asymptotic-theory

Asymptotic Theory

Rigorous framework for statistical inference and efficiency in modern methodology

Use this skill when working on: asymptotic properties of estimators, influence functions, semiparametric efficiency, double robustness, variance estimation, confidence intervals, hypothesis testing, M-estimation, or deriving limiting distributions.

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Efficiency Bounds

Semiparametric Efficiency Theory

Cramér-Rao Lower Bound: For any unbiased estimator, $$\text{Var}(\hat{\theta}) \geq \frac{1}{nI(\theta)}$$

where $I(\theta)$ is the Fisher information.

Semiparametric Efficiency Bound: The variance of the efficient influence function: $$V_{eff} = E[\phi^*(\theta_0)^2]$$

where $\phi^*$ is the efficient influence function (EIF).

Influence Function Notation: $IF(O; \theta, P)$ represents the influence of observation $O$ on parameter $\theta$ under distribution $P$: $$IF(O; \theta, P) = \lim_{\epsilon \to 0} \frac{T((1-\epsilon)P + \epsilon \delta_O) - T(P)}{\epsilon}$$

Semiparametric Variance: For RAL estimators, $$\sqrt{n}(\hat{\theta} - \theta_0) \xrightarrow{d} N(0, E[IF(O)^2])$$

Estimating Equations: M-estimators solve $\sum_{i=1}^n \psi(O_i; \theta) = 0$, with asymptotic variance: $$V = \left(\frac{\partial}{\partial \theta} E[\psi(O; \theta)]\right)^{-1} E[\psi(O; \theta)\psi(O; \theta)^T] \left(\frac{\partial}{\partial \theta} E[\psi(O; \theta)]\right)^{-T}$$

Efficiency for Mediation Estimands

Estimand	Efficient Influence Function	Efficiency Bound
ATE	$\phi_{ATE} = \frac{A}{\pi}(Y-\mu_1) - \frac{1-A}{1-\pi}(Y-\mu_0) + \mu_1 - \mu_0 - \psi$	$V_{ATE} = E[\phi_{ATE}^2]$
NDE	Complex (VanderWeele & Tchetgen, 2014)	Higher than ATE
NIE	Complex (VanderWeele & Tchetgen, 2014)	Higher than ATE

# Compute semiparametric efficiency bound
compute_efficiency_bound <- function(data, estimand = "ATE") {
  n <- nrow(data)

  if (estimand == "ATE") {
    # Estimate nuisance functions
    ps_model <- glm(A ~ X, data = data, family = binomial)
    pi_hat <- predict(ps_model, type = "response")

    mu1_model <- lm(Y ~ X, data = subset(data, A == 1))
    mu0_model <- lm(Y ~ X, data = subset(data, A == 0))

    mu1_hat <- predict(mu1_model, newdata = data)
    mu0_hat <- predict(mu0_model, newdata = data)

    # Efficient influence function
    psi_hat <- mean(mu1_hat - mu0_hat)
    phi <- with(data, {
      A/pi_hat * (Y - mu1_hat) -
      (1-A)/(1-pi_hat) * (Y - mu0_hat) +
      mu1_hat - mu0_hat - psi_hat
    })

    # Efficiency bound = variance of EIF
    list(
      efficiency_bound = var(phi),
      standard_error = sqrt(var(phi) / n),
      eif_values = phi
    )
  }
}