Modern Stochastics: Theory and Applications

We deal with Cox proportional hazards model where a lifetime $T\ge 0$ has the following intensity function Here we say that positive random variable ξ has intensity function $\tilde{\lambda }(\cdot )$ if

In (1.1) covariate X is a random vector distributed in ${\mathbb{R}}^{k}$, $\lambda (\cdot )\in \varTheta _{\lambda }\subset C[0,\tau ]$ is the baseline hazard function and β is a parameter from $\varTheta _{\beta }\subset {\mathbb{R}}^{k}$. We observe only censored value $Y:=\min \{T,C\}$, where censor C is distributed in $[0,\tau ]$. Survival function of C, $G_{C}(u)=1-F_{C}(u)$, is unknown but we know τ. Censorship indicator $\varDelta :=\mathbb{I}_{\{T\le C\}}$ is observed as well. X is not observed directly, instead a surrogate data $W=X+U$ is observed, where U has known and finite moment generating function $M_{U}(\beta ):=\mathbf{E}{e}^{{\beta }^{\mathsf{T}}U}$. Here E stands for expectation. A couple $(T,X)$, censor C and measurement error U are stochastically independent. We mention that recently measurement error models become quite popular, e.g., in [9] an autoregressive model with measurement error was studied.

Consider independent copies of the model $(X_{i},T_{i},C_{i},Y_{i},\varDelta _{i})$, $i=1,\dots ,n$. Based on $(Y_{i},\varDelta _{i},W_{i})$, $i=1,\dots ,n$, we estimate true values of β and $\lambda (\cdot )$ that we denote by $\beta _{0}$ and $\lambda _{0}(\cdot )$, respectively. The latter is estimated on $[0,\tau ]$ only.

There are a lot of papers on estimation of $\beta _{0}$ and cumulative hazard $\varLambda (t)={\int _{0}^{t}}\lambda (t)\hspace{0.1667em}\mathrm{d}t$. In [1] general ideas are presented based on partial likelihood. Same model but with measurement errors is considered in [4], where, based on Corrected Score method, consistent and asymptotically normal estimators are constructed for regression parameter and cumulative hazard function. Another approach is proposed in [6] where doubly censored data are considered without measurement error. Here cumulative hazard is estimated, and strong consistency and asymptotic normality of maximum likelihood estimators are proven. However, sometimes it is necessary to know the behaviour of baseline hazard function $\lambda (\cdot )$ itself, not cumulative hazard (see [10]). Our model is presented in [2] and [5] where baseline hazard function is assumed to belong to a parametric space while we consider $\lambda (\cdot )$ from a compact set of $C[0,\tau ]$.

If values of $X_{i}$ were measured without measurement error, we could use Maximum Likelihood Estimator (MLE) which maximizes the log-likelihood function where

Since $X_{i}$ is contaminated, we have to correct our objective function for measurement error. Due to suggestion of Augustin [2] we construct a new objective function q such that

Then the corrected log-likelihood function is where

As an estimator of true parameters $(\lambda _{0},\beta _{0})$, we use a couple $(\lambda _{n},\beta _{n})$ which maximizes (1.2).

Introduce further assumptions.

In [7] the strong consistency of $(\lambda _{n},\beta _{n})$ is proven and the rate of convergence is presented. Our goal is to provide asymptotic normality for $\beta _{n}$ and $\lambda _{n}$. The paper is organised as follows. Section 2 states the main results on the asymptotic normality. Section 3 suggests the procedure for computation of the estimates. Section 4 proves the stochastic boundedness results. Section 5 proves auxiliary results, Section 6 gives the proof of the main result, and Section 7 concludes.

For a sequence of random variables $\{x_{n}\}$, notation $x_{n}=O_{p}(1)$ means that $\{x_{n}\}$ is stochastically bounded. We assume that censor C has pdf $f_{C}$ (this is a technical assumption that can be easily avoided). According to [7], Section 3, conditional density of $(Y,\varDelta )$ given X at point $(\lambda _{0},\beta _{0})$ equals where $f_{T}$ is conditional pdf of T given X and $G_{T}$ is conditional survival function:

Let Z be a normed linear space. For a function $f:\hspace{2.5pt}Z\to \mathbb{R}$ we denote ${f}^{(n)}(x_{0})$ its n-th Fréchet derivative at a point $x_{0}\in Z$. ${f}^{(n)}(x_{0})$ is n-linear form and for $h_{1},\dots ,h_{n}\in Z$ we denote $\langle {f}^{(n)}(x_{0}),(h_{1},\dots ,h_{n})\rangle $ the action of ${f}^{(n)}(x_{0})$. If $h_{1}=\cdots =h_{n}$ we simply write $\langle {f}^{(n)}(x_{0}),{(h_{1})}^{n}\rangle $ where it does not cause ambiguity. If a functional F acts on a product space $Z_{1}\times Z_{2}$ then elements of this space are denoted as $(h_{1},h_{2})\in Z_{1}\times Z_{2}$ and $\langle F,(h_{1},h_{2})\rangle $ stands for the action of F on $(h_{1},h_{2})$. For $x,y\in Z$, the following set is called an interval that connects x and y

