HW3

Equation 5: \[ \widehat{\text{ATE}}_{\text{AIPW}} = \widehat{\theta}^1_{\text{AIPW}} - \widehat{\theta}^0_{\text{AIPW}} = \frac{1}{n} \sum_{i=1}^n \left[ \left( \hat{\mu}(1, X_i) + \frac{A_i}{\hat{\pi}(X_i)} \left( Y_i - \hat{\mu}(1, X_i) \right) \right) - \left( \hat{\mu}(0, X_i) + \frac{1 - A_i}{1 - \hat{\pi}(X_i)} \left( Y_i - \hat{\mu}(0, X_i) \right) \right) \right] \]

Equation 6:Answer \[ \widehat{\text{ATE}}_{\text{AIPW}} = \frac{1}{n} \sum_{i=1}^n \left[ \hat{\mu}(1, X_i) - \hat{\mu}(0, X_i) + \frac{A_i}{\hat{\pi}(X_i)} (Y_i - \hat{\mu}(1, X_i)) - \frac{1 - A_i}{1 - \hat{\pi}(X_i)} (Y_i - \hat{\mu}(0, X_i)) \right] \]