Inference in a survival cure model with mismeasured covariates using a simulation-extrapolation approach

SUMMARY

In many situations in survival analysis, it may happen that a fraction of individuals will never experience the event of interest: they are considered to be cured. The promotion time cure model takes this into account. We consider the case where one or more explanatory variables in the model are subject to measurement error, which should be taken into account to avoid biased estimators. A general approach is the simulation-extrapolation algorithm, a method based on simulations which allows one to estimate the effect of measurement error on the bias of the estimators and to reduce this bias. We extend this approach to the promotion time cure model. We explain how the algorithm works, and we show that the proposed estimator is approximately consistent and asymptotically normally distributed, and that it performs well in finite samples. Finally, we analyse a database in cardiology: among the explanatory variables of interest is the ejection fraction, which is known to be measured with error.

1. INTRODUCTION

When analysing time-to-event data, it often happens that a certain proportion of subjects will never experience the event of interest. For example, in medical studies where one is interested in the time until recurrence of a certain disease, it is known that, for some diseases, some patients will never suffer a relapse. In studies in econometrics on duration of unemployment, some unemployed people will never find a new job, and in sociological studies on the age at which a person marries, some people will stay unmarried for their whole life. Other examples can be found in finance, marketing, demography, and education, where each time there is a certain proportion of subjects whose time to event is infinite; they are said to be cured. Since classical survival models implicitly assume that all individuals will eventually experience the event of interest, they cannot be used in such contexts, as they would lead to incorrect results such as overestimation of the survival of the non-cured subjects. This is why specific models, called cure models, have been developed.

In order to model the impact of a set of covariates on the time-to-event variable, two main streams of cure models, as well as proposals that overarch both, can be found in the literature. The first is the so-called mixture cure model, which supposes that the conditional survival function is , where is the probability of being cured for a given vector of covariates , and is the conditional survival function of the non-cured subjects, where is another set of covariates, possibly with common components. This model has been studied by, among others, Boag (1949), Berkson & Gage (1952), Farewell (1982), Kuk & Chen (1992), Taylor (1995), Peng & Dear (2000), Sy & Taylor (2000), Peng (2003) and Lu (2008). A second class of models is based on an adaptation of the Cox (1972) model to allow for a cure fraction. It is called the class of promotion time cure models and supposes that

$$\begin{array}{c}S(t\mid x)=\exp \left\{-\theta (x)F(t)\right\},\hfill \end{array}$$ (1)

where is a proper baseline cumulative distribution function and captures the effect of the covariates on the conditional survival function. Unlike the mixture cure model, this formulation has a proportional hazards structure. One often chooses , where the first component of the -dimensional covariate is supposed to be 1, in order to include an intercept in the model. The Cox model without cure fraction does not include an intercept, since it supposes that tends to infinity when tends to infinity, and an intercept would therefore not be identifiable. References on the promotion time cure model include Yakovlev & Tsodikov (1996), Tsodikov (1998a,b, 2001), Chen et al. (1999), Ibrahim et al. (2001), Tsodikov et al. (2003), Zeng et al. (2006) and Carvalho Lopes & Bolfarine (2012).

In this paper we consider the promotion time cure model (1) in which we leave unspecified. We suppose that the survival time is subject to random right censoring, i.e., instead of observing we observe and , where the censoring time is independent of given . An immediate consequence is that for the censored observations, we do not observe whether they are cured or not cured, the latter observations being called susceptible.

In addition to being exposed to censoring, the data can also be subject to another type of incompleteness. As is often the case in practice, some continuous covariates are subject to measurement error. For instance, in medical studies the error can be caused by imprecise medical instruments, and in econometric studies variables like welfare or income often cannot be measured precisely. Although measurement error is rarely taken into account, ignoring it can lead to incorrect conclusions (Carroll et al., 2006). In order to deal with this measurement error, some assumptions about its form are necessary. We consider a classical additive measurement error model for the continuous covariates, so that we have, for the whole vector of covariates,

$$\begin{array}{c}W=X+U,\hfill \end{array}$$ (2)

where is the vector of observed covariates and is the vector of measurement errors. We further assume that is independent of and follows a continuous distribution with mean zero and known covariance matrix , where the elements of corresponding to covariates with no measurement error, including the noncontinuous covariates and possibly some continuous ones, are set to 0. It is also assumed that and are independent given . When is assumed to be normally distributed, (2) is the measurement error model studied, for example, by Cook & Stefanski (1994) and Ma & Yin (2008).

Methods designed to deal with measurement error in the covariates can be classified into structural modelling and functional modelling approaches (Carroll et al., 2006). In structural modelling, the distribution of the unobservable covariates must be modelled, usually parametrically, while in functional modelling, no assumptions are made regarding the distribution of . When the distributional assumptions are met, the approaches of the first type yield higher efficiency. However, an obvious advantage of methods of the second type is their robustness with respect to possible misspecification of the distribution of .

In this paper we use the so-called simulation-extrapolation, or simex, approach to correct for the measurement error. The basic idea of simex has two steps. In the first we consider increasing levels of measurement error, and simulate a large number of datasets for each level. At each level we estimate the vector of regression coefficients ignoring the measurement error. In the second step we extrapolate the estimators corresponding to the different levels of error to the situation where the covariates are observed without error. This algorithm, proposed by Cook & Stefanski (1994), has a number of advantages; see §6. The method has been considered in many different contexts. In survival analysis, it has been used in the Cox model (Carroll et al., 2006), the Cox model with nonlinear effect of mismeasured covariates in a 2006 Johns Hopkins University working paper by Crainiceanu et al., the multivariate Cox model (Greene & Cai, 2004) and the frailty model (Li & Lin, 2003), but, as far as we know, not in cure models. It has also been applied to nonparametric regression (Carroll et al., 1999) and to general semiparametric problems (Apanasovich et al., 2009) where if is mismeasured and is measured exactly, then the loglikelihood is of the form or , with an unknown function .

To the best of our knowledge, the problem considered in this paper has previously been addressed only by Ma & Yin (2008), who also studied a promotion time cure model with right-censored responses and mismeasured covariates. But instead of using the simex approach, they introduced a corrected score approach to deal with the measurement error in the covariates. Their approach yields consistent and asymptotically normal estimators when the measurement error variance is known and the error is normally distributed. However, their method only works for the specification , while the simex algorithm can be used for any parametric version of . Moreover, they do not study non-Gaussian measurement error in detail.

2. METHODOLOGY

Suppose that we have independent and identically distributed right-censored observations . We denote by the distinct ordered event times, so that . We use model (1), where we consider for some given function . Two examples are and . We present the simex algorithm for the case where the error is normally distributed. However, as mentioned in §1, this is not essential, as the algorithm below remains valid without any modification, as long as the extrapolation function is correctly specified.

The general idea of the simex algorithm consists in adding successively increasing amounts of artificial noise to the covariates subject to measurement error, estimating the model without taking the measurement error into account, and extrapolating back to the case of no measurement error. Two types of parameters have to be chosen: the levels of added noise and the number of simulations for each value of . Some common values are and (Cook & Stefanski, 1994; Carroll et al., 1996).

The simex algorithm for the promotion time cure model is as follows.

For , , and for , we generate independent and identically distributed independently of the observed data and construct for each individual , where is the known covariance matrix of the error term, as defined in §1. The covariance matrix of the contaminated is

$$\text{var}\left(W_{i,\lambda ,b}\mid X_i\right)=\text{var}\left(W_i\mid X_i\right)+\lambda V=V+\lambda V=(1+\lambda )V,$$

which converges to the zero matrix as converges to . We replace by in the promotion time cure model, giving

$$S\left(t\mid W_{i,\lambda ,b}\right)=\exp \left\{-F(t)\eta (W_{i,\lambda ,b}^{\text{T}}{\beta }_{\lambda })\right\}\text{.}$$

When the are known, this model is the standard promotion time cure model. We obtain the estimates of , by using a naive estimation method that does not take the measurement error into account.

