Following eq. (12) and using the martingale integral representation $\sqrt{n}(\stackrel{\u02c6}{w}(Y)-w(Y))$, we have
$\frac{\mathrm{\delta}}{\stackrel{\u02c6}{w}(Y)}\left[m(\mathbf{X},D,Y)-{\mathrm{\beta}}_{0}\right]=\frac{\mathrm{\delta}}{w(Y)}\left[m(\mathbf{X},D,Y)-{\mathrm{\beta}}_{0}\right]+{\int}_{0}^{s}\frac{\mathrm{\kappa}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}+{o}_{p}(1)$where
$\mathrm{\kappa}(t)=\mathbb{E}\left[\frac{\mathrm{\delta}\phantom{\rule{thinmathspace}{0ex}}\mathbb{1}(Y>t)[m(\mathbf{X},D,Y)]{\int}_{t}^{Y}{S}_{C}(v)\phantom{\rule{thinmathspace}{0ex}}dv}{{w}^{2}(Y)}\right]$and
$m(\mathbf{X},D,Y)=\frac{D[log(Y)-{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})]}{\mathrm{\pi}(\mathbf{X},\mathrm{\alpha})}-\frac{(1-D)[log(Y)-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})]}{(1-\mathrm{\pi}(\mathbf{X},\mathrm{\alpha}))}+{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta}).$Hence, the generic elements of the class of influence functions ${\mathcal{G}}^{(\mathrm{A}\mathrm{F}\mathrm{T})}$
${\mathcal{G}}^{(\mathrm{A}\mathrm{F}\mathrm{T})}=\left\{\mathrm{\phi}(Y,D,\mathbf{X}):\frac{\mathrm{\delta}}{w(Y)}\left[m(\mathbf{X},D,Y)-{\mathrm{\beta}}_{0}\right]+{\int}_{0}^{s}\frac{\mathrm{\kappa}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\},$can be written as $\{{\stackrel{\u02c6}{V}}_{0}(\mathrm{\theta})-{\stackrel{\u02c6}{V}}_{1}(\mathrm{\theta})+{\stackrel{\u02c6}{V}}_{2}(\mathrm{\theta})\}$ where
${V}_{0}(\mathrm{\theta},\mathrm{\alpha})=\frac{(1-D)\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})]}{(1-\mathrm{\pi}(\mathbf{X},\mathrm{\alpha}))w(Y)}+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{0}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}$
${V}_{1}(\mathrm{\theta},\mathrm{\alpha})=\frac{D\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})]}{\mathrm{\pi}(\mathbf{X},\mathrm{\alpha})w(Y)}+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{1}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}$
${V}_{2}(\mathrm{\theta},\mathrm{\alpha})=\frac{\mathrm{\delta}}{w(Y)}[{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})-\mathrm{\beta}]+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{2}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)},$with
${\mathrm{\kappa}}_{d}(t)=\mathbb{E}\left[\frac{I(D=d)\mathrm{\delta}I(Y>t)[log(Y)-{\mathrm{\mu}}_{d}(\mathbf{X},\mathrm{\theta})]{\int}_{t}^{Y}{S}_{C}(v)dv}{p(D=d|\mathbf{X},\mathrm{\alpha}){w}^{2}(Y)}\right],\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}d\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}0,1$
${\mathrm{\kappa}}_{2}(t)=\mathbb{E}\left[\frac{\mathrm{\delta}I(Y>t)[{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})-\mathrm{\beta}]{\int}_{t}^{Y}{S}_{C}(v)dv}{{w}^{2}(Y)}\right].$In order to show that ${\mathcal{G}}^{(\mathrm{A}\mathrm{F}\mathrm{T})}$ results in an unbiased estimator, we need to show that $\mathbb{E}[{V}_{0}(\mathrm{\theta},\mathrm{\alpha})]=\mathbb{E}[{V}_{1}(\mathrm{\theta},\mathrm{\alpha})]=\mathbb{E}[{V}_{2}(\mathrm{\theta},\mathrm{\alpha})]=0$. For the first expectation, we have
$\begin{array}{rl}\mathbb{E}[{V}_{0}(\mathrm{\theta},\mathrm{\alpha})]& =\mathbb{E}\left[\frac{(1-D)\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})]}{(1-\mathrm{\pi}(\mathbf{X},\mathrm{\alpha}))w(Y)}+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{0}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right]\\ & =\mathbb{E}\left[\frac{(1-D)\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})]}{(1-\mathrm{\pi}(\mathbf{X},\mathrm{\alpha}))w(Y)}\right]+\mathbb{E}\left[{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{0}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right]\end{array}$(14)The first expectation on the RHS of eq. (14) is
$\begin{array}{l}=\mathbb{E}\left[{\displaystyle {\int}_{0}^{\infty}{\displaystyle {\int}_{0}^{\infty}f}}(Y=y,A=a,\delta =1,D=1|X=x)\times \frac{D[\mathrm{log}(y)-{\mu}_{1}(x,\theta )]}{\pi (x,\alpha )w(Y)}dady\right]\\ =\mathbb{E}\left[{\displaystyle {\int}_{0}^{\infty}\frac{1}{\pi (x,\alpha )}}\frac{f(Y=y|D=1,X=x)}{{\mu}_{1}(x,\theta )}w(y)\frac{{\mu}_{1}(x,\theta )}{\mu (x,\theta )}\pi (x,\alpha )\right]\times \frac{[\mathrm{log}(y)-{\mu}_{1}(x,\theta )]}{w(y)}dy]\\ =\mathbb{E}\left[\frac{1}{\mu (x,\theta )}{\displaystyle {\int}_{0}^{\infty}f}(y|x,D=1)[\mathrm{log}(Y)-{\mu}_{1}(x,\theta )]dy\right]=0,\end{array}$where ${\mathrm{\mu}}_{d}(\mathbf{x},\mathrm{\theta})=\int yf(y|D=d,\mathbf{x})\phantom{\rule{thinmathspace}{0ex}}dy$ and $\mathrm{\mu}(\mathbf{x},\mathrm{\theta})={\int}_{0}^{\mathrm{\infty}}p({T}^{\mathrm{p}\mathrm{o}\mathrm{p}}\ge a|\mathbf{x},\mathrm{\theta})$. The second equality follows from
$p(Y\in (t,t+dt),\mathrm{\delta}=1|d,\mathbf{x})=\frac{f(t|d,\mathbf{x})w(t)dt}{{\mathrm{\mu}}_{d}(\mathbf{x},\mathrm{\theta})},$and
${p}_{LB}(D=1|\mathbf{X}=\mathbf{x})=\frac{{\mathrm{\mu}}_{1}(\mathbf{x},\mathrm{\theta})p({D}^{\mathrm{p}\mathrm{o}\mathrm{p}}=1|{\mathbf{X}}^{\mathrm{p}\mathrm{o}\mathrm{p}}=\mathbf{x})}{\mathrm{\mu}(\mathbf{x},\mathrm{\theta})},$where ${p}_{LB}(D=1|\mathbf{X}=\mathbf{x})$ is the propensity score estimated from the length-biased sample and $\mathrm{\pi}(\mathbf{x},\mathrm{\alpha})=p({D}^{\mathrm{p}\mathrm{o}\mathrm{p}}=1|{\mathbf{X}}^{\mathrm{p}\mathrm{o}\mathrm{p}}=\mathbf{x})$ is the true propensity score. The second expectation of the RHS of eq. (14) is also equal to zero since $\mathbb{E}[{M}_{C}(s)]=0$. Similarly, we can show that $E[{V}_{1}(\mathrm{\theta},\mathrm{\alpha})]=E[{V}_{2}(\mathrm{\theta},\mathrm{\alpha})]=0$.

