Cox proportional hazards model with measurement errors is considered. In Kukush and Chernova (2017), we elaborated a simultaneous estimator of the baseline hazard rate λ(·) and the regression parameter β, with the unbounded parameter set Θ=Θλ×Θβ, where Θλ is a closed convex subset of C[0,τ] and Θβ is a compact set in Rm. The estimator is consistent and asymptotically normal. In the present paper, we construct confidence intervals for integral functionals of λ(·) and a confidence region for β under restrictions on the error distribution. In particular, we handle the following cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.

Asymptotic normalityconfidence regionconsistent estimatorCox proportional hazards modelmeasurement errorssimultaneous estimation of baseline hazard rate and regression parameterIntroduction

Survival analysis models time to an event of interest (e.g., lifetime). It is a powerful tool in biometrics, epidemiology, engineering, and credit risk assessment in financial institutions. The proportional hazards model proposed in Cox (1972) [3] is a widely used technique to characterize a relation between survival time and covariates.

Our model is presented in Augustin (2004) [1] where the baseline hazard function λ(·) is assumed to belong to a parametric space, while we consider λ(·) belonging to a closed convex subset of C[0,τ]. In practice covariates are often contaminated by errors, so we deal with errors-in-variables model. Kukush et al. (2011) [5] derive a simultaneous estimator of the baseline hazard rate λ(·) and the regression parameter β and prove the consistency of the estimator. At that, the parameter set Θλ for the baseline hazard rate is assumed to be bounded and separated away from zero. The asymptotic normality of the estimator is shown in Chimisov and Kukush (2014) [2]. In [7, 6] we construct an estimator (λˆn(1)(·),βˆn(1)) of λ(·) and β over the parameter set Θ=Θλ×Θβ, where n is the sample size and Θλ is a subset of C[0,τ], which is unbounded from above and not separated away from zero. The estimator is consistent and can be modified to be asymptotically normal.

The goal of present paper is to construct confidence intervals for integral functionals of λ(·) and a confidence region for β based on the estimators from [7, 6]. We impose certain restrictions on the error distribution. Actually we handle three cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.

The paper is organized as follows. Section 2 describes the observation model, gives main assumptions, defines an estimator under an unbounded parameter set, and states the asymptotic normality result from [7, 6]. Sections 3 and 4 present the main results: a confidence region for the regression parameter and confidence intervals for integral functionals of the baseline hazard rate. Section 5 provides a method to compute auxiliary consistent estimates, and Section 6 concludes.

Throughout the paper, all vectors are column ones, E stands for the expectation, Var stands for the variance, and Cov for the covariance matrix. A relation holds eventually if it is valid for all sample sizes n starting from some random number, almost surely.

The model and estimator

Let T denote the lifetime and have the intensity function
λ(t|X;λ0,β0)=λ0(t)exp(β0⊤X),t≥0.
A covariate X is a time-independent random vector distributed in Rm, β is a parameter belonging to Θβ⊂Rm, and λ(·)∈Θλ⊂C[0,τ] is a baseline hazard function.

We observe censored data, i.e., instead of T only a censored lifetime Y:=min{T,C} and the censorship indicator Δ:=I{T≤C} are available, where the censor C is distributed on a given interval [0,τ]. The survival function of censor GC(u):=1−FC(u) is unknown. The conditional pdf of T given X is
fT(t|X)=λ(t|X;λ0,β0)exp(−∫0tλ(t|X;λ0,β0)ds).
The conditional survival function of T given X equals
GT(t|X)=exp(−∫0tλ(s|X;λ0,β0)ds)=exp(−eβ0⊤X∫0tλ0(s)ds).

We deal with an additive error model, which means that instead of X, a surrogate variable
W=X+U
is observed. We suppose that a random error U has known moment generating function MU(z):=Eez⊤U, where ||z|| is bounded according to assumptions stated below. A couple (T,X), censor C, and measurement error U are stochastically independent.

Introduce assumptions from [7, 6].

Θλ⊂C[0,τ] is the following closed convex set of nonnegative functions
Θλ:={f:[0,τ]→R|f(t)≥0,∀t∈[0,τ]and|f(t)−f(s)|≤L|t−s|,∀t,s∈[0,τ]},
where L>00$]]> is a fixed constant.

Θβ⊂Rm is a compact set.

EU=0 and for some fixed ϵ>00$]]>,
Ee2D‖U‖<∞,withD:=maxβ∈Θβ‖β‖+ϵ.

Ee2D‖X‖<∞, where D is defined in (iii).

τ is the right endpoint of the distribution of C, that is P(C>τ)=0\tau )=0$]]> and for all ϵ>00$]]>, P(C>τ−ϵ)>0\tau -\epsilon )>0$]]>.

The covariance matrix of random vector X is positive definite.

Denote
Θ=Θλ×Θβ.

The couple of true parameters (λ0,β0) belongs to Θ given in (1), and moreover λ0(t)>00$]]>, t∈[0,τ].

β0 is an interior point of Θβ.