For , , we obtain .

We then choose an extrapolant, e.g., linear, quadratic or fractional, for each parameter, i.e., for each element of the vector , as a function of the s: depending on a vector of parameters . In the case of the quadratic extrapolant, one obtains

$$\begin{array}{ccc}{\hat{\beta }}_{{\lambda }_k,d}=g_{{\beta }_d}({\gamma }_{{\beta }_d},{\lambda }_k)+{\pi }_{d,k}\hfill & \hspace{0.17em}=\hfill & {\gamma }_{{\beta }_d,1}+{\gamma }_{{\beta }_d,2}{\lambda }_k+{\gamma }_{{\beta }_d,3}{\lambda }_k^2+{\pi }_{d,k}\hfill \\ & & \hspace{0.17em}(d=1,\dots ,D;k=1,\dots ,K),\hfill \end{array}$$

where are the error terms in the extrapolant model, assumed to be independent and to have mean zero. We fit these parametric models for each in order to obtain . In practice, this function is often an approximation of the true extrapolation function, which will then yield an estimator that converges in probability to some constant approximately equal to the true parameter (Cook & Stefanski, 1994). There are some cases in semiparametric models where the exact extrapolant is known: Cox regression, with and normally distributed and homoscedastic (Prentice, 1982), and the partially linear model , , with independent (Liang et al., 1999).

Finally, we obtain the simex estimated values

$$\begin{array}{ccc}rCl{\hat{\beta }}_{\text{SIMEX}}\hfill & \hspace{0.17em}=\hfill & \hspace{0.17em}\underset{\lambda \to -1}{lim}{g}_{\beta }({\hat{\gamma }}_{\beta },\lambda )\text{.}\hfill \end{array}$$

We use the results of Zeng et al. (2006) and Ma & Yin (2008) to estimate the model parameters and when there is no measurement error in the covariates. They show that the loglikelihood of the promotion time cure model without measurement error is

$$\begin{array}{cc}\ell =\sum _{i=1}^n\hfill & [{\delta }_{i}I(Y_i<\mathrm{\infty })\left\{-F(Y_i)\eta (X_i^{\mathrm{T}}\beta )+\log p_i+\log \eta (X_i^{\mathrm{T}}\beta )\right\}\hspace{0.17em}\hfill \\ & \hspace{0.17em}+(1-{\delta }_i)I(Y_i<\mathrm{\infty })\left\{-F(Y_i)\eta (X_i^{\mathrm{T}}\beta )\right\}-I(Y_i=\mathrm{\infty })\eta (X_i^{\mathrm{T}}\beta )],\hfill \end{array}$$ (3)

where is the jump size of at . We use instead of the particular case considered by the authors. As Zeng et al. (2006) explain, it can be shown that the nonparametric maximum likelihood estimator for is a function with point masses at the distinct observed failure times only. If denotes the jump size of at , then . Moreover, the authors also explain that, in order for this semiparametric model to be identifiable in , we need a threshold , called the cure threshold, such that all censored individuals with a censoring time greater than this threshold are treated as if they were known to be cured, i.e., . In practice, the estimated baseline cumulative distribution function is forced to be beyond the largest observed failure time, . This implies that no event can occur after this time: the cure threshold is then determined to be .

The parameters can then be estimated by solving the score equations related to the likelihood in which the baseline cumulative distribution function is replaced by a step function.

If interest also lies in estimating the baseline cumulative distribution function , exactly the same simex procedure can be applied to the , yielding the . In order to ensure that their sum is equal to , each of them is divided by their sum: . Finally, we obtain .

3. ASYMPTOTIC PROPERTIES

We present some theorems regarding consistency and asymptotic normality of the simex estimators of the regression parameters and the baseline cumulative distribution function . Theorem 1 states their consistency; its proof can be found in the Supplementary Material. Theorem 2 establishes their asymptotic normality and is proved in the Appendix. Both results rely on the assumption that the true extrapolation function is known, which is rarely the case in practice. As explained by Cook & Stefanski (1994) and mentioned in §2, since the extrapolation function used in the algorithm is often an approximation to the true one, we will obtain an estimator which converges in probability to some constant that is approximately equal to the true parameter. This is sometimes called approximate consistency. In this case, in all the results that follow, and are replaced in the results by and , the limiting values with this extrapolant. In the parametric case, when is scalar, with being the measurement error variance, the bias of a polynomial extrapolant of order is , see Cook & Stefanski (1994, p. 1317), although they only consider .

Here, we assume that the that are generated in the simulation step follow a truncated Gaussian distribution with large truncation limits, which will always be the case in practice. We also assume that the expectation of the loglikelihood has a unique maximizer, whether or not there is measurement error in the covariates.

Theorem 1.

Under the regularity conditions (C1)–(C4) of Zeng et al. (2006), by replacing that appears there by () for each , if the measurement error variance and the true extrapolant function are known, then, with probability 1,

$$\begin{array}{c}c\Vert {\hat{\beta }}_{\mathrm{S}\mathrm{I}\mathrm{M}\mathrm{E}\mathrm{X}}-{\beta }_{\mathrm{T}\mathrm{R}\mathrm{U}\mathrm{E}}\Vert \to 0,\underset{t\in {\mathbb{R}}^{+}}{sup}|{\hat{F}}_{\mathrm{S}\mathrm{I}\mathrm{M}\mathrm{E}\mathrm{X}}(t)-F_{\mathrm{T}\mathrm{R}\mathrm{U}\mathrm{E}}(t)|\to 0,n\to \mathrm{\infty }\text{.}\hfill \end{array}$$

Theorem 2.

Under the regularity conditions (C1)–(C4) of Zeng et al. (2006), by replacing that appears there by () for each , if the measurement error variance and the true extrapolant function are known, then converges in distribution to , where is given by (A1) in the Appendix. Moreover, converges weakly to a zero-mean Gaussian process whose covariance function is given by (A2) in the Appendix.

The regularity conditions (C1)–(C4) of Zeng et al. (2006) are the usual constraints pertaining to the independence of the right-censoring variable, the boundedness of the covariate, the compactness of the set of possible values for , the differentiability of the baseline cumulative distribution function and the monotonicity and differentiability of the link function .

The variance of the simex estimator can be estimated using the method introduced by Stefanski & Cook (1995) and summarized, for example, in Carroll et al. (2006). The variance estimator can be computed as , where is the extrapolation function corresponding to , where is the estimated covariance matrix of when using the variance estimator corresponding to the naive estimation method, and is the extrapolation function corresponding to , i.e., the empirical covariance matrix of .

4. SIMULATION STUDIES

4.1. Settings

The objective of our first three simulation studies is to investigate the properties of the proposed estimator in finite samples and to compare it with the naive method based on (3), which does not take measurement error into account, and the corrected score method of Ma & Yin (2008).

In the next three subsections, we focus on our proposed estimator. In §4.5 and in the Supplementary Material, we present the results of simulation studies for examining the effect of the choice of the extrapolation function and of the grid of values of on the simex estimator. Subsections 4.6 and 4.7 contain results pertaining to the robustness of the simex estimator with respect to misspecification of the error distribution and variance.

An extensive simulation study investigating the robustness of both the simex algorithm and the corrected score approach of Ma & Yin (2008) with respect to the assumptions of normal distribution and known variance of the error can be found in a 2016 article by A. Bertrand et al., available at http://hdl.handle.net/2078.1/171508.

