30 problems

Let $\bmu$ and $\bDelta$ be the preliminary estimators, let $\hn$ be the preliminary density estimator, and let the untruncated one-step estimator be the estimator obtained from th…

Let Horvitz--Thompson estimators be constructed under complex sampling designs, and let the paper's fundamental equation be the equation referred to as $(\mathrm{general})$. Extens…

Let $r_{mu,n}$ denote the convergence rate of the outcome-regression estimator, let $gamma_0$ govern the simulated propensity-score distribution, and let $b_n$ be the clipping thre…

Let autoDML and autoTML denote the automatic debiased machine-learning and targeted maximum-likelihood estimators considered in the paper, respectively, with stabilization referrin…

Let $\pi^\ast$ denote the true propensity score and let $\tilde{\pi}$ be its estimator. Let $r^\ast$ denote the true outcome regression, and suppose that only the outcome-regressio…

The phenomenon refers to the possibility that product bias from nuisance-function estimation can dominate coverage errors even when estimating the asymptotic variance more accurate…

Fixed-design validity conjecture. For the fixed design, these theoretical results are still valid under appropriate regularity conditions.

Conjecture on the convergence rate. This rate may not be sharp for the estimator $\widehat{\boldsymbol{\beta}}$; there may be room for improvement.

Conjecture on the convergence rate. This rate may not be optimal; the best rate may be $O_p(n^{-1/2})$.

Conjecture on the order of HAL empirical process differences. This term is a $j$-th order difference of $\tilde{P}_n$ and $P_0$.

Epsilon-relaxation conjecture. Using an $\epsilon$-relaxation is unnecessary in this setting.

Let $f_0\in\mathcal{P}_0$ be a symmetric log-concave density in the location model, and consider the untruncated one-step estimator. The untruncated one-step rate conjecture. The u…

Let $f_0\in\mathcal{P}_0$ be a symmetric log-concave density in the location model with location parameter $\theta_0$, and let $\widehat{\theta}_{\mathrm{MLE}}$ denote its maximum…

Let $f_0\in\mathcal{P}_0$ be a symmetric log-concave location-model density with location parameter $\theta_0$, and let $\widehat{\theta}_{\mathrm{MLE}}$ denote the maximum likelih…

Let $Q_0$ denote the true parameter and $G_0$ a nuisance parameter, and let $Q_n$ be an undersmoothed highly adaptive lasso maximum likelihood estimator (HAL-MLE). An HAL-TMLE is a…

Let $Y$ be a random variable, let $X$ take values in a compact set $\mathcal{X}\subset\mathbb{R}^q$, and let $\beta_0$ be an interior point of a bounded set…

Residual pseudo-likelihood validity conjecture. Other procedures using $\widehat{\alpha}_n$ will be valid under these assumptions.

Pseudo-likelihood instability conjecture. The method of maximum pseudo-likelihood estimation can be problematic in this setting.

Let $P_n^0$ be an initial estimate, and let $P_n^*$ denote the estimate obtained by the one-step targeted maximum likelihood estimation (TMLE) algorithm. Let $R_2(P_n^*,P_0)$ be th…

One-step local least favorable submodel conjecture. Under regularity conditions, the efficient score equation will be solved in one step using the local least favorable submodel, u…

Equivalence conjecture. Both estimators have the same asymptotic properties.

A sequential HAL-TMLE estimates a target parameter through recursively defined nuisance parameters, where each regression is fitted using a highly adaptive lasso (HAL) regression a…

The paper considers a cross-validated targeted maximum likelihood estimator based on an approximating family indexed by a smoothing parameter, with assumptions A9 and A12 concernin…

Given covariates $\mathbf{x}^*$ and $\mathbf{z}^*$, consider estimating the conditional quantile function $Q_{Y_i|\mathbf{x}^*,\mathbf{z}^*}(\tau)$ by … where $\hat{\bolds{\beta}}$…

Let the $1^\star$-TMLE and the 1-TMLE be the two estimators for the mean of an outcome missing at random, and suppose that the observed finite-sample gains of the $1^\star$-TMLE ov…