufficient statistic, g depends on θ through T, h depends only on data

UMVUE of theta for Uniform(0, theta)

θ^_{UMVUE} = \frac{n + 1}{n} max X_{i}

— n = sample size, max X_i = largest order statistic (complete sufficient stat)

One-sample z test statistic for a mean

Z = \frac{X ˉ - μ _{0}}{σ / n}

— X-bar = sample mean, μ₀ = hypothesized mean, σ = population SD, n = sample size

Two-sided p-value for a z test

p = 2 \cdot P (∣ Z ∣ \geq ∣ z_{obs} ∣)

— z_obs = observed test statistic, probability computed under H₀

Likelihood ratio test statistic

- 2 ln (L_{0} / L_{1}) \sim χ_{k}^{2}

— L₀ = restricted-model likelihood, L₁ = full-model likelihood, k = number of restrictions

Sample size for target power in a one-sided z test

n = \frac{σ ^{2} ( z _{1 - α} + z _{1 - β} ) ^{2}}{( μ _{0} - μ _{1} ) ^{2}}

— σ = SD, α = Type I rate, β = Type II rate, μ₀ = null mean, μ₁ = alternative mean

Nelson-Aalen cumulative hazard estimator

H^(t) = \sum_{t_{j} \leq t} s_{j} / n_{j}

— sⱼ = events at time tⱼ, nⱼ = risk-set size (counts only those with dᵢ < tⱼ ≤ yᵢ)

Conditional density under left-truncation at a deductible

f_{X ∣ X > d} (x) = f (x) / S (d)

for

x > d

— f = unconditional density, S(d) = survival at deductible d

Likelihood for a right-censored sample

L (θ) = \prod_{i = 1}^{n} f (y_{i})^{δ_{i}} S (y_{i})^{1 - δ_{i}}

— f = density, S = survival, δᵢ = 1 if uncensored, 0 if right-censored at yᵢ

Likelihood contribution with left truncation and right censoring

L_{i} (θ) = [f (x_{i} ∣ θ) / S (d_{i} ∣ θ)]^{δ_{i}} [S (u_{i} ∣ θ) / S (d_{i} ∣ θ)]^{1 - δ_{i}}

— δ=1 if observed, d=truncation, u=censoring point

Fisher information for a single observation

I (θ) = E [(\partial lo g f / \partial θ)^{2}] = - E [\partial^{2} lo g f / \partial θ^{2}]

— f = density, θ = parameter

Cramer-Rao lower bound for an unbiased estimator

Var (θ^) \geq 1/ [n I (θ)]

— n = sample size, I(θ) = Fisher information per observation, θ = parameter

Mean squared error decomposition

MSE (θ^) = Var (θ^) + [Bias (θ^)]^{2}

— Bias(θ̂) = E[θ̂] − θ, Var = sampling variance of the estimator

CDF of the sample minimum

F_{(1)} (x) = 1 - [1 - F (x)]^{n}

— F = parent CDF, n = sample size, X_{(1)} = minimum

Unbiased sample variance with Bessel's correction

S^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - X ˉ)^{2}

— n = sample size, X_i = i-th observation,

X ˉ

= sample mean, S² unbiased for σ²

Computational shortcut for sum of squared deviations

\sum_{i = 1}^{n} (X_{i} - X ˉ)^{2} = \sum_{i = 1}^{n} X_{i}^{2} - n X ˉ^{2}

— n = sample size, X_i = i-th observation,

X ˉ

= sample mean

Density of the k-th order statistic

f_{(k)} (x) = \frac{n !}{( k - 1 )! ( n - k )!} F (x)^{k - 1} [1 - F (x)]^{n - k} f (x)

— F = CDF, f = pdf, n = sample size, k = rank

CDF of the sample maximum

F_{(n)} (x) = [F (x)]^{n}

— F = parent CDF, n = sample size, X_{(n)} = maximum

Standard error of the sample mean

SE (X ˉ) = S / n

— S = sample standard deviation, n = sample size; uses σ/√n when σ known

Sample mean

X ˉ = \frac{1}{n} \sum_{i = 1}^{n} X_{i}

— n = sample size, X_i = i-th observation,

X ˉ

= sample mean (unbiased for μ)

Uniform order statistic as a Beta distribution

U_{(k)} \sim Beta (k, n - k + 1)

E [U_{(k)}] = k / (n + 1)