We make some more notations. Let

Denote where $K(u)=\frac{\lambda _{0}(u)}{b(u)}$. Also introduce a sequence of random vectors with i.i.d. summands

Let $\varSigma _{\beta }=4\mathrm{Cov}(\zeta _{1})$, $m(\varphi _{\lambda })={\int _{0}^{\tau }}\hspace{-0.1667em}\varphi _{\lambda }(u)a(u)G_{C}(u)\hspace{0.1667em}\mathrm{d}u$, ${\varSigma _{\varphi }^{2}}=4\mathrm{Var}[\langle {q^{\prime }}(Y,\varDelta ,W,\lambda _{0},\beta _{0}),\varphi \rangle ]$ with $\varphi =(\varphi _{\lambda },\varphi _{\beta })\in C[0,\tau ]\times {\mathbb{R}}^{k}$.

Note that the corollary immediately follows from the theorem after f is substituted by $\frac{f}{G_{C}}$.

Since $\varTheta _{\lambda }$ is infinite-dimensional, computation of $(\lambda _{n},\beta _{n})$ is not a parametric problem in general setting. We refer to the ideas of I.J. Schoenberg [11]. We will show that maximum of (1.2) is attained on a linear spline with nodes located at points $Y_{i}$, $i=1,\dots ,n$ and some other points that can be calculated.

Let $i_{1},\dots ,i_{n}\in 1,\dots ,n$ be such a numbering that $Y_{i_{1}}\le \cdots \le Y_{i_{n}}$, i.e., $(Y_{i_{1}},\dots ,Y_{i_{n}})$ is a variational series of $(Y_{1},\dots ,Y_{n})$. Alongside with $(\lambda _{n},\beta _{n})$ we consider $(\overline{\lambda }_{n},\beta _{n})$, where $\overline{\lambda }_{n}$ is the following function. We set $\overline{\lambda }_{n}(Y_{i_{k}})=\lambda (Y_{i_{k}})$, $k=1,\dots ,n$. For each interval $[Y_{i_{k}},Y_{i_{k+1}}]$, $k=1,\dots ,n-1$, perform the next procedure. Draw straight lines and where L is defined in (i).

Denote $B_{i_{k}}$ the intersection of ${L_{i_{k}}^{1}}(t)$ and ${L_{i_{k}}^{2}}(t)$. $B_{i_{0}}:=0$, $B_{i_{n}}:=\tau $, $Y_{i_{0}}:=0$, $Y_{i_{n+1}}:=\tau $. We set

Note that $\lambda _{n}\ge \overline{\lambda }_{n}$ because $\lambda _{n}\in \varTheta _{\lambda }$. Then

Thus, one can easily see that implying $\lambda _{n}=\overline{\lambda }_{n}$ so that we conclude with the following statement.

Using maximization in (3.3) makes computation of $(\lambda _{n},\beta _{n})$ inconvenient. Thus, we propose to modify the estimators. As soon as condition (viii) is satisfied and estimator $(\lambda _{n},\beta _{n})$ is strongly consistent, one can induce that eventually $\overline{\lambda }(B_{i_{k}})>a$, and thus, eventually there is no need in finding maximum in (3.3). Therefore, instead of $(\lambda _{n},\beta _{n})$ we propose to consider a couple $(\widehat{\lambda }_{n},\widehat{\beta }_{n})$ with $\widehat{\beta }_{n}\in \varTheta _{\beta }$ that maximizes $Q_{n}$ under restrictions:

Evaluating $(\widehat{\lambda }_{n},\widehat{\beta }_{n})$ is a parametric problem. We mention that eventually $(\widehat{\lambda }_{n},\widehat{\beta }_{n})=(\lambda _{n},\beta _{n})$. We summarise with the next statement.

The proof is based on the three lemmas. Using integration by parts one can easily prove the following.

Crucial step of the proof of Theorem 5 is the following.

Let $A_{n}(\omega )$ be a collection of assertions (here ω stands for elementary event). We say that $\{A_{n}\}$ hold eventually if for almost all ω there exists $N_{\omega }$ such that for all $n>N_{\omega }$, $A_{n}(\omega )$ holds.

We use the ideas of [3].

Let $\theta _{n}=(\lambda _{n},\beta _{n})$, $\theta _{0}=(\lambda _{0},\beta _{0})$, $\varTheta =\varTheta _{\lambda }\times \varTheta _{\beta }$. Denote $\varphi =(\varphi _{\lambda },\varphi _{\beta })$ an admissible shift such that there exists $\delta >0$ with $\theta _{0}\pm \delta \varphi \in \varTheta $. We demand that (vii)–(viii) hold. Note that φ can be a random element and depend on n. However, $\| \varphi \| $ should be bounded from above a.s.