In the following, for the simex algorithm, we used and except in the simulations regarding the choice of this grid, and a quadratic extrapolant except in §4.5. The variables appearing in the simulation step of the algorithm are always taken to be Gaussian. For each setting, 500 simulated datasets were analysed.

4.2. One mismeasured covariate

The first set of simulation studies that we conduct is the first in Ma & Yin (2008). They assume that the follow-up is infinite, and that the censoring distribution and the failure distribution have infinite support, so that each individual is known to be either cured, dead or censored, when , or , respectively. The censored individuals for whom are actually cured. In such a case, a cure threshold is not needed for estimation. Each subject has a probability of of having an infinite censoring time. Because of the infinite follow-up, this does not correspond to a realistic case, but is useful for assessing the proposed method in an ideal situation.

The model under study is

$$\begin{array}{ccc}rClS(t\mid X_1,X_2)\hfill & \hspace{0.17em}=\hfill & \exp \left\{-\exp \left({\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2\right)F(t)\right\}\hfill \end{array}$$

for and we generate the data from this model with , , , and , ; is subject to measurement error so that is observed, where , with the only nonzero element of . Moreover, the censoring time is independent of and of given , and the finite censoring times follow an exponential distribution with mean .

Eight different settings are obtained by considering two possible values for sample size, or , variance of the measurement error, or , and mean of the finite censoring times, or . The average cure rate, i.e., the rate of observations for which , is , the average proportion of subjects with , considered cured for the estimation, is ; and the average censoring rate is when and when .

The results for the four settings with are summarized in Table 1, while those corresponding to can be found in the Supplementary Material.

Table 1.

Empirical bias, empirical and estimated variances, coverages and mean squared errors for the settings with one mismeasured covariate, when

			Ma & Yin method			Naive method			Simex method
		Estimate
200	0.1	Bias	-1.0	3.2	-0.2	4.3	-9.2	0.2	-0.4	1.5	-0.1
		Emp. var.	5.3	13.2	4.1	4.7	9.9	4.0	5.2	12.8	4.1
		Est. var.	5.3	13.0	3.6	4.6	9.8	3.6	5.2	12.4	3.6
		95% cv	94.6	94.8	93.6	93.6	93.4	93.8	94.2	94.2	93.8
		MSE	5.3	13.3	4.1	4.8	10.8	4.0	5.2	12.8	4.1
200	0.2	Bias	-3.0	8.5	-0.5	14.2	-31.5	.7	3.6	-7.6	0.1
		Emp. var.	6.9	22.6	4.4	4.1	7.5	4.0	5.4	14.3	4.2
		Est. var.	7.4	22.7	3.9	4.0	7.4	3.6	5.2	12.7	3.7
		95% cv	96.0	96.8	93.2	88.4	78.8	93.4	93.8	92.4	93.0
		MSE	7.0	23.4	4.4	6.1	17.5	4.0	5.5	14.8	4.2
300	0.1	Bias	-1.1	3.7	-1.4	4.2	-8.4	-1.1	-0.5	2.3	-1.4
		Emp. var.	3.9	9.4	2.6	3.4	7.1	2.5	3.8	9.0	2.6
		Est. var.	3.5	8.7	2.4	3.1	6.5	2.4	3.4	8.3	2.4
		95% cv	95.6	94.4	96.6	93.4	93.2	96.6	95.4	94.2	96.6
		MSE	3.9	9.5	2.6	3.6	7.8	2.5	3.8	9.1	2.6
300	0.2	Bias	-2.1	6.8	-1.8	14.2	-31.0	-0.6	3.6	-6.9	-1.3
		Emp. var.	4.9	14.5	2.7	3.0	5.1	2.5	3.9	9.7	2.6
		Est. var.	4.6	13.9	2.6	2.7	4.9	2.4	3.4	8.4	2.5
		95% cv	96.6	95.8	95.8	85.2	71.6	95.4	93.8	93.4	95.8
		MSE	5.0	14.9	2.7	5.0	14.7	2.5	4.1	10.2	2.6

Emp. var., empirical variance; Est. var., estimated variance; 95% cv, coverage probabilities of 95% confidence intervals computed based on the asymptotic normal distribution; MSE, mean squared error. All numbers were multiplied by 100.

The empirical and estimated variances are always quite close to each other, while both the corrected score and the simex approaches yield coverage probabilities close to the nominal . Compared to the naive estimation method, both correction methods decrease the bias in the intercept and the parameter corresponding to the mismeasured covariate, but at the cost of a larger variance. Although the simex algorithm and the method of Ma & Yin (2008) cannot really be distinguished on the basis of the bias, the former leads to a smaller variance for and when , and to similar variances when . This results in a mean squared error which is, when , the smallest for simex, compared to the naive and corrected score methods. When the measurement error variance is smaller, the naive method yields a smaller mean squared error than both correction methods. This is to be expected since bias correction methods have a larger variance than the naive method.

4.3. Two mismeasured covariates

We now introduce, in the previous setting, an additional covariate with measurement error. In this case, , . We generate the data with , , , , and , , ; and are subject to measurement error so that and are observed, where , and and are uncorrelated.

The average censoring rate is when and when . The average proportion of subjects considered cured for the estimation is , while the average cure rate is .

The results for the four settings with are summarized in Table 2, while those corresponding to can be found in the Supplementary Material.

Table 2.

Empirical bias, empirical and estimated variances, coverages and mean squared errors for the settings with two mismeasured covariates, when

	Ma & Yin method				Naive method				Simex method
Estimate
, ,
Bias	0.1	2.0	1.6	-1.2	4.9	-10.7	-0.7	-0.6	0.7	0.1	1.2	-1.1
Emp. var.	7.0	15.4	1.7	4.0	6.1	11.5	1.6	3.9	6.8	14.8	1.7	4.0
Est. var.	6.1	13.8	1.5	3.9	5.2	10.2	1.4	3.8	5.9	13.1	1.5	3.9
95% cv	93.2	93.8	95.0	94.8	92.2	93.4	93.0	95.0	93.0	94.6	95.0	95.0
MSE	7.0	15.4	1.7	4.0	6.4	12.6	1.6	3.9	6.8	14.8	1.7	4.0
, ,
Bias	-1.9	8.1	3.0	-1.8	13.3	-33.4	-5.6	0.3	4.7	-9.4	0.7	-0.9
Emp. var.	9.8	27.9	2.2	4.6	5.6	8.8	1.5	3.9	7.6	17.3	1.9	4.3
Est. var.	8.6	25.2	1.9	4.3	4.6	7.7	1.3	3.8	6.0	13.6	1.6	4.1
95% cv	94.2	96.2	95.8	94.2	88.8	75.8	90.2	94.0	91.8	92.4	94.8	94.2
MSE	9.8	28.6	2.3	4.6	7.4	20.0	1.8	3.9	7.9	18.2	1.9	4.3
, ,
Bias	2.0	0.8	1.2	-1.3	6.6	-11.5	-1.0	-0.8	2.4	-0.7	0.9	-1.2
Emp. var.	4.1	9.9	0.9	2.4	3.6	7.5	0.9	2.3	4.0	9.6	0.9	2.4
Est. var.	3.9	9.0	1.0	2.6	3.4	6.8	0.9	2.5	3.8	8.6	1.0	2.6
95% cv	94.4	94.2	96.4	95.2	92.4	93.6	95.8	95.6	93.8	94.2	96.4	95.2
MSE	4.1	9.9	1.0	2.4	4.0	8.9	0.9	2.3	4.1	9.6	0.9	2.4
, ,
Bias	0.8	4.7	2.2	-1.6	14.9	-34.0	-5.9	0.2	6.6	-10.1	0.5	-1.0
Emp. var.	5.3	16.3	1.2	2.6	3.2	5.9	0.8	2.4	4.4	11.2	1.1	2.5
Est. var.	5.2	14.9	1.2	2.8	3.0	5.1	0.9	2.5	3.9	8.9	1.1	2.7
95% cv	95.0	95.6	96.2	95.4	86.8	66.2	90.6	94.8	90.6	91.2	95.8	95.0
MSE	5.3	16.6	1.3	2.7	5.4	17.4	1.2	2.4	4.8	12.2	1.1	2.5

Emp. var., empirical variance; Est. var., estimated variance; 95% cv, coverage probabilities of 95% confidence intervals computed based on the asymptotic normal distribution; MSE, mean squared error. All numbers were multiplied by 100.

