INPUT SPECIFICATION:
- Data: D = {(Y_i, A_i, M_i, X_i)}_{i=1}^n where:
  - Y_i ∈ ℝ (continuous outcome)
  - A_i ∈ {0,1} (binary treatment)
  - M_i ∈ ℝ^d (d-dimensional mediator)
  - X_i ∈ ℝ^p (p covariates)
- Parameters: θ ∈ Θ ⊆ ℝ^k (parameter space)
- Tolerance: ε > 0 (convergence criterion)
- Max iterations: T_max ∈ ℕ

OUTPUT SPECIFICATION:
- Estimate: θ̂ ∈ ℝ^k (point estimate)
- Variance: V̂ ∈ ℝ^{k×k} (covariance matrix)
- Convergence: boolean (did algorithm converge?)
- Iterations: t ∈ ℕ (iterations used)

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# Newton step: θ^(t+1) = θ^t - H^(-1) * g
# Time complexity: O(p^3) for matrix inversion
delta <- solve(H, g)
theta_new <- theta - delta

# Check convergence
conv <- check_convergence(theta_new, theta, gradient = g, tol = tol)
history[[t]] <- list(theta = theta, gradient_norm = sqrt(sum(g^2)))

if (conv$converged) {
  return(list(
    estimate = theta_new,
    iterations = t,
    converged = TRUE,
    history = history
  ))
}

theta <- theta_new

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ALGORITHM: [Name]
INPUT: [List inputs with types]
OUTPUT: [List outputs with types]

1. [Initialize]
2. [Main loop or procedure]
   2.1 [Sub-step]
   2.2 [Sub-step]
3. [Return]

ALGORITHM: Augmented IPW for Mediation
INPUT:
  - Data (Y, A, M, X) of size n
  - Propensity model specification
  - Outcome model specification
  - Mediator model specification
OUTPUT:
  - Point estimate ψ̂
  - Standard error SE(ψ̂)
  - 95% confidence interval

1. ESTIMATE NUISANCE FUNCTIONS
   1.1 Fit propensity score: π̂(x) = P̂(A=1|X=x)
   1.2 Fit mediator density: f̂(m|a,x)
   1.3 Fit outcome regression: μ̂(a,m,x) = Ê[Y|A=a,M=m,X=x]

2. COMPUTE PSEUDO-OUTCOMES
   For i = 1 to n:
     2.1 Compute IPW weight: w_i = A_i/π̂(X_i) + (1-A_i)/(1-π̂(X_i))
     2.2 Compute outcome prediction: μ̂_i = μ̂(A_i, M_i, X_i)
     2.3 Compute augmentation term
     2.4 φ_i = w_i(Y_i - μ̂_i) + [integration term]

3. ESTIMATE AND INFERENCE
   3.1 ψ̂ = n⁻¹ Σᵢ φ_i
   3.2 SE = √(n⁻¹ Σᵢ (φ_i - ψ̂)²)
   3.3 CI = [ψ̂ - 1.96·SE, ψ̂ + 1.96·SE]

4. RETURN (ψ̂, SE, CI)

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TIME COMPLEXITY:
- Initialization: O(...)
- Per iteration: O(...)
- Total (T iterations): O(...)
- Convergence typically in T = O(...) iterations

SPACE COMPLEXITY:
- Data storage: O(n × p)
- Working memory: O(...)
- Output: O(...)

CONVERGENCE:
- Type: [Linear/Superlinear/Quadratic]
- Rate: [Expression]
- Conditions: [What must hold]
- Stopping criterion: [When to stop]
- Typical iterations: [Order of magnitude]

ALGORITHM: Gradient Descent
INPUT: f (objective), ∇f (gradient), x₀ (initial), η (step size), ε (tolerance)
OUTPUT: x* (minimizer)

1. k ← 0
2. WHILE ‖∇f(xₖ)‖ > ε:
   2.1 xₖ₊₁ ← xₖ - η∇f(xₖ)
   2.2 k ← k + 1
3. RETURN xₖ

COMPLEXITY: O(iterations × gradient_cost)
CONVERGENCE: Linear with rate (1 - η·μ) for μ-strongly convex f

ALGORITHM: Newton-Raphson
INPUT: f, ∇f, ∇²f, x₀, ε
OUTPUT: x*

1. k ← 0
2. WHILE ‖∇f(xₖ)‖ > ε:
   2.1 Solve ∇²f(xₖ)·d = -∇f(xₖ) for direction d
   2.2 xₖ₊₁ ← xₖ + d
   2.3 k ← k + 1
3. RETURN xₖ

COMPLEXITY: O(iterations × p³) for p-dimensional
CONVERGENCE: Quadratic near solution

ALGORITHM: Expectation-Maximization
INPUT: Data Y, model parameters θ₀, tolerance ε
OUTPUT: MLE θ̂

1. θ ← θ₀
2. REPEAT:
   2.1 E-STEP: Compute Q(θ'|θ) = E[log L(θ'|Y,Z) | Y, θ]
   2.2 M-STEP: θ_new ← argmax_θ' Q(θ'|θ)
   2.3 Δ ← |θ_new - θ|
   2.4 θ ← θ_new
3. UNTIL Δ < ε
4. RETURN θ

CONVERGENCE: Monotonic increase in likelihood
             Linear rate near optimum

ALGORITHM: Nonparametric Bootstrap
INPUT: Data X of size n, statistic T, B (number of replicates)
OUTPUT: SE estimate, CI

1. FOR b = 1 to B:
   1.1 Draw X*_b by sampling n observations with replacement from X
   1.2 Compute T*_b = T(X*_b)
2. SE_boot ← SD({T*_1, ..., T*_B})
3. CI_percentile ← [quantile(T*, 0.025), quantile(T*, 0.975)]
4. RETURN (SE_boot, CI_percentile)

COMPLEXITY: O(B × cost(T))
NOTES: B ≥ 1000 for SE, B ≥ 10000 for percentile CI

ALGORITHM: Parametric Bootstrap
INPUT: Data X, parametric model M, B replicates
OUTPUT: SE estimate

1. Fit θ̂ = MLE(X, M)
2. FOR b = 1 to B:
   2.1 Generate X*_b ~ M(θ̂)
   2.2 Compute θ̂*_b = MLE(X*_b, M)
3. SE_boot ← SD({θ̂*_1, ..., θ̂*_B})
4. RETURN SE_boot

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