Adaptive-TMLE for the Average Treatment Effect based on Randomized Controlled Trial Augmented with Real-World Data

The decision to integrate or discard external data in the analysis of an RCT often hinges on whether pooling introduces bias. A simple “test-then-pool” strategy involves first conducting a hypothesis test to determine the similarity between the RCT and RWD before deciding to pool them together or rely solely on the RCT data [8]. The threshold of the test statistic above which the hypothesis test would be rejected could be determined data-adaptively [9]. However, methods of this type can be limited by the small sample sizes typical of RCTs, potentially resulting in underpowered hypothesis tests [10]. Additionally, using the “test-then-pool” strategy, either an efficiency gain is realized, or it is not, without gradation. Bayesian dynamic borrowing methods adjust the weight of external data based on the bias it introduces [11, 12, 13, 14, 15]. Similar frequentist approaches data-adaptively choose between data sources by balancing the bias-variance trade-off, often resulting in a weighted combination of the RCT-only and the pooled-ATE estimands [16, 17, 18]. In particular, the experiment-selector cross-validated targeted maximum likelihood estimator (ES-CVTMLE) method [16] also allow the incorporation of a negative control outcome (NCO) in the selection criteria. The gain in efficiency of these methods largely depends on the bias magnitude, with larger biases diminishing efficiency gains. Other methods incorporate a bias correction by initially generating a biased estimate, then learning a bias function, and finally adjusting the biased estimate to achieve an unbiased estimate of the causal effect, which has a similar favor to the method we are proposing [19, 20, 21]. However, the key aspect distinguishing our work and theirs is that their techniques rely on the independence between potential outcomes and the study indicator $S$ given observed covariates, an assumption that could easily be violated in reality. For instance, subjects might adhere more strictly to assigned treatment regimens had they been enrolled in the RCT, so being in the RCT or not modifies the treatment effects. These methods also focus primarily on unmeasured confounding bias, yet other biases, such as differences in outcome measurement or adherence levels between RCT and RWD settings, may also occur. The reliance on this independence assumption also limits their applicability in scenarios involving surrogate outcomes in the RWD, in which case the outcomes in the RCT and RWD are either measured differently or are fundamentally different variables, a common scenario one would encounter in practice. We define our bias directly as the difference between the target estimand and the pooled estimand, therefore it covers any kind of bias, including unmeasured confounding bias. Thus, our method can be applied more broadly, even in cases where the outcomes differ in the two studies.

This article is organized as follows. In Section 2 we formally define the estimation problem in terms of the statistical model and target estimand that identifies the desired average treatment effect. In Section 3, we decompose our target estimand into a difference between a pooled-ATE estimand $\tilde{\Psi}$ and a bias estimand $\Psi^{\#}$. In Sections 4 and 5, we provide the key ingredients for constructing A-TMLEs of $\Psi^{\#}$ and $\tilde{\Psi}$ respectively. In Section 4, we present the semiparametric regression working model for the conditional effect of the study indicator on the outcome and the corresponding projection parameter $\Psi^{\#}_{{\cal M}_{w,2}}$ that replaces the outcome regression by its $L^{2}$-projection onto the working model $\mathcal{M}_{w,2}$. This then defines a working-model-specific target estimand on the original statistical model and thus defines a new estimation problem that would approximate the desired estimation problem if the working model approximates the true data density. If the study indicator has zero impact on the outcome, then the pooled-ATE estimand that just combines the data provides a valid estimator, thereby allowing full integration of the external data into the estimator; If the study indicator has an impact explained by a parametric form (e.g. a linear combination of a finite set of spline basis functions), then the estimand still integrates the data but carries out a bias correction according to this parametric form. As the parametric form becomes nonparametric, the estimand becomes the estimand that ignores the outcome data in the external study when estimating the outcome regression, corresponding with an efficient estimator. We derive the canonical gradient and the corresponding TMLE of the projection parameter $\Psi^{\#}_{{\cal M}_{w,2}}$. Similarly, in Section 5, we present a semiparametric regression working model for the pooled-ATE estimand $\tilde{\Psi}$ and define the corresponding projection parameter. In this case the semiparametric regression model learns the conditional effect of the treatment (instead of the study indicator) on the outcome, conditioned on treatment and baseline covariates, while not conditioning on the study indicator. The projection is defined as the $L^{2}$-projection of the true outcome regression $E_{P}(Y\mid A,W)$ (ignoring the study indicator) onto the working model $\mathcal{M}_{w,1}$. We present the canonical gradient and the corresponding TMLE of $\tilde{\Psi}_{{\cal M}_{w,1}}$. We summarize our A-TMLE algorithm in Section 6. Specifically, using a synthetic example, we illustrate how one might apply A-TMLE to augment RCT with external data in practice. In Section 7 we analyze the A-TMLE analogue to [6], to make this article self-contained. The analysis can be applied to both components of the target parameter, thereby also providing an asymptotic linearity theorem for the resulting A-TMLE of the target parameter. Specifically, we prove that the A-TMLEs of $\Psi^{\#}$ and $\tilde{\Psi}$ are $\sqrt{n}$-consistent, asymptotically normal with super-efficient variances. Under an asymptotic stability condition for the working model, they are also asymptotically linear with efficient influence functions that equal the efficient influence functions of the limit of the submodel (oracle model). In Section 8, we carry out simulation studies to evaluate the performance of our proposed A-TMLE against other methods including ES-CVTMLE [16], PROCOVA (a covariate-adjustment method) [22], a regular TMLE for the target estimand, and a TMLE using RCT data alone. We conclude with a discussion in Section 9.