The three methods perform similarly as far as is concerned. None clearly has lowest bias overall. For larger values of the measurement error variances, the method of Ma & Yin (2008) is the best for and , while simex is preferred for . However, when also taking the variance of the estimators into account, the mean squared error indicates that the naive method is preferable for small values of and , while simex outperforms the corrected score approach and the naive method for larger values of and .

4.4. A more realistic case

In practice, neither the failure times nor the censoring times can be infinite. Consequently, none of the cured subjects are observed to be cured. The cure threshold, i.e., the largest observed event time, as mentioned in §2, is thus needed for the estimation of the model parameters. Moreover, depending on the context, the censoring and cure rates can be much larger than the values considered in the two previous settings. We therefore consider the model

$$\begin{array}{c}cS(t\mid X_1,X_2)=\exp \left\{-\exp (-0.3+X_1-0.5X_2)F(t)\right\},t>0,\hfill \end{array}$$

where , ; is subject to measurement error so that is observed, where .

For the baseline cumulative distribution function , we use an exponential distribution with mean which is truncated at . Consequently, the maximum event time is . The censoring times are independent of the covariates and are generated from an exponential distribution with mean , which is truncated at .

Four different settings are obtained by considering two possible values for sample size, or , and variance of the measurement error, or . The average censoring rate is and the average proportion of cured subjects is , while the average observed cure rate is . The results are summarized in Table 3.

Table 3.

Empirical bias, empirical and estimated variances, coverages and mean squared errors for the realistic settings with one mismeasured covariate

			Ma & Yin method			Naive method			Simex method
		Estimate
200	0.1	Bias	-2.1	1.3	-2.2	3.8	-10.6	-2.1	-1.4	-0.3	-2.2
		Emp.	12.6	21.2	6.7	11.2	16.2	6.7	12.4	20.6	6.7
		Est.	11.4	22.1	6.3	9.8	16.7	6.2	10.7	19.1	6.3
		95% cv	94.4	97.2	94.8	94.2	95.4	95.0	93.4	96.2	94.8
		MSE	12.6	21.2	6.8	11.4	17.3	6.7	12.4	20.6	6.8
200	0.25	Bias	-5.6	9.3	-2.6	19.9	-42.9	-1.9	7.4	-18.6	-2.0
		Emp.	18.5	45.0	7.1	9.4	10.0	6.7	12.1	20.7	6.8
		Est.	21.8	57.8	6.9	7.9	10.5	6.2	9.7	14.6	6.3
		95% cv	97.2	97.4	96.0	88.6	74.2	95.4	92.7	88.6	95.1
		MSE	18.8	45.9	7.2	13.3	28.4	6.7	12.6	24.2	6.8
300	0.1	Bias	-3.1	2.5	-1.3	2.5	-9.4	-1.2	-2.5	1.3	-1.3
		Emp.	10.0	16.7	4.3	9.2	12.8	4.3	9.8	16.4	4.3
		Est.	7.9	14.4	4.1	7.2	11.0	4.1	7.6	12.5	4.1
		95% cv	93.2	94.8	95.6	93.0	92.4	95.8	92.8	93.8	95.6
		MSE	10.1	16.8	4.4	9.3	13.6	4.3	9.9	16.4	4.4
300	0.25	Bias	-6.8	10.5	-1.9	18.7	-41.2	-1.1	6.0	-15.8	-1.4
		Emp.	15.3	37.1	4.7	7.4	8.2	4.3	9.9	17.4	4.4
		Est.	14.4	37.5	4.6	5.6	6.9	4.1	6.9	9.7	4.2
		95% cv	96.0	96.8	95.6	85.0	63.2	95.8	89.7	83.9	95.6
		MSE	15.8	38.2	4.7	10.9	25.2	4.3	10.3	19.9	4.4

Emp., empirical variance; Est., estimated variance; 95% cv, coverage probabilities of 95% confidence intervals computed based on the asymptotic normal distribution; MSE, mean squared error. All numbers were multiplied by 100.

As might be expected, differences between the methods appear only for and, in some cases, for . In terms of the bias, both correction methods are preferable to the naive one. When , simex is the best for , while the method of Ma & Yin (2008) is the best for this parameter when . When the mean squared error of the naive estimator is the smallest. When , the mean squared error of the simex estimator is the smallest, while the method of Ma & Yin (2008) yields the largest mean squared error.

4.5. Impact of the choice of the extrapolation function

In order to compare the performance of simex with different choices of extrapolant, we consider the setting in the previous subsection, and we estimate the model parameters using, in addition to the quadratic extrapolant, a linear and a cubic extrapolant. Table 4 reports the results.

Table 4.

Empirical bias, empirical and estimated variances, coverages and mean squared errors for the realistic settings, for simex with three different extrapolation functions

			Simex (linear)			Simex (quadratic)			Simex (cubic)
		Estimate
200	0.1	Bias	0.0	-3.0	-2.2	-1.4	-0.3	-2.2	-1.4	0.0	-2.2
		Emp. var.	12.0	19.2	6.7	12.4	20.6	6.7	12.5	21.2	6.7
		Est. Var.	10.2	17.9	6.3	10.7	19.1	6.3	11.0	19.4	6.3
		95% cv	93.6	95.6	94.8	93.4	96.2	94.8	93.2	95.0	94.4
		MSE	12.0	19.3	6.8	12.4	20.6	6.8	12.5	21.2	6.8
200	0.25	Bias	14.5	-32.5	-1.9	7.4	-18.6	-2.0	3.6	-10.6	-1.9
		Emp. var.	10.4	14.1	6.7	12.1	20.7	6.8	13.8	26.5	6.8
		Est. Var.	8.8	11.2	6.2	9.7	14.6	6.3	16.0	18.2	6.3
		95% cv	90.9	80.8	95.4	92.7	88.6	95.1	93.1	89.9	95.7
		MSE	12.5	24.7	6.7	12.6	24.2	6.8	14.0	27.6	6.9
300	0.1	Bias	-1.1	-1.6	-1.3	-2.5	1.3	-1.3	-2.5	1.9	-1.2
		Emp. var.	9.5	15.2	4.3	9.8	16.4	4.3	10.8	17.1	4.3
		Est. Var.	8.0	11.8	4.1	7.6	12.5	4.1	7.8	12.8	4.1
		95% cv	92.8	93.8	95.6	92.8	93.8	95.6	92.8	93.6	95.4
		MSE	9.5	15.2	4.3	9.9	16.4	4.4	10.8	17.1	4.4
300	0.25	Bias	13.1	-30.0	-1.2	6.0	-15.8	-1.4	1.9	-6.8	-1.6
		Emp. var.	8.4	11.7	4.4	9.9	17.4	4.4	11.4	22.4	4.5
		Est. Var.	8.1	7.4	4.1	6.9	9.7	4.2	8.1	12.1	4.2
		95% cv	87.2	72.9	95.6	89.7	83.9	95.6	91.8	85.2	95.2
		MSE	10.1	20.7	4.4	10.3	19.9	4.4	11.4	22.9	4.5

Emp. var., empirical variance; Est. var., estimated variance; 95% cv, coverage probabilities of 95% confidence intervals computed based on the asymptotic normal distribution; MSE, mean squared error. All numbers were multiplied by 100.

