Right-censored nonparametric regression with measurement error

Abstract

This study focuses on estimating a nonparametric regression model with right-censored data when the covariate is subject to measurement error. To achieve this goal, it is necessary to solve the problems of censorship and measurement error ignored by many researchers. Note that the presence of measurement errors causes biased and inconsistent parameter estimates. Moreover, non-parametric regression techniques cannot be applied directly to right-censored observations. In this context, we consider an updated response variable using the Buckley–James method (BJM), which is essentially based on the Kaplan–Meier estimator, to solve the censorship problem. Then the measurement error problem is handled using the kernel deconvolution method, which is a specialized tool to solve this problem. Accordingly, three denconvoluted estimators based on BJM are introduced using kernel smoothing, local polynomial smoothing, and B-spline techniques that incorporate both the updated response variable and kernel deconvolution.The performances of these estimators are compared in a detailed simulation study. In addition, a real-world data example is presented using the Covid-19 dataset.

Appendices

Appendix A1. Proof of Lemma 4.1

Differentiating the ${\text {AMISE}}\left\{ {\hat{m}}_{h}(x)\right\}$ with respect to h and setting the derivative equal to zero yields

$$\begin{aligned} \frac{\partial }{\partial h} A {\text {MISE}}\left\{ {\hat{m}}_{h}(x)\right\} =\frac{-n[R(K) V(x)]}{(n h)^{2}}+\frac{4 h^{3} V(K)^{2} R\left( m^{\prime \prime }\right) 4}{16}=0 \end{aligned}$$

(A2.1)

Setting Eq. (A2.1) equal to zero we obtain the following equation,

$$\begin{aligned} n h^{5} V(K)^{2} R\left( m^{\prime \prime }\right) =R(K) V(x) \end{aligned}$$

By taking simple algebraic operations it is seen that the optimal value of parameter h is

$$\begin{aligned} h_{o p t}=\left[ \frac{R(K) V(x)}{V(K)^{2} R\left( m^{\prime \prime }\right) n}\right] ^{1 / 5}, \end{aligned}$$

as claimed.

Appendix A2. Proof of Lemma 4.2

To prove Lemma 4.2, one needs to strong assumptions and conditions with regard to nonparametric smoothing, measurement errors, censorship and, the Buckley–James procedure. We provided the restrictions as follows:

Conditions for the Buckley–James (BJ) Method

By following the study of James and Smith (1984) and Meier (1975), let $\Xi =sup\{\xi :F(\xi )<1\}<\infty$ and suppose that $N(\xi )$ is the mean of the number of data points (censored or not) for which $\varepsilon ^*=Y^*-m(W)>\xi$. Finally, $\varepsilon _C=C-m(W)$. Accordingly, to reach the variance equation given in Lemma 4.2 the following assumptions need to be ensured:

BJ1. $N(\xi ) \rightarrow \infty$ when $n \rightarrow \infty$ for all $\xi <\Xi$.

BJ2. $\sum _{i=1}^{n}(W_i-{\bar{W}})^2\rightarrow \infty$ when $n \rightarrow \infty$.

BJ3. $lim sup_{n\rightarrow \infty }\{\frac{\sum _{i=1}^{n} F(\varepsilon _{C_i}) \mid W_i-{\bar{W}}\mid }{\sum _{i=1}^{n} (W_i-{\bar{W}})^2}\}<\infty$.

BJ4. To ensure the BJ3, following condition is needed: $\liminf _{n\rightarrow \infty } (1/n_c)\sum _i^n(W_i-{\bar{W}})^2>0$ where $n_c$ is no. the censored observations.

In addition to these conditions, assumptions (A4) and (B1–B5) should be ensured regarding the $\hat{{\textbf{m}}}_h(W)\rightarrow \hat{{\textbf{m}}}_h(X)$ based on the corresponding smoothing matrix. Note that assumptions (A1–A3) are also needed to take into account the censorship. Under the given restrictions, the following proof can be written for $MSSE({\hat{\textbf{m}}})$:

$$\begin{aligned} {\text {MSSE}}\left( {\hat{m}}_{h}\right)&=\sum \limits _{i=1}^{n} E\left[ \left\{ {\hat{m}}_{h}\left( W_{i}\right) -m\left( W_{i}\right) \right\} ^{2}\right] =\sum _{i=1}^{m} {\text {MSE}}\left[ {\hat{m}}_{h}\left( W_{i}\right) \right] \\&=\sum \limits _{i=1}^{n}\left\{ E\left( {\hat{m}}_{h}\left( W_{i}\right) \right) -m\left( W_{i}\right) \right\} ^ {2}+{\text {Var}}\left[ {\hat{m}}_{h}\left( W_{i}\right) \right] \\&=\sum \limits _{i=1}^{n} \{E\left( {\textbf{S}}_{h} {\textbf{Y}}_{i}^{*}\right) -m(W_{i})\}^{2}+Var({\textbf{S}}_{h} {\textbf{Y}}_{i}^{*}) \\&=\sum \limits _{i=1}^{n}\{E\left( {\textbf{S}}_{h} {\textbf{Y}}_{i}^{*}\right) -m(W_{i})\}^{2}+{\text {tr}}\left[ {\text {Cov}}\left( {\textbf{S}}_{h} {\textbf{Y}}_{i}^{*}\right) \right] \\&=\Vert (\hat{{\varvec{m}}_{h}}-{\varvec{m}})\Vert ^{2}+{\text {tr}}\left[ {\text {Cov}}\left( {\textbf{S}}_{h} {\textbf{Y}}^{*}\right) \right] \\&=\left\| \left( {\textbf{S}}_{h}-{\textbf{I}}\right) {\varvec{m}}\right\| ^{2}+{\text {tr}}\left[ {\textbf{S}}_{h} {\text {Cov}}({\textbf{Y}}^{*}) {\textbf{S}}_{h}^{\prime }\right] \end{aligned}$$

Due to censorship and transformed response variable ${\textbf{Y}}^*$, the variance ${\text {Cov}}({\textbf{Y}}^{*})$ is written as given in (4.20) $\sigma _{*}^{2} {\textbf{I}}=\sigma ^2_\varepsilon -Var(Y \mid Y>C){\bar{F}}(C)$ produces

$$\begin{aligned} {\text {MSSE}}\left( \widehat{{\varvec{m}}}_{h}\right) =\left\| \left( {\textbf{S}}_{h}-{\textbf{I}}\right) {\varvec{m}}\right\| ^{2}+\sigma _{*}^{2} {\text {tr}}\left( {\textbf{S}}_{h} {\textbf{S}}_{h}^{\prime }\right) \end{aligned}$$

(A3.1)

As claimed, Eq. (4.19) has been proven.