We assume a structural causal model (SCM): $S=f_{S}(U_{S})$; $W=f_{W}(S,U_{W})$; $A=f_{A}(S,W,U_{A})$; $Y=f_{Y}(S,W,A,U_{Y})$, where $U=(U_{S},U_{W},U_{A},U_{Y})$ is a vector of exogenous errors. The joint distribution of $(U,O)$ is parametrized by a vector of functions $f=(f_{S},f_{W},f_{A},f_{Y})$ and the error distribution $P_{U}.$ The SCM ${\cal M}^{F}$ is defined by assumptions on these functions and the error distribution. Here ${\cal M}^{F}$ denotes the set of full data distributions $P_{U,O}$ that satisfy these assumptions on $f$ and $P_{U}$. The SCM allows us to define the potential outcomes $Y_{1}=f_{Y}(S,W,A=1,U_{Y})$ and $Y_{0}=f_{Y}(S,W,A=0,U_{Y})$ by intervening on the treatment node $A$.

We make the following three assumptions:

1. A1

   $(Y_{0},Y_{1})\perp A\mid S=1,W$ (randomization in the RCT);

2. A2

   $0<P(A=1\mid S=1,W)<1,P_{W}\text{-a.e.}$ (positivity of treatment assignment in the RCT);

3. A3

   $P(S=1\mid W)>0,P_{W}\text{-a.e.}$ (positivity of RCT enrollment in the pooled population).

Assumption A1 is a causal assumption and is non-testable, assumptions A2 and A3 are statistical assumptions. Note that assumptions A1 and A2 are satisfied in an RCT (or a well-designed observational study). Importantly, we do not make the assumption that $E(Y_{a}\mid S=0,W)=E(Y_{a}\mid W)$, sometimes referred to as the “mean exchangeability over $S$” [5]. In other words, for the combined study, $A$ might not be conditionally independent of $Y_{0},Y_{1}$, given $W$, due to the bias introduced from the external data. Specifically, we are concerned that $E_{W}E(Y_{1}-Y_{0}\mid W)\not=E_{W}E(Y\mid W,A=1)-E_{W}E(Y\mid W,A=0)$, due to $E_{W}E(Y_{1}-Y_{0}\mid W,S=0)\not=E_{W}E(Y\mid S=0,W,A=1)-E_{W}E(Y\mid S=0,W,A=0)$. Beyond these three assumptions, we will make additional assumptions on $g_{0}(1\mid S,W)\equiv P(A=1\mid S,W).$ In particular, if $S=1$ corresponds with an RCT, then $g_{0}(1\mid S=1,W)$ would just be the randomization probability; If the $S=1$-study is not an RCT, we might still be able to make a conditional independence assumption $g_{0}(A\mid 1,W)=g_{0}(A\mid 1,W_{1})$ for some subset $W_{1}$ of $W$. In some cases, we might also be able to assume that $S$ is independent of $W$, i.e. $P_{W\mid S=1}=P_{W\mid S=0}.$ This independence could be achieved by sampling the external subjects from the same target population as in the $S=1$-study. As we will see later in the identification step, for our target parameter it is crucial that the support of $P_{W\mid S=0}$ is included in the support of $P_{W\mid S=1}$ so that assumption A3 holds. In this article we will not make assumptions on the joint distribution of $(S,W)$ beyond assumption A3, but our results are not hard to generalize to the case that we assume $W\perp S$. In the important case that we augment an RCT with external controls only we have that $g_{0}(1\mid 0,W)=0$, so that the only purpose of the $S=0$-study is to augment the control arm of the RCT. To briefly summarize, we do not make any additional assumptions beyond the same set of assumptions for a standard RCT. The only additional assumption we require is A3 and could be made plausible in the selection of the external subjects.