It can be shown that ${V}_{0}(\mathrm{\theta},\mathrm{\alpha})$ and ${V}_{1}(\mathrm{\theta},\mathrm{\alpha})$ are uncorrelated. Hence the asymptotic variance of the estimator is given by
$\mathrm{\eta}(\mathrm{\theta},\mathrm{\alpha})=\mathbb{E}[{V}_{1}^{\ast 2}(\mathrm{\theta},\mathrm{\alpha})+{V}_{0}^{\ast 2}(\mathrm{\theta},\mathrm{\alpha})+{V}_{2}^{\ast 2}(\mathrm{\theta},\mathrm{\alpha})-{V}_{0}^{\ast 2}(\mathrm{\theta},\mathrm{\alpha}){V}_{2}^{\ast 2}(\mathrm{\theta})+{V}_{1}^{\ast 2}(\mathrm{\theta},\mathrm{\alpha}){V}_{2}^{\ast 2}(\mathrm{\theta},\mathrm{\alpha})].$where
$\mathbb{E}\left[{V}_{1}^{2}(\mathrm{\theta},\mathrm{\alpha})\right]=\mathbb{E}\left[{\left\{\frac{D\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})]}{\mathrm{\pi}(\mathbf{X},\mathrm{\alpha})w(Y)}+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{1}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\}}^{2}\right]$
$\mathbb{E}\left[{V}_{0}^{2}(\mathrm{\theta},\mathrm{\alpha})\right]=\mathbb{E}\left[{\left\{\frac{(1-D)\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})]}{(1-\mathrm{\pi}(\mathbf{X},\mathrm{\alpha}))w(Y)}+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{0}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\}}^{2}\right]$
$\mathbb{E}\left[{V}_{2}^{2}(\mathrm{\theta},\mathrm{\alpha})\right]=\mathbb{E}\left[{\left\{\frac{\mathrm{\delta}}{w(Y)}[{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})-\mathrm{\beta}]+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{2}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\}}^{2}\right],$and
$\begin{array}{rl}\mathbb{E}\left[{V}_{1}(\mathrm{\theta},\mathrm{\alpha}){V}_{2}(\mathrm{\theta},\mathrm{\alpha})\right]& =\mathbb{E}\left[\left\{\frac{D\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})]}{(\mathrm{\pi}(\mathbf{X},\mathrm{\alpha}))w(Y)}+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{1}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\}\right)\\ & \phantom{\rule{1em}{0ex}}\times \left(\left\{\frac{\mathrm{\delta}}{w(Y)}[{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})-\mathrm{\beta}]+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{2}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\}\right]\\ \mathbb{E}\left[{V}_{0}(\mathrm{\theta},\mathrm{\alpha}){V}_{2}(\mathrm{\theta},\mathrm{\alpha})\right]& =\mathbb{E}\left[\left\{\frac{(1-D)\mathrm{\delta}[log(Y)-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})]}{(1-\mathrm{\pi}(\mathbf{X},\mathrm{\alpha}))w(Y)}+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{0}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\}\right)\\ & \phantom{\rule{1em}{0ex}}\times \left(\left\{\frac{\mathrm{\delta}}{w(Y)}[{\mathrm{\mu}}_{1}(\mathbf{X},\mathrm{\theta})-{\mathrm{\mu}}_{0}(\mathbf{X},\mathrm{\theta})-\mathrm{\beta}]+{\int}_{0}^{s}\frac{{\mathrm{\kappa}}_{2}(t)d{M}_{C}(t)}{{S}_{C}(t){S}_{R}(t)}\right\}\right].\end{array}$

Table 5: Accelerated failure time simulation study when the propensity score is misspecified. PSR is the estimator based on eq. (9).

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