In these simulations, varying the extrapolation function used in the simex algorithm has no effect on the estimation of . This is not the case for the other two parameters. When the measurement error variance is rather low, the smallest bias for is obtained by the linear extrapolant, while there is no clear conclusion regarding . However, for the largest variance, the higher the extrapolation order, the smaller the bias. In terms of mean squared error, the lowest order of extrapolation yields the best results for , but the differences among extrapolants are quite limited. For , the mean squared error increases with the order of the extrapolation when the measurement error variance is low, while the quadratic extrapolant outperforms the other ones when the variance is larger.

These findings are consistent with the general behaviour of the simex estimator: the lower the order of the extrapolant, the more conservative the correction (Cook & Stefanski, 1994). Consequently, the extrapolation function has to be chosen in the context of the bias-variance trade-off; the quadratic extrapolant has seemed to be a good compromise in many cases (Cook & Stefanski, 1994; Carroll et al., 2006) and is widely used (He et al., 2007; Li & Lin, 2003).

4.6. Robustness with respect to misspecification of the error distribution

In this subsection, the effect of a misspecification of the measurement error distribution is investigated through another simulation study, again using the settings of §4.4 with and the quadratic extrapolant. The measurement error is now generated using two distributions which are very different from the Gaussian: a uniform and a chi-squared distribution, with standard deviation and , assumed to be known. When , we see in Table 5 that, for this setting, the misspecification has no impact on the mean squared error; the impact on the bias is very limited except, to some extent, for with the chi-squared distribution. For the largest value of the measurement error standard deviation, , the estimation of is not influenced by the true distribution. The mean squared errors of and increase slightly. With the uniform distribution, the biases of these two parameters stay nearly constant, while with a chi-squared error, they increase quite markedly.

Table 5.

Empirical bias, empirical and estimated variances, coverages and mean squared errors for the simulations investigating the robustness of simex with respect to a misspecification of the error distribution

True distribution	Estimate
Gaussian	Bias	-0.014	-0.3	-2.2	7.2	-18.2	-2.0
	Emp. var.	12.4	20.6	6.7	12.1	21.0	6.8
	Est. var.	10.8	19.1	6.3	9.7	14.8	6.3
	95% cv	93.4	95.8	94.8	92.7	88.8	95.1
	MSE	12.4	20.6	6.8	12.7	24.3	6.8
Chi-squared	Bias	-0.4	-2.0	-2.1	11.6	-25.1	-2.0
	Emp. var.	12.8	21.0	6.8	12.4	19.9	7.0
	Est. var.	10.8	18.6	6.3	9.2	12.8	6.3
	95% cv	93.2	93.8	94.0	90.2	83.3	94.4
	MSE	12.8	21.0	6.9	13.8	26.2	7.0
Uniform	Bias	-1.0	-0.7	-2.0	7.1	-17.7	-1.9
	Emp. var.	12.6	20.2	6.9	13.5	21.9	7.1
	Est. var.	10.7	19.1	6.3	82.6	15.2	6.3
	95% cv	93.1	95.0	94.4	91.3	86.1	94.2
	MSE	12.7	20.2	7.0	14.0	25.0	7.1

Emp. var., empirical variance; Est. var., estimated variance; 95% cv, coverage probabilities of 95% confidence intervals computed based on the asymptotic normal distribution; MSE, mean squared error. All numbers were multiplied by 100.

4.7. Robustness with respect to misspecification of the error variance

We now investigate the behaviour of our estimator when the measurement error variance is misspecified. More precisely, we consider the same setting as in §4.4 with , where the error is simulated with standard deviations and . However, in the estimation process, the variance is misspecified as for and as for . The extrapolation function is quadratic.

The results, reported in Table 6, show that, in these simulations, a misspecification of the measurement error variance has no impact on the estimation of . Both and are influenced, in terms of both the bias and the mean squared error. The latter increases with the value of the variance assumed in the estimation procedure, although this increase is less marked when switching from the underspecified variance to the true one, compared to when switching from the true variance to the overspecified one. When the true value of the measurement error variance is low, the lowest bias is obtained when the correct variance is assumed; when the true variance is higher, the bias decreases when the specified variance increases.

Table 6.

Empirical bias, empirical and estimated variances, coverages and mean squared errors for the simulations investigating the robustness of simex with respect to a misspecification of the error variance

0.05	Bias	2.5	-8.1	-2.1	0.20	11.6	-26.2	-2.1
	Emp. var.	11.5	17.1	6.7		11.4	17.2	6.7
	Est. Var.	11.2	17.3	6.2		9.2	13.8	6.3
	95% cv	94.4	95.8	94.8		92.3	86.3	95.4
	MSE	11.5	17.8	6.7		12.8	24.0	6.8
0.1	Bias	-01.4	-0.3	-2.2	0.25	7.2	-18.2	-2.0
	Emp. var.	12.4	20.6	6.7		12.1	21.0	6.8
	Est. Var.	10.8	19.1	6.3		9.7	14.8	6.3
	95% cv	93.4	95.8	94.8		92.7	88.8	95.1
	MSE	12.4	20.6	6.8		12.7	24.3	6.8
0.15	Bias	-8.0	13.0	-2.2	0.30	2.9	-9.3	-2.1
	Emp. var.	14.1	26.6	6.8		13.3	26.5	7.0
	Est. Var.	11.7	21.8	6.3		10.1	15.9	6.3
	95% cv	93.2	92.4	94.6		92.9	88.8	94.8
	MSE	14.8	28.3	6.8		13.4	27.4	7.0

Emp. var., empirical variance; Est. var., estimated variance; 95% cv, coverage probabilities of 95% confidence intervals computed based on the asymptotic normal distribution; MSE, mean squared error. All numbers were multiplied by 100.

5. AORTIC INSUFFICIENCY DATABASE

We illustrate our methodology on data from patients suffering from aortic insufficiency, a cardiovascular disease. Between 1995 to 2013, 393 patients underwent echocardiography for severe aortic insufficiency at the Brussels Saint-Luc University Hospital, Belgium. These data were collected by one of the authors, C. de Meester, and include information from the diagnosis of the pathology, between 1981 and 2013. %The main event of interest in this study is death from AI. Although aortic insufficiency can be lethal, it is known that a proportion of patients will never die from it. Since the patients considered in this study have no or limited other known morbidity, those who survive for a sufficiently long period after the diagnosis can be considered as long-term survivors. % from their heart disease. The main objective of this study is to investigate the link between the ejection fraction measured at baseline and the survival of the patients. The ejection fraction is the ratio of the difference between the end-diastolic and end-systolic volumes over the end-diastolic volume and therefore measures the fraction of blood which leaves the heart each time it contracts. It is typically high for healthy individuals and is one of the main indicators appearing in the guidelines used to decide whether a patient should be operated on (Bonow et al., 1998; Vahanian et al., 2007). However, the ejection fraction is measured with error (Otterstad et al., 1997), and this should be taken into account when evaluating its impact on survival.