Let ${\cal M}$ be the set of possible distributions $P$ of $O$ that satisfies the statistical assumptions A2 and A3. Then, ${\cal M}=\{P_{P_{U,O}}:P_{U,O}\in{\cal M}^{F}\}$ is the model implied by the full data model ${\cal M}^{F}$. We can factorize the density of $O$ according to the time-ordering as follows:

$$p(s,w,a,y)=p_{S}(s)p_{W}(w\mid s)g_{A}(a\mid s,w)q_{Y}(y\mid s,w,a),$$

where $q_{Y}(y\mid s,w,a)\equiv p_{Y}(y\mid s,w,a).$ Our statistical model only makes assumptions on $g_{A}$ and leaves the other factors in the likelihood unspecified.

We could consider two candidate target parameters. The first one is

$$\Psi^{F}(P_{O,U})=E_{W}E(Y_{1}-Y_{0}\mid S=1,W).$$

Note that this parameter measures the conditional treatment effect in the RCT, while it takes an average with respect to the combined covariate distribution $p_{W}=p_{W\mid S=1}p_{S=1}+p_{W\mid S=0}p_{S=0}$. Alternatively, we could take the average with respect to the RCT covariate distribution $p_{W\mid S=1}$, in which case we have the target parameter given by

$$\Psi^{F}_{2}(P_{O,U})=E(Y_{1}-Y_{0}\mid S=1)=E_{W}[E(Y_{1}-Y_{0}\mid S=1,W)\mid S=1].$$

In the following subsections, we will discuss the their identification results and compare their efficient influence functions.

Under assumptions A1, A2 and A3, the full-data target causal parameter $\Psi^{F}$ is identified by the following statistical target parameter $\Psi:{\cal M}\rightarrow\hbox{${\rm I\kern-1.99997ptR}$}$ defined by

$$\Psi(P_{0})=E_{0}[E_{0}(Y\mid S=1,W,A=1)-E_{0}(Y\mid S=1,W,A=0)].$$

Note that this estimand is only well-defined if $\min_{a\in\{0,1\}}P(S=1,W=w,A=a)>0$ for $P_{W}$-a.e. So, if the $S=0$-study has a covariate distribution with a support not included in the support of $P_{W\mid S=1}$, then the estimand is not well-defined. Therefore, we need assumption A2. As we will see in the next subsection, the efficient influence function of $\Psi$ indeed involves inverse weighting by $P(S=1\mid W)$. Similarly, $\Psi^{F}_{2}$ is identified by

$$\Psi_{2}(P_{0})=E_{0}\{[E_{0}(Y\mid S=1,W,A=1)-E_{0}(Y\mid S=1,W,A=0)]\mid S=1\}.$$

Thus, $\Psi^{F}(P_{U,O})=\Psi(P_{P_{U,O}})$ for any $P_{U,O}$ in our statistical model $\mathcal{M}.$ Note that the estimand $\Psi_{2}(P_{0})$ only relies on $\min_{a\in\{0,1\}}P(S=1,W=w,A=a)>0$ for $P_{W\mid S=1}$-a.e, which thus always holds by assumption on the $S=1$-study.

We could construct an efficient estimator of $\Psi:{\cal M}\rightarrow\hbox{${\rm I\kern-1.99997ptR}$}$. However, it would still lack power since it would only be using the $S=1$-observations for learning the conditional treatment effect, and similarly for $\Psi_{2}$. This point is further discussed in the next subsection by analyzing and comparing the nonparametric efficiency bounds of $\Psi$ and $\Psi_{2}.$ Therefore, we instead pursue a finite sample super-efficient estimator through adaptive-TMLE whose gain in efficiency is adapted to the underlying unknown (but learnable) complexity of $P_{0}$.