λ0∈Θλϵ for some ϵ>00$]]>, with
Θλϵ:={f:[0,τ]→R|f(t)≥ϵ,∀t∈[0,τ]and|f(t)−f(s)|≤(L−ϵ)|t−s|,∀t,s∈[0,τ]}.

P(C>0)=10)=1$]]>.

Consider independent copies of the model (Xi,Ti,Ci,Yi,Δi,Ui,Wi), i=1,…,n. Based on triples (Yi,Δi,Wi), i=1,…,n, we estimate true parameters β0 and λ0(t), t∈[0,τ]. Following Augustin (2004) [1], we use the corrected partial log-likelihood function
Qncor(λ,β):=1n∑i=1nq(Yi,Δi,Wi;λ,β),
with
q(Y,Δ,W;λ,β):=Δ·(logλ(Y)+β⊤W)−exp(β⊤W)MU(β)∫0Yλ(u)du.

The estimator [7, 6] of the baseline hazard rate λ(·) and parameter β is defined as follows.

Fix a sequence {εn} of positive numbers, with εn↓0, as n→∞. The corrected estimator (λˆn(1),βˆn(1)) of (λ,β) is a Borel measurable function of observations (Yi,Δi,Wi), i=1,…,n, with values in Θ and such that
Qncor(λˆn(1),βˆn(1))≥sup(λ,β)∈ΘQncor(λ,β)−εn.

Theorem 3 from [7, 6] proves that under conditions (i) to (vii) the corrected estimator (λˆn(1),βˆn(1)) is a strongly consistent estimator of the true parameters (λ0,β0). In the proof of Theorem 3 from [7, 6], it is shown that eventually and for R large enough, the upper bound on the right-hand side of (2) can be taken over the set ΘR:=ΘλR×Θβ, with
ΘλR:=Θλ∩B¯(0,R),
where B¯(0,R) denotes the closed ball in C[0,τ] with center in the origin and radius R. Thus, we assume that for all n≥1,
Qncor(λˆn(1),βˆn(1))≥sup(λ,β)∈ΘRQncor(λ,β)−εn
and (λˆn(1),βˆn(1))∈ΘR. Notice that ΘR is a compact set in C[0,τ].

Definition 2 from [7, 6] provides, based on (λˆn(1),βˆn(1)), a modified estimator (λˆn(2),βˆn(2)) which is consistent and asymptotically normal.

The modified corrected estimator (λˆn(2),βˆn(2)) of (λ,β) is a Borel measurable function of observations (Yi,Δi,Wi), i=1,…,n, with values in Θ and such that
(λˆn(2),βˆn(2))=argmax{Qncor(λ,β)|(λ,β)∈Θ,μλ≥12μλˆn(1)},ifμλˆn(1)>0;(λˆn(1),βˆn(1)),otherwise,0;\\{} ({\hat{\lambda }_{n}^{(1)}},{\hat{\beta }_{n}^{(1)}}),\hspace{1em}& \text{otherwise},\end{array}\right.\]]]>
where μλ:=mint∈[0,τ]λ(t).

Below we use notations from [2]. Let
a(t)=E[Xeβ0⊤XGT(t|X)],b(t)=E[eβ0⊤XGT(t|X)],Λ(t)=∫0tλ0(t)dt,p(t)=E[XX⊤eβ0⊤XGT(t|X)],T(t)=p(t)b(t)−a(t)a⊤(t),K(t)=λ0(t)b(t),A=E[XX⊤eβ0⊤X∫0Yλ0(u)du],M=∫0τT(u)K(u)Gc(u)du.
For i=1,2,…, introduce random variables
ζi=−Δia(Yi)b(Yi)+exp(β0⊤Wi)MU(β0)∫0Yia(u)K(u)du+∂q∂β(Yi,Δi,Wi,β0,λ0),
with
∂q∂β(Y,Δ,W;λ,β)=Δ·W−MU(β)W−E(Ueβ⊤U)MU(β)2exp(β⊤W)∫0Yλ(u)du.
Let
Σβ=4·Cov(ζ1),m(φλ)=∫0τφλ(u)a(u)GC(u)du,σφ2=4·Var⟨q′(Y,Δ,W,λ0,β0),φ⟩=4·Varξ(Y,Δ,W),
with
ξ(Y,Δ,W)=Δ·φλ(Y)λ0(Y)−exp(β0⊤W)MU(β0)∫0Yφλ(u)du+Δ·φβ⊤W−φβ⊤MU(β0)W−E[Ueβ0⊤U]MU(β0)2exp(β0⊤W)∫0Yλ0(u)du,
where φ=(φλ,φβ)∈C[0,τ]×Rm and q′ denotes the Fréchet derivative. ([<xref ref-type="bibr" rid="j_vmsta94_ref_007">7</xref>, <xref ref-type="bibr" rid="j_vmsta94_ref_006">6</xref>]).

Assume conditions (i) – (x). Then M is nonsingular andn(βˆn(2)−β0)→dNm(0,M−1ΣβM−1).Moreover, for any Lipschitz continuous function f on[0,τ],n∫0τ(λˆn(2)−λ0)(u)f(u)GC(u)du→dN(0,σφ2(f)),whereσφ2(f)=σφ2withφ=(φλ,φβ),φβ=−A−1m(φλ)andφλis a unique solution inC[0,τ]to the Fredholm integral equationφλ(u)K(u)−a⊤(u)A−1m(φλ)=f(u),u∈[0,τ].