After a median follow-up of years, only 58 patients had died, and the Kaplan–Meier estimate of the survival curve for these patients shows a clear plateau after about 17 years, as can be seen in Fig. 1. As explained in §2, the cure threshold is the largest observed event time: all patients surviving up to years are considered as not being at risk of dying of their aortic insufficiency. To take into account the presence of cured patients and the measurement error in the covariate of interest, we apply the promotion time cure model estimated with the simex algorithm with the quadratic extrapolant. We compare our results with those obtained by the method of Ma & Yin (2008), as well as with those from a naive promotion time cure model ignoring measurement error. In our data the ejection fraction takes values between 019 and 084, median 056, and based on previous work (Otterstad et al., 1997) we consider a standard deviation of the measurement error of 005 and 010. Our model is adjusted for other patient characteristics, measured without error, namely: gender, with male; age at diagnosis, with median 52 and range 17–88, standardized for the analysis; and surgery strategy chosen by the cardiologist for this patient, with of the patients having no surgery, surgery within the first three months and surgery after the first three months. See Table 7.

Fig. 1.

Kaplan–Meier estimate (solid) and 95% pointwise confidence limits (dashed) of the survival curve for the patients from the aortic insufficiency database.

Table 7.

Regression coefficient estimates, estimated standard deviations and confidence intervals based on the asymptotic normal distribution for the aortic insufficiency data

Estimate	EF	Gender	Age	Surgery	Surgery
			(standardized)	months	months
Naive	-1.22	0.63	1.23	0.14	-0.73
(Estimated SD)	(1.38)	(0.28)	(0.20)	(0.37)	(0.40)
95% C.I. (lower bound)	-3.92	0.08	0.85	-0.60	-1.51
95% C.I. (upper bound)	1.48	1.19	1.62	0.87	0.05
Ma & Yin method ()	-1.50	0.63	1.23	0.12	-0.73
(Estimated SD)	(1.69)	(0.29)	(0.20)	(0.38)	(0.40)
95% C.I. (lower bound)	-4.80	0.07	0.85	-0.62	-1.51
95% C.I. (upper bound)	1.81	1.19	1.61	0.86	0.04
Ma & Yin method ()	-3.94	0.63	1.17	-0.03	-0.77
(Estimated SD)	(4.21)	(0.31)	(0.21)	(0.43)	(0.40)
95% C.I. (lower bound)	-12.19	0.02	0.76	-0.89	-1.55
95% C.I. (upper bound)	4.31	1.24	1.58	0.82	0.01
Simex ()	-1.45	0.63	1.23	0.12	-0.73
(Estimated SD)	(1.48)	(0.29)	(0.20)	(0.38)	(0.40)
95% C.I. (lower bound)	-4.35	0.07	0.85	-0.62	-1.50
95% C.I. (upper bound)	1.45	1.19	1.61	0.86	0.05
Simex ()	-2.09	0.62	1.22	0.10	-0.73
(Estimated SD)	(1.71)	(0.29)	(0.20)	(0.38)	(0.39)
95% C.I. (lower bound)	-5.44	0.06	0.84	-0.64	-1.50
95% C.I. (upper bound)	1.26	1.19	1.61	0.85	0.04

SD, standard deviation; C.I., confidence interval; EF, ejection fraction.

We also estimated the model with the logarithm of the ejection fraction instead of this variable in its natural scale, which allows one to take into account the potential case in which the measurement error would be multiplicative rather than additive. The qualitative conclusions are identical: both correction methods yield a larger negative estimated effect of the ejection fraction, compared to the naive method. The estimated coefficients of the covariates without measurement error hardly change.

The parameter most affected by taking the measurement error into account is the coefficient of the ejection fraction. Both methods correct in the same direction. However, the simex approach with a quadratic extrapolant yields a more conservative correction, as reported by Carroll et al. (2006). This smaller correction is associated with a smaller estimated standard deviation, which is consistent with what was observed in the simulation study, and hence a narrower confidence interval. Correcting for the measurement error increases the size of the estimated effect of the ejection fraction. In the promotion time cure model, a negative coefficient implies an increase in the cure probability and in survival at all times, when the value of the covariate increases. The results hence indicate that, all other things being equal, the higher the ejection fraction, the higher the cure probability and the better the survival for the susceptible subjects. This is consistent with expectations and with existing guidelines, which advise performing surgery when the ejection fraction is below a given threshold (Bonow et al., 1998; Vahanian et al., 2007). As far as the surgery strategy is concerned, our results indicate better survival for patients having undergone surgery more than three months after the discovery of the disease, and the worst for those with surgery within the first three months, although the effect is reduced when measurement error is taken into account. These results should, however, be interpreted carefully. First, the patients having undergone surgery more than three months after the discovery of the disease have, by definition, lived at least three months after the discovery of their disease. Second, and probably more importantly, the two groups are not comparable at baseline, as the decision of whether to operate immediately was taken according to existing guidelines, based on the prognosis of the patients. Therefore, the worse survival for patients having surgery within the first three months can be explained by the fact that 80% of these patients met at least one of the guideline criteria for surgery, including the presence of symptoms in 62% of them. The survival of severe aortic insufficiency patients with symptoms is worse than for those without (Dujardin et al., 1999), as also observed in post-operative survival (Klodas et al., 1997).

We also considered introducing an interaction between the ejection fraction level and the surgery strategy. However, an interaction term between a mismeasured covariate and a correctly measured one is actually a mismeasured covariate whose variance depends on the latter covariate. The simex algorithm can easily be tuned to accommodate such a case and yield parameter estimates; however, our asymptotic results do not then hold. Nevertheless, the bootstrap could be used to perform inference on the estimated parameters. It is unclear how to modify the method of Ma & Yin (2008) to allow a dependence between an error term and a covariate. In Table 8, we report the results for this model. When the measurement error is not taken into account, one of the interaction terms is significant, and its introduction modifies the significance of other parameters. The estimated parameters obtained with simex are also reported: as before, the correction leads to estimated effects of higher size for the mismeasured covariates, but also for the covariates included in the interaction terms. We observe a negative estimated effect of the ejection fraction on the survival for patients with surgery after more than three months, as well as for those without surgery: this means, as in the previous model, that a higher value of the ejection fraction is associated with better survival. This effect is less impressive for patients without surgery. According to the naive estimates, there is no significant effect of the ejection fraction in the patients having undergone surgery within the first three months, probably because these patients are operated on due to the presence of symptoms, as explained in the previous paragraph, independently of their ejection fraction.

Table 8.

Regression coefficient estimates, estimated standard deviations and confidence intervals based on the asymptotic normal distribution when interaction terms are included in the model for the aortic insufficiency data

Estimate	EF	Gender	Age	Surgery	Surgery	EF Surgery	EF Surgery
			(stand.)	months	months	months	months
Naive	-6.73	0.71	1.27	-4.33	-2.36	8.56	3.17
(Estimated SD)	(2.16)	(0.27)	(0.19)	(1.43)	(1.38)	(2.74)	(2.72)
95% C.I. (lower bound)	-11.04	0.17	0.88	-7.19	-5.11	3.08	-2.28
95% C.I. (upper bound)	-2.41	1.26	1.65	-1.48	0.40	14.05	8.62
Simex ()	-7.59	0.72	1.27	-5.10	-2.63	9.96	3.68
Simex ()	-10.56	0.77	1.27	-7.01	-3.10	13.62	4.65

SD, standard deviation; C.I., confidence interval; EF, ejection fraction.