For completeness, we will present canonical gradients $D_{\Psi,P}$ of $\Psi:{\cal M}\rightarrow\hbox{${\rm I\kern-1.99997ptR}$}$ and $D_{\Psi_{2},P}$ of $\Psi_{2}:{\cal M}\rightarrow\hbox{${\rm I\kern-1.99997ptR}$}$, even though we will not utilize them in the construction of our A-TMLE.

The proof can be found in Appendix A. To compare the nonparametric efficiency bound of $\Psi(P_{0})$ and $\Psi_{2}(P_{0}),$ let

$$\sigma^{2}_{P}(s,w,a)=E_{P}[(Y-\bar{Q}_{P}(S,W,A))^{2}\mid S=s,W=w,A=a].$$

We note that

	$\displaystyle PD^{2}_{\Psi,P}(O)$	$\displaystyle=E_{P}(\bar{Q}_{P}(1,W,1)-\bar{Q}_{P}(1,W,0)-\Psi(P))^{2}$
		$\displaystyle+E_{P}\left[\frac{1}{P(S=1\mid W)}\left(\frac{\sigma^{2}_{P}(1,W,1)}{g_{P}(1\mid 1,W)}+\frac{\sigma^{2}_{P}(1,W,0)}{g_{P}(0\mid 1,W)}\right)\right].$

Similarly,

	$\displaystyle PD^{2}_{\Psi_{2},P}(O)$	$\displaystyle=E_{P}\left[\frac{P(S=1\mid W)}{P^{2}(S=1)}(\bar{Q}_{P}(1,W,1)-\bar{Q}_{P}(1,W,0)-\Psi_{2}(P))^{2}\right]$
		$\displaystyle+E_{P}\left[\frac{P(S=1\mid W)}{P^{2}(S=1)}\left(\frac{\sigma^{2}_{P}(1,W,1)}{g_{P}(1\mid 1,W)}+\frac{\sigma^{2}_{P}(1,W,0)}{g_{P}(0\mid 1,W)}\right)\right].$

Thus, the variance of the $W$-component of $D_{\Psi,P}$ is a factor of $\sim 1/P(S=1)$ smaller than the variance of the $W$-component of $D_{\Psi_{2},P}$. However, the variance of the $Y$-component of $D_{\Psi,P}$ involves the factor $1/P(S=1\mid W)$ versus the factor $\sim 1/P(S=1)$ in $D_{\Psi_{2},P,Y}$. Therefore, if $S$ is independent of $W$, then it follows that the variance of $D_{\Psi,P}$ is smaller than the variance of $D_{\Psi_{2},P}$, due to a significantly smaller variance of its $W$-component, while having identical $Y$-components. On the other hand, if $S$ is highly dependent on $W$, then the inverse weighting $1/P(S=1\mid W)$ could easily cause the variance of the $Y$-component of $D_{\Psi,P}$ to be significantly larger than the variance of the $Y$-component of $D_{\Psi_{2},P}$. Generally speaking, especially when $P(A=1\mid S=1,W)$ depends on $W$, the variance of the $Y$ component dominates the variance of the $W$ component. Therefore without controlling the dependence of $S$ and $W$, one could easily have that the variance of $D_{\Psi,P}$ is larger than the variance of $D_{\Psi_{2},P}$. Hence, if one wants to use $\Psi$ instead of $\Psi_{2}$, then one wants sample external subjects such that $S$ approximately independent of $W$. For this article, we will focus our attention on the target causal parameter $\Psi(P)$, which, as we argued above, would be the preferred choice if $P(S=1\mid W)\approx P(S=1)$. However, our results can be easily generalized to $\Psi_{2}(P)$.