Consider the function $f(t)=Q_{n}(\theta _{n}+t(\theta _{0}-\theta _{n}\pm \delta \varphi ))$, $0\le t\le 1$. It is well-defined (due to the convexity of Θ) and attains its maximum at point $t=0$. Therefore, $\langle {Q^{\prime }_{n}}(\theta _{n}),\theta _{0}-\theta _{n}\pm \delta \varphi \rangle \le 0$ and where $\varDelta \theta _{n}:=\theta _{n}-\theta _{0}$.

Taylor’s expansion at point $(\lambda _{0},\beta _{0})$ implies for some $\widehat{\theta }_{n}$ and $\tilde{\theta }_{n}$ from interval $[\theta _{0},\theta _{n}]$.

Relying on this proposition we will be able to show that $\sqrt{n}\| \beta _{n}-\beta _{0}\| $ and $\sqrt{n}{\int _{0}^{\tau }}\hspace{-0.1667em}(\lambda _{n}-\lambda _{0})(u)G_{C}(u)\hspace{0.1667em}\mathrm{d}u$ are stochastically bounded and then prove the asymptotic normality of $\sqrt{n}\langle {Q^{\prime\prime }_{n}}(\theta _{0}),(\varDelta \theta _{n},\varphi )\rangle $.

Denote $\varTheta _{-}=\varTheta -\varTheta $. It is clear that it is compact and convex. Before proving the proposition, we show the following.

Returning to inequality (5.1), because $\varDelta \theta _{n}$ converges to zero a.s., one can conclude the following.

We present the main result of this section.

We assume that condition (ix) is satisfied. Thus A is positive definite and, consequently, invertible. Since $T(u)$ is positive definite, M is positive definite as well and therefore, invertible.

Note that due to conditions (vii) and (viii), Theorem 12 is valid for all non-random shifts $\varphi \in {\mathbb{R}}^{k}\times Lip_{1}([0,\tau ])$, where $Lip_{1}([0,\tau ])$ is a class of Lipschitz continuous functions. Take the shift as follows: $\varphi =({\varphi _{\beta }^{\mathsf{T}}}a(u)K(u),\varphi _{\beta })$, where $\varphi _{\beta }$ is a fixed vector of ${\mathbb{R}}^{k}$. For this shift one can rewrite $\langle {Q^{\prime }_{n}}(\theta _{0}),\varphi \rangle $ as

By the CLT applied to $\xi _{n}$ one can see that the limit distribution of $\sqrt{n}\xi _{n}$ is in fact $N_{k}(0,\frac{1}{4}\varSigma _{\beta })$. Note that we have already faced with the shift φ in Corollary 11. In particular, (5.9) yields that $\langle {q^{\prime\prime }_{\infty }}(\theta _{0}),(\varDelta \theta _{n},\varphi )\rangle $ can be rewritten as

Theorem 12 and Cramér-Wold’s theorem yield that Since M is invertible, the convergence (2.1) is proved.

Now, for a fixed shift $\varphi _{\lambda }$ take such $\varphi _{\beta }$ that

Hence, $\varphi _{\beta }=-{A}^{-1}m(\varphi _{\lambda })$. From (5.6) it follows that

In view of Theorem 12 and the remark at the beginning of the proof, in order to show the convergence (2.4), one should show that the equation (2.3) has a Lipschitz continuous solution $\varphi _{\lambda }$. But if $\varphi _{\lambda }$ is a solution to (2.3) then for some constant $C\in {\mathbb{R}}^{k}$ and thus, is Lipschitz continuous. After substitution (6.1) in (2.3) we obtain

Let $S={\int _{0}^{\tau }}\hspace{-0.1667em}f(u){A}^{-1}a(u)K(u)G_{C}(u)\hspace{0.1667em}\mathrm{d}u$ and $P(u)=\mathbf{E}[X{X}^{\mathsf{T}}\exp ({\beta _{0}^{\mathsf{T}}}X)G_{T}(u|X)]$. We show that it is possible to choose C so that

After transposing both sides and multiplying by A, we have

Transformation of $R:=A-{\int _{0}^{\tau }}\hspace{-0.1667em}a(u){a}^{\mathsf{T}}(u)K(u)G_{C}(u)\hspace{0.1667em}\mathrm{d}u$ leads to

In the proof of Corollary 11 it was shown that $P(u)-\frac{a(u){a}^{\mathsf{T}}(u)}{b(u)}=\frac{T(u)}{b(u)}$ is a positive definite matrix. Therefore, R is positive definite and invertible. Hence, (2.3) has a unique solution and convergence (2.4) holds. This completes the proof.

Here we studied properties of the Corrected MLE $(\lambda _{n},\beta _{n})$ proposed by Kukush et al. [7] in Cox proportional hazards model with measurement error. Asymptotic normality was obtained for $\beta _{n}$ and integral functionals of $\lambda _{n}$. We also present estimator $(\widehat{\lambda }_{n},\widehat{\beta }_{n})$ that inherits properties of $(\lambda _{n},\beta _{n})$ and transforms the maximization problem to a parametric one.

In future we intend to provide simulations in this model.