6. DISCUSSION

The simex algorithm has several advantages that make it very appealing, especially in applied problems. First, since it allows one to graphically represent the effect of the measurement error and of the correction on the bias, it helps justify the need for a correction. Secondly, its intuitive nature makes it appealing in applied problems, particularly to users not familiar with the issue of measurement error. Finally, the scope of the correction can be tuned, making a conservative correction possible. Compared to the alternative approach introduced by Ma & Yin (2008), simex can be applied to a broader class of models, since can take any parametric form, including non-penalized fixed-knot B-splines. Also, when using the simex approach, the additive error can have any distribution, whereas Ma & Yin (2008) only study the normal case in detail. Moreover, the practical implementation of the simex method is easier, since it only requires software to estimate the parameters of the model without measurement error.

SUPPLEMENTARY MATERIAL

Supplementary material available at Biometrika online includes the proof of Theorem 1 and further simulation results.

Appendix

Proof of Theorem 2

For showing the asymptotics under this model, we follow the approach proposed by Zeng et al. (2006), using . In the case of no measurement error, the loglikelihood function is

$$\begin{array}{ccc}rCl\tilde{\ell }(\beta ,F)\hfill & \hspace{0.17em}=\hfill & I(Y<\mathrm{\infty })(\delta \log f+\delta \log \left[-G^{\prime }\left\{\eta (X^{\mathrm{T}}\beta )F(Y)\right\}\eta (X^{\mathrm{T}}\beta )\right]\hspace{0.17em}\hfill \\ & & \hspace{0.17em}+(1-\delta )\log G\left\{\eta (X^{\mathrm{T}}\beta )F(Y)\right\})+I(Y=\mathrm{\infty })\log G\left\{\eta (X^{\mathrm{T}}\beta )\right\}\text{,}\hfill \end{array}$$

where is the density function corresponding to . Then, the true maximizes the expected loglikelihood over the class a cumulative distribution function, for some compact set .

With measurement error, we define

$$\begin{array}{ccc}rCl{\ell }_{\lambda }(\beta ,F)\hfill & \hspace{0.17em}=\hfill & I(Y<\mathrm{\infty })(\delta \log f+\delta \log \left[-G^{\prime }\left\{\eta (W_{\lambda }^{\mathrm{T}}\beta )F(Y)\right\}\eta (W_{\lambda }^{\mathrm{T}}\beta )\right]\hspace{0.17em}\hfill \\ & & \hspace{0.17em}+(1-\delta )\log G\left\{\eta (W_{\lambda }^{\mathrm{T}}\beta )F(Y)\right\})+I(Y=\mathrm{\infty })\log G\left\{\eta (W_{\lambda }^{\mathrm{T}}\beta )\right\},\hfill \end{array}$$

where with , and we suppose that has a unique maximizer . Therefore, we can follow exactly the same reasoning as in Zeng et al. (2006), replacing by in all their calculations.

For a fixed and a fixed , it follows from equation (A.7) in Zeng et al. (2006) that

$$\begin{array}{ccc}rCl\hfill & & \hspace{0.17em}({\hat{\beta }}_{\lambda ,b}-{\beta }_{\lambda }{)}^{\mathrm{T}}{h}_1+{\int }_0^{\mathrm{\infty }}{h}_2\mathrm{d}({\hat{F}}_{\lambda ,b}-F_{\lambda })\hfill \\ & & \hspace{0.17em}=-(P_n-P)\left[{\ell }_{\lambda ,\beta }({\beta }_{\lambda },F_{\lambda }{)}^{\mathrm{T}}{\mathrm{\Omega }}_{\lambda ,\beta }^{-1}(h_1,h_2)+{\ell }_{\lambda ,F}({\beta }_{\lambda },F_{\lambda })\left\{\int {\mathrm{\Omega }}_{\lambda ,F}^{-1}(h_1,h_2)\mathrm{d}{F}_{\lambda }\right\}\right]+o_{\mathrm{p}}(n^{-1/2})\hfill \\ & & \hspace{0.17em}=n^{-1}\sum _{i=1}^n{\psi }_{\lambda }(T_i,W_{i,\lambda ,b},h_1,h_2)+o_{\mathrm{p}}(n^{-1/2}),\hfill \end{array}$$

uniformly over all . Here, is the empirical measure of independent and identically distributed observations, is the expectation, is the derivative of with respect to , is the derivative of along the path , for a small constant , and is the inverse of the linear operator defined in Appendix A.2 in Zeng et al. (2006). Finally,

$$\begin{array}{c}c{S}_0=\left\{h_1\in {\mathbb{R}}^D:\Vert h_1\Vert \le 1\right\}\times \left\{h_2:{\mathbb{R}}^{+}\to \mathbb{R}:\Vert h_2{\Vert }_V\le 1,{\int }_0^{\mathrm{\infty }}{h}_2(y)\mathrm{d}{F}_{\lambda }(y)=0\right\},\hfill \end{array}$$

with the total variation of defined as the supremum over all finite partitions ,

$$\Vert h_2{\Vert }_V=\underset{0=t_1<t_2<\cdots <t_{m+1}=\mathrm{\infty }}{sup}\sum _{i=1}^m|h_2(t_{i+1})-h_2(t_i)|\text{.}$$

Of course, for all .

Next, for fixed , the class is Donsker (Zeng et al., 2006), and hence the class

$$\begin{array}{c}c\left\{(t,w_1,\dots ,w_B)\to B^{-1}\sum _{b=1}^B{\psi }_{\lambda }(t,w_b,h_1,h_2):(h_1,h_2)\in S_0\right\}\hfill \end{array}$$

is also Donsker, since sums of Donsker classes are Donsker; see van der Vaart & Wellner (1996), Lemma 2.10.6. It now follows that the process

$$\begin{array}{cc}{n}^{1/2}\{({\hat{\beta }}_{\lambda }-{\beta }_{\lambda }{)}^{\mathrm{T}}{h}_1\hspace{0.17em}\hfill & \hspace{0.17em}+\hspace{0.17em}{\int }_0^{\mathrm{\infty }}{h}_2\mathrm{d}({\hat{F}}_{\lambda }-F_{\lambda })\}\hfill \\ & \hspace{0.17em}=n^{1/2}\left[B^{-1}\sum _{b=1}^B({\hat{\beta }}_{\lambda ,b}-{\beta }_{\lambda }{)}^{\mathrm{T}}{h}_1+{\int }_0^{\mathrm{\infty }}{h}_2\mathrm{d}\left\{\frac{1}{B}\sum _{b=1}^B({\hat{F}}_{\lambda ,b}-F_{\lambda })\right\}\right]\hfill \\ & \hspace{0.17em}=n^{-1/2}\sum _{i=1}^n{B}^{-1}\sum _{b=1}^B{\psi }_{\lambda }(T_i,W_{i,\lambda ,b},h_1,h_2)+o_{\mathrm{p}}(1)\hfill \end{array}$$

converges weakly to a zero-mean Gaussian process GP indexed by ; see Zeng et al. (2006) after equation (A.7).

The covariance between GP and GP is

$$\begin{array}{c}cE\{\left({\ell }_{\lambda ,\beta }({\beta }_{\lambda },F_{\lambda }{)}^{\mathrm{T}}{\mathrm{\Omega }}_{\lambda ,\beta }^{-1}(h_1,h_2)+{\ell }_{\lambda ,F}({\beta }_{\lambda },F_{\lambda })\left[\int {\mathrm{\Omega }}_{\lambda ,F}^{-1}\left\{h_1,Q_{F_{\lambda }}(h_2)\right\}\mathrm{d}{F}_{\lambda }\right]\right)\hspace{0.17em}\\ \hspace{0.17em}\times \left({\ell }_{\lambda ,\beta }({\beta }_{\lambda },F_{\lambda }{)}^{\mathrm{T}}{\mathrm{\Omega }}_{\lambda ,\beta }^{-1}(h_1^{\ast },h_2^{\ast })+{\ell }_{\lambda ,F}({\beta }_{\lambda },F_{\lambda })\left[\int {\mathrm{\Omega }}_{\lambda ,F}^{-1}\left\{h_1^{\ast },Q_{F_{\lambda }}(h_2^{\ast })\right\}\mathrm{d}{F}_{\lambda }\right]\right)\}\text{.}\end{array}$$

However, since for any in the class

$$\begin{array}{c}cS=\left\{h_1\in {\mathbb{R}}^D:\Vert h_1\Vert \le 1\right\}\times \left\{h_2:{\mathbb{R}}^{+}\to \mathbb{R}:\Vert h_2{\Vert }_V\le 1\right\}\hfill \end{array}$$

we have , where , we can also consider this process as a process indexed by .