Since $\bar{Q}_{0}(S,W,A)=\bar{Q}_{0}(0,W,A)+S\tau_{0}(W,A)$, a working model ${\cal T}_{n}$ for $\tau_{0}$ corresponds with a semiparametric regression working model ${\cal Q}_{w,n}=\{\theta+S\tau_{w,n,\beta}:\beta,\theta\}$ for $\bar{Q}_{0}=E_{0}(Y\mid S,W,A)$, where $\theta$ is an unspecified function of $W,A$. The working model ${\cal Q}_{w,n}$ further implies a working model ${\cal M}_{w,2,n}=\{P\in{\cal M}:\bar{Q}_{P}\in{\cal Q}_{w,n}\}\subset{\cal M}$ for the observed data distribution $P_{0}$. Let $\tau_{w,n,\beta(P)}\in{\cal T}_{n}$ be a projection of $\tau_{P}$ onto this working model ${\cal T}_{n}$ with respect to some loss function. We will also use the notation $\tau_{w,n,P}$ for $\tau_{w,n,\beta(P)}$. This generally corresponds with mapping a $P\in{\cal M}$ into a projection $\Pi_{w,n}(P)\in{\cal M}_{w,2,n}$, and setting $\tau_{w,n,P}\equiv\tau_{\Pi_{w,n}(P)}$. For the squared-error loss, and a semiparametric regression working model ${\cal Q}_{w,n}$ for $\bar{Q}_{P}$, we recommend the projection used in [6]:

$$\bar{Q}_{w,n,P}\equiv\Pi_{{\cal Q}_{w,n},P}(\bar{Q}_{P})=\arg\min_{\bar{Q}\in{\cal Q}_{w,n}}P\left(\bar{Q}_{P}-\bar{Q}\right)^{2}.$$

Interestingly, as shown in [6], we have

$$\tau_{w,n,P}=\arg\min_{\tau\in\tau_{n}}E_{P}L_{\tilde{\theta}_{P},\Pi_{P}}(\tau),$$

where

$$L_{\tilde{\theta}_{P},\Pi_{P}}(\tau)\equiv\left(Y-\tilde{\theta}_{P}(W,A)-(S-\Pi_{P}(1\mid W,A))\tau\right)^{2},$$

and $\tilde{\theta}_{P}(W,A)=E_{P}(Y\mid W,A)$. This insight provides us with a nice loss function for learning a working model ${\cal T}_{n}$, by using, for example, a highly adaptive lasso minimum-loss estimator (HAL-MLE) [7] for this loss function with estimators $\tilde{\theta}_{n}$ of $\tilde{\theta}_{0}$ and $\Pi_{n}$ of $\Pi_{0}$. Specifically, given a rich linear model $\{\tau_{\beta}=\sum_{j}\beta_{j}\phi_{j}\}$ with a large set of spline basis functions $\{\phi_{j}:j\}$, we compute the lasso-estimator

$$\beta_{n}=\arg\min_{\beta,\parallel\beta\parallel_{1}\leq C_{n}}\sum_{i=1}^{n}\left(Y_{i}-\tilde{\theta}_{n}(W_{i},A_{i})-(S_{i}-\Pi_{n}(1\mid W_{i},A_{i}))m_{\beta}(W_{i},A_{i})\right)^{2},$$

where $m_{\beta}(W,A)$ is a linear combination of HAL basis functions and $C_{n}$ is an upper bound on the sectional variation norm [23]. The working model for $\tau_{P}$ is then given by ${\cal T}_{n}=\{\sum_{j,\beta_{n}(j)\not=0}\beta_{j}\phi_{j}:\beta\}$, i.e. linear combinations of the basis functions with non-zero coefficients. In addition, it has been shown in [6] that

$$\tau_{w,n,P}=\arg\min_{\tau\in\mathcal{T}_{n}}E_{P}\Pi_{P}(1-\Pi_{P})(1\mid A,W)(\tau_{P}-\tau)^{2}.$$

Put in other words, the squared-error projection of $\bar{Q}_{P}$ onto the semi-parametric regression working model corresponds with a weighted squared-error projection of $\tau_{P}$ onto the corresponding working model for $\tau_{P}$. The weights $\Pi_{P}(1-\Pi_{P})(1\mid W,A)$ stabilize the efficient influence function for $\beta(P)$, showing that the loss function $L_{\tilde{\theta},\Pi}(\tau)$ for $\tau_{P}$ yields relatively robust estimators of $\tau_{0}$.