— n = sample size, k = rank, U = Uniform(0,1) sample

Z-test statistic for a single mean with known variance

Z = \frac{X ˉ - μ _{0}}{σ / n}

—

X ˉ

= sample mean,

μ_{0}

= hypothesized mean,

σ

= known population SD, n = sample size

T-test statistic for a single mean with unknown variance

T = \frac{X ˉ - μ _{0}}{s / n} \sim t_{n - 1}

—

X ˉ

= sample mean,

μ_{0}

= hypothesized mean, s = sample SD, n = sample size

F-test statistic for the ratio of two variances

F = \frac{s _{1}^{2}}{s _{2}^{2}} \sim F_{n_{1} - 1, n_{2} - 1}

—

s_{1}^{2}, s_{2}^{2}

= sample variances (larger on top),

n_{1}, n_{2}

= sample sizes

Chi-square test statistic for a single variance

W = \frac{( n - 1 ) s ^{2}}{σ _{0}^{2}} \sim χ_{n - 1}^{2}

— n = sample size,

s^{2}

= sample variance,

σ_{0}^{2}

= hypothesized variance

Lognormal mean and variance

E [X] = e^{μ + σ^{2} /2}, Var (X) = e^{2 μ + σ^{2}} (e^{σ^{2}} - 1)

— μ, σ = mean and SD of log X

Aggregate loss mean and variance (general two-term)

E [S] = E [N] E [X], Var (S) = E [N] Var (X) + Var (N) E [X]^{2}

— N = claim count, X = iid severity

Negative binomial mean and variance

E [N] = r β, Var (N) = r β (1 + β)

— r = shape, β = scale; variance exceeds mean by factor (1+β)

(a,b,0) class recursion

p_{k} / p_{k - 1} = a + b / k

— p_k = probability of k claims, a and b = family-specific constants (Poisson, NegBin, Binomial)

Extended Linear Models 35 items

Pearson chi-square dispersion estimate

ϕ^= X^{2} / (n - p)

where

X^{2} = \sum (r_{i}^{P})^{2}

— n = sample size, p = parameter count, r_i^P = Pearson residual

Deviance of a GLM

D = 2 [ℓ (y; y) - ℓ (μ^; y)]

— ℓ(y;y) = saturated log-likelihood, ℓ(μ̂;y) = fitted log-likelihood

Pearson residual for a GLM

r_{i}^{P} = (y_{i} - μ^_{i}) / V (μ^_{i})

— y_i = observed, μ̂_i = fitted mean, V(μ̂_i) = variance function at μ̂_i

McFadden pseudo R-squared

R_{McF}^{2} = 1 - ℓ_{model} / ℓ_{null}

— ℓ_model = fitted log-likelihood, ℓ_null = intercept-only log-likelihood

Incremental pure-premium GLM with base-rate offset

ln E [P_{i}] = ln (B_{i}) + β_{0} + \sum_{j} β_{j} x_{ij}

— P = pure premium, B = current base premium, β = log-relativities to the base

Annualized Poisson claim frequency from an exposure-offset model

λ^_{i} = μ_{i} / E_{i} = exp (β_{0} + \sum_{j} β_{j} x_{ij})

— λ = per-exposure rate, μ = expected count, E = earned exposure

Population-averaged prediction across a control variable

μ ˉ = \sum_{k} p_{k} g^{- 1} (η_{k})

— p_k = population share of control level k, η_k = linear predictor at level k, g = link function

Linear predictor in a GLM with an offset term

η_{i} = o_{i} + β_{0} + \sum_{j} β_{j} x_{ij}

— o = known offset (coef fixed at 1), β = estimated coefficients, x = predictors, η = linear predictor

Score equation under the canonical link

X^{⊤} (y - μ) = 0

— X = design matrix, y = response vector, μ = fitted mean vector at the MLE

GLM response variance with dispersion and exposure weight

Var (Y_{i}) = ϕ V (μ_{i}) / w_{i}

— φ = dispersion, V = variance function, μ = mean, w = exposure weight

Log-link GLM with exposure offset

ln μ_{i} = ln (exposure_{i}) + x_{i}^{⊤} β

— μ = mean response, exposure = policy-years at risk, x = covariates, β = coefficients

Exponential family density form

f (y; θ, ϕ) = exp {(y θ - b (θ)) / a (ϕ) + c (y, ϕ)}

— θ = canonical parameter, φ = dispersion, b = cumulant function, a,c = known functions

Likelihood ratio statistic for nested GLMs

Λ = 2 (ℓ_{1} - ℓ_{0}) \sim ˙ χ_{Δ p}^{2}

— ℓ₁ = full-model log-likelihood, ℓ₀ = reduced-model log-likelihood, Δp = extra parameters