Finally, we take a finite grid . The foregoing reasoning based on a single value of can be redone in exactly the same way for the vector . At the end we have that

$$\begin{array}{c}c{n}^{1/2}\left\{\begin{array}{c}({\hat{\beta }}_{{\lambda }_1}-{\beta }_{{\lambda }_1}{)}^{\mathrm{T}}{h}_1+{\int }_0^{\mathrm{\infty }}{h}_2\mathrm{d}({\hat{F}}_{{\lambda }_1}-F_{{\lambda }_1})\hfill \\ ⋮\hfill \\ ({\hat{\beta }}_{{\lambda }_K}-{\beta }_{{\lambda }_K}{)}^{\mathrm{T}}{h}_1+{\int }_0^{\mathrm{\infty }}{h}_2\mathrm{d}({\hat{F}}_{{\lambda }_K}-F_{{\lambda }_K})\hfill \end{array}\right\}\hfill \end{array}$$

converges to a -dimensional Gaussian process of mean zero. The covariance function between the th and th components () is

$$\begin{array}{c}cE\{\left({\ell }_{{\lambda }_i,\beta }({\beta }_{{\lambda }_i},F_{{\lambda }_i}{)}^{\mathrm{T}}{\mathrm{\Omega }}_{{\lambda }_i,\beta }^{-1}(h_1,h_2)+{\ell }_{{\lambda }_i,F}({\beta }_{{\lambda }_i},F_{{\lambda }_i})\left[\int {\mathrm{\Omega }}_{{\lambda }_i,F}^{-1}\left\{h_1,Q_{F_{{\lambda }_i}}(h_2)\right\}\mathrm{d}{F}_{{\lambda }_i}\right]\right)\hspace{0.17em}\\ \hspace{0.17em}\times \left({\ell }_{{\lambda }_j,\beta }({\beta }_{{\lambda }_j},F_{{\lambda }_j}{)}^{\mathrm{T}}{\mathrm{\Omega }}_{{\lambda }_j,\beta }^{-1}(h_1^{\ast },h_2^{\ast })+{\ell }_{{\lambda }_j,F}({\beta }_{{\lambda }_j},F_{{\lambda }_j})\left[\int {\mathrm{\Omega }}_{{\lambda }_j,F}^{-1}\left\{h_1^{\ast },Q_{F_{{\lambda }_j}}(h_2^{\ast })\right\}\mathrm{d}{F}_{{\lambda }_j}\right]\right)\}\text{.}\end{array}$$

We consider two particular cases. First, consider the class

$$\begin{array}{c}c\left\{(h_1,h_2)\in S:h_1=(0,\dots ,0,1,0,\dots ,0)\text{\hspace{0.5em}and\hspace{0.5em}}{h}_2\equiv 0\right\}\hfill \end{array}$$

where is a vector containing at the th position () and elsewhere. Then, we get weak convergence of the vector to a multivariate normal random variable of dimension , , where . The second class that we consider is

$$\begin{array}{c}c\left\{(h_1,h_2)\in S:h_1=0\text{\hspace{0.5em}and\hspace{0.5em}}{h}_2(\cdot )=I(\cdot \le t),t\in {\mathbb{R}}^{+}\right\}\text{.}\hfill \end{array}$$

Then, we get weak convergence of to a Gaussian process indexed by , where .

We will now prove the asymptotic normality of . Suppose that can be specified using a parametric model depending on a vector of parameters . Assuming that is the true extrapolation function, we have that and , where solves, by the least-squares estimation method,

$$\begin{array}{c}c{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}\left\{g_{\beta }({\gamma }_{\beta },\mathrm{\Lambda })-\hat{\beta }(\mathrm{\Lambda })\right\}=0\hfill \end{array}$$

and is the matrix of partial derivatives of the elements of with respect to the elements of . We then have that

$$\begin{array}{c}c{n}^{1/2}({\hat{\gamma }}_{\beta }-{\gamma }_{\beta })={\left\{{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda })\right\}}^{-1}{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}{n}^{1/2}\left\{\hat{\beta }(\mathrm{\Lambda })-\beta (\mathrm{\Lambda })\right\}+o_{\mathrm{p}}(1)\hfill \end{array}$$

converges to . Because and , using the delta method we have that

$$\begin{array}{c}c{n}^{1/2}({\hat{\beta }}_{\text{SIMEX}}-{\beta }_{\text{TRUE}})⟶{\dot{g}}_{\beta }({\gamma }_{\beta },-1){\left\{{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda })\right\}}^{-1}{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}N(0,{\mathrm{\Sigma }}_{\beta }),\hfill \end{array}$$

with variance

$$\begin{array}{c}c\mathrm{\Sigma }={\dot{g}}_{\beta }({\gamma }_{\beta },-1){\left\{{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda })\right\}}^{-1}{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}{\mathrm{\Sigma }}_{\beta }\hfill \\ \times {\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }){\left\{{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda }{)}^{\mathrm{T}}{\dot{g}}_{\beta }({\gamma }_{\beta },\mathrm{\Lambda })\right\}}^{-1}{\dot{g}}_{\beta }({\gamma }_{\beta },-1{)}^{\mathrm{T}}\text{.}\hfill \end{array}$$ (A.1)

Finally, we show that converges weakly to a Gaussian process. For a fixed , suppose that is determined by a parametric model depending on a parameter vector . Under the assumption that this is the true extrapolation function, we have that and , where is a solution of

$$\begin{array}{c}c{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda }{)}^{\mathrm{T}}\left\{g_t({\gamma }_t,\mathrm{\Lambda })-\hat{F}(\mathrm{\Lambda },t)\right\}=0\hfill \end{array}$$

and . It now follows that

$$\begin{array}{c}c{n}^{1/2}({\hat{\gamma }}_t-{\gamma }_t)={\left\{{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda }{)}^{\mathrm{T}}{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda })\right\}}^{-1}{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda }{)}^{\mathrm{T}}{n}^{1/2}\left\{\hat{F}(\mathrm{\Lambda },t)-F(\mathrm{\Lambda },t)\right\}+o_{\mathrm{p}}(1)\hfill \end{array}$$

for all , and hence the process indexed by converges to the Gaussian process

$$\begin{array}{c}c{\left\{{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda }{)}^{\mathrm{T}}{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda })\right\}}^{-1}{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda }{)}^{\mathrm{T}}\mathcal{G}\text{.}\hfill \end{array}$$

Since by definition and , using the delta method we obtain that as ,

$$\begin{array}{c}c{n}^{1/2}({\hat{F}}_{\text{SIMEX}}-F_{\text{TRUE}})⟶{\dot{g}}_t({\gamma }_t,-1{)}^{\mathrm{T}}{\left\{{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda }{)}^{\mathrm{T}}{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda })\right\}}^{-1}{\dot{g}}_t({\gamma }_t,\mathrm{\Lambda }{)}^{\mathrm{T}}\mathcal{G}\text{.}\hfill \end{array}$$ (A.2)