Given the above definition of the semiparametric regression working model, $\tau_{w,n,P}=\tau_{w,n,\beta(P)}$ for $P\in\mathcal{M}$, for the conditional effect of the study indicator $S$ on the outcome $Y$, we can now define a corresponding working-model-specific projection parameter for $\Psi^{\#}(P)$. Specifically, let $\Psi^{\#}_{{\cal M}_{w,2,n}}:{\cal M}\rightarrow\hbox{${\rm I\kern-1.99997ptR}$}$ be defined as

$$\Psi^{\#}_{{\cal M}_{w,2,n}}(P)\equiv E_{P}\Pi_{P}(0\mid W,0)\tau_{w,n,\beta(P)}(W,0)-E_{P}\Pi_{P}(0\mid W,1)\tau_{w,n,\beta(P)}(W,1).$$

We will sometimes use the notation $\Psi^{\#}(P_{W},\Pi_{P},\tau_{w,n,P})$ to emphasize its reliance on the nuisance parameters $(P_{W},\Pi_{P},\tau_{w,n,P})$. The following lemma reviews the above stated results.

Let’s first find the canonical gradient of the $\beta(P)$-component.

The proof can be found in Appendix A. We then need to find the canonical gradient of the working-model-specific projection parameter $\Psi^{\#}_{{\cal M}_{w,2}}$ at $P$.

The proof can be found in Appendix A.

A plug-in estimator of the projection parameter $\Psi^{\#}_{{\cal M}_{w,2,n}}(P_{0})$ requires an estimator of $\Pi_{0}$, $\beta_{0}$, $P_{W,0}$, and a model selection method, as discussed earlier, so that the semiparametric regression working model ${\cal M}_{w,2,n}$ is known, including the working model $\{\tau_{w,n,\beta}:\beta\}$ for $\tau_{P_{0}}$. We then need to solve each component of the canonical gradient of $\Psi^{\#}_{{\cal M}_{w,2,n}}(P_{0})$. First, for the $W$-component, we use the empirical distribution $P_{W,n}$ of $W$ as the estimator of $P_{W,0}$. Therefore, $P_{n}D^{\#}_{\mathcal{M}_{w,2,n},W,\beta,\Pi,P_{W,n}}(O)=0$ for any $\beta,\Pi$. Then, for the $\beta$-component, we need to construct an initial estimator $\tilde{\theta}_{n}$ of $\tilde{\theta}_{0}(W,A)=E_{0}(Y\mid W,A)$ and $\Pi_{n}$ of $\Pi_{0}(s\mid W,A)=P_{0}(S=s\mid W,A)$. Then, with these two nuisance estimates, we can compute the MLE over the working model for $\tau_{0}$ with respect to our loss function $L_{\tilde{\theta}_{n},\Pi_{n}}(\tau)$:

$$\beta_{n}=\arg\min_{\beta}P_{n}L_{\tilde{\theta}_{n},\Pi_{n}}(\tau_{w,n,\beta}).$$

We have then solved $P_{n}D^{\#}_{{\cal M}_{w,2,n},\beta,\beta_{n},\Pi_{n}}(O)=0$. Finally, for the $\Pi$-component, note that we have the representation: $D^{\#}_{{\cal M}_{w,2,n},\Pi,P}(O)=C(g,\beta)(S-\Pi_{P}(1\mid W,A))$, where we refer to $C(g,\beta)$ as the clever covariate. With an estimator $g_{n}$ of $g_{0}$, we can compute a targeted estimator $\Pi_{n}^{*}$ solving $P_{n}C(g_{n},\beta_{n})(S-\Pi_{n}^{*}(1\mid W,A))=0$. The desired TMLE is then given by $\Psi^{\#}_{\mathcal{M}_{w,2,n}}(P_{n}^{*})\equiv\Psi^{\#}_{{\cal M}_{w,2,n}}(P_{W,n},\Pi_{n}^{*},\beta_{n})$. For inference with respect to the projection parameter $\Psi^{\#}_{{\cal M}_{w,2,n}}(P_{0})$, we use that

$$\Psi^{\#}_{{\cal M}_{w,2,n}}(P_{n}^{*})-\Psi^{\#}_{{\cal M}_{w,2,n}}(P_{0})=P_{n}D^{\#}_{{\cal M}_{w,2,n},P_{0}}+o_{P}(n^{-1/2}),$$

while for inference for the original target parameter $\Psi^{\#}(P_{0})$, we also use that $\Psi^{\#}_{{\cal M}_{w,2,n}}(P_{0})-\Psi^{\#}(P_{0})=o_{P}(n^{-1/2})$, under reasonable regularity conditions as discussed in [6].