Akaike information criterion for GLM selection

AIC = - 2 ℓ + 2 p

— ℓ = maximized log-likelihood, p = number of fitted parameters; lower is better

Elastic net penalized GLM objective

β^= ar g min_{β} {- ℓ (β) + λ [α ∥ β ∥_{1} + (1 - α) ∥ β ∥_{2}^{2}]}

— λ = penalty strength, α = L1/L2 mix (1 = lasso, 0 = ridge)

Bayesian information criterion for GLM selection

BIC = - 2 ℓ + p ln n

— ℓ = maximized log-likelihood, p = parameters, n = sample size; lower is better

Extended linear model linear predictor and link

η = β_{0} + \sum_{j = 1}^{p} β_{j} x_{j}, g (μ) = η

— η = linear predictor, β = coefficients, x = design columns, g = link, μ = mean response

Continuous-by-continuous interaction model

η = β_{0} + β_{1} x_{1} + β_{2} x_{2} + β_{3} (x_{1} x_{2})

— x1, x2 = continuous predictors, β3 = interaction coefficient measuring departure from additivity

Degrees of freedom for a categorical-by-categorical interaction

d f_{in t} = (k_{1} - 1) (k_{2} - 1)

— k1, k2 = number of levels in the two categorical predictors; added on top of (k1-1)+(k2-1) main-effect df

Effective slope on x1 under a continuous interaction

\partial η / \partial x_{1} = β_{1} + β_{3} x_{2}

— β1 = main-effect slope, β3 = interaction coefficient, x2 = partner predictor value

Standardized residual

r_{i} = \frac{e _{i}}{s 1 - h _{ii}}

— e_i = raw residual, s = residual std error, h_ii = leverage of point i

Added variable plot residuals for predictor X_j

e_{y ∣ - j} = y - y^_{(- j)}, e_{j ∣ - j} = X_{j} - X^_{j, (- j)}

— hats with (-j) = fitted values from regressions that exclude X_j

Hat matrix for ordinary least squares

H = X (X^{T} X)^{- 1} X^{T}

— X = design matrix; h_ii = i-th diagonal of H is the leverage of observation i

Cook's distance for an observation

D_{i} = \frac{r _{i}^{2}}{p + 1} \cdot \frac{h _{ii}}{1 - h _{ii}}

— r_i = standardized residual, h_ii = leverage, p = number of predictors

Deviance-based pseudo R-squared for a GLM

R_{dev}^{2} = 1 - \frac{D _{model}}{D _{null}}

— D_model = deviance of fitted GLM, D_null = deviance of intercept-only model

Adjusted R-squared for linear regression

R_{adj}^{2} = 1 - (1 - R^{2}) \frac{n - 1}{n - k - 1}

— n = sample size, k = number of slope parameters (intercept excluded)

Coefficient of determination for OLS with intercept

R^{2} = 1 - \frac{SSE}{SST} = \frac{SSR}{SST}

— SSE = error sum of squares, SST = total sum of squares, SSR = regression sum of squares

Scaled deviance of a GLM from log-likelihoods

D^{*} = 2 (ℓ_{sat} - ℓ_{model})

, with unscaled

D = ϕ D^{*}

— ℓ_sat = saturated log-likelihood, ℓ_model = fitted log-likelihood, φ = dispersion

Freedman-Diaconis rule for histogram bin width

h = 2 \cdot I QR / n^{1/3}

— h = bin width, IQR = interquartile range of the data, n = sample size

Interquartile range

I QR = Q_{3} - Q_{1}

— Q1 = first quartile (25th percentile), Q3 = third quartile (75th percentile)

Tukey upper outlier fence for a box plot

Upper fence = Q_{3} + 1.5 \cdot I QR

— Q3 = third quartile, IQR = interquartile range; points above are flagged outliers

Tukey lower outlier fence for a box plot

Lower fence = Q_{1} - 1.5 \cdot I QR

— Q1 = first quartile, IQR = interquartile range; points below are flagged outliers

F statistic for analysis of deviance with estimated dispersion

F = (Δ D /Δ df) / ϕ^

— ΔD = deviance reduction from added terms, Δdf = added parameters, φ̂ = estimated dispersion

Wald z statistic for a GLM coefficient

z = β^_{j} / SE (β^_{j})

— β̂_j = MLE of coefficient j, SE = standard error of the estimate

Pearson chi-square goodness-of-fit statistic for a GLM

X^{2} = \sum (y_{i} - μ^_{i})^{2} / V (μ^_{i})

— y_i = observation, μ̂_i = fitted mean, V = variance function

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