Confidence regions for the regression parameter

Denote as EX[·] the conditional expectation given a random variable X. Remember that MU(z)=Eez⊤U. For simplicity of notation, we write Mk,β instead of MU((k+1)β). Using differentiation in z one can easily prove the following.

The equalities hold true:ez⊤X=EX[ez⊤W]MU(z),Xez⊤X=1MU(z)(EX[Wez⊤W]−E[Uez⊤U]MU(z)EX[ez⊤W]),XX⊤ez⊤X=1MU(z)(EX[WW⊤ez⊤W]−2E[Uez⊤U]MU(z)EX[W⊤ez⊤W]−−(E[UU⊤ez⊤U]MU(z)−2E[Uez⊤U]·E[U⊤ez⊤U]MU2(z))EX[ez⊤W]).

Now, we state conditions on measurement error U under which one can construct unbiased estimators for a(t), b(t) and p(t), t∈[0,τ].

Suppose that for anyβ∈ΘβandA>00$]]>,∑k=0∞ak+1(β)k!Ak<∞,

withak+1(β):=E‖U‖2e(k+1)β⊤UMk,β.

Then there exist functionsB(·,·),A(·,·)andP(·,·)which satisfy deconvolution equations:

EX[B(W,t)]=exp(β⊤X−Λ(t)eβ⊤X),

EX[A(W,t)]=Xexp(β⊤X−Λ(t)eβ⊤X),

EX[P(W,t)]=XX⊤exp(β⊤X−Λ(t)eβ⊤X);t∈[0,τ].

We find solutions to the equations in a form of series expansions using the idea from Stefanski (1990) [8].

(a) Utilizing Taylor decomposition of the right-hand side, we obtain
exp(β⊤X−Λ(t)eβ⊤X)=∑k=0∞gk(X,t),gk(X,t):=(−1)kk!Λk(t)e(k+1)β⊤X.

Using Lemma 1 take for k≥0Bk(W,t)=(−1)kk!Mk,βΛk(t)e(k+1)β⊤W,
so that EX[Bk(W,t)]=gk(X,t), t∈[0,τ]. If we ensure that
∑k=0∞EX|Bk(W,t)|<∞,
then B(W,t)=∑k=0∞Bk(W,t) is a solution to the first equation. We have
∑k=0∞EX|Bk(W,t)|=∑k=0∞Λk(t)k!e(k+1)β⊤X=exp(β⊤X+Λ(t)eβ⊤X)<∞.
Here no additional restriction on U is needed.

(b) Similarly, we show that A(W,t)=∑k=0∞Ak(W,t), with
Ak(W,t):=(−1)kk!Mk,βΛk(t)[W−E[Ue(k+1)β⊤U]Mk,β]e(k+1)β⊤W,
is a solution to the second equation, if ∑k=0∞EX‖Ak(W,t)‖<∞. We have
∑k=0∞EX‖Ak(W,t)‖=∑k=0∞Λk(t)k!Mk,βEX||X+U−E[Ue(k+1)β⊤U]Mk,β||e(k+1)β⊤(X+U)≤‖X‖exp(β⊤X+Λ(t)eβ⊤X)+2∑k=0∞Λk(t)k!E‖U‖e(k+1)β⊤UMk,βe(k+1)β⊤X.
The latter sum is finite due to condition (6). Therefore, there exists a solution to the second equation.

(c) Finally, for the third equation we put
Pk(W,t)=(−1)kΛk(t)k!Mk,β[WW⊤e(k+1)β⊤W−2E[Ue(k+1)β⊤U]Mk,βW⊤e(k+1)β⊤W−(E[UU⊤e(k+1)β⊤U]Mk,β−2E[Ue(k+1)β⊤U]·E[U⊤e(k+1)β⊤U]Mk,β2)e(k+1)β⊤W].

The matrix P(W,t)=∑k=0∞Pk(W,t) is a solution to the third equation if
∑k=0∞EX‖Pk(W,t)‖<∞.
Hereafter ‖Q‖ is the Euclidean norm of a matrix Q. We have
∑k=0∞EX‖Pk(W,t)‖≤∑k=0∞Λk(t)k![EX[‖W‖2e(k+1)β⊤W]Mk,β+2E[‖U‖e(k+1)β⊤U]·EX[‖W‖e(k+1)β⊤W]Mk,β2+E[‖U‖2e(k+1)β⊤U]·EXe(k+1)β⊤WMk,β2+2(E‖U‖e(k+1)β⊤U)2·EXe(k+1)β⊤WMk,β3].
The right-hand side of (8) is a sum of four series which can be bounded similarly based on condition (6). E.g., for the last of the four series we have:
(E‖U‖e12(k+1)β⊤Ue12(k+1)β⊤U)2≤E‖U‖2e(k+1)β⊤U·Mk,β,EXe(k+1)β⊤W=Mk,β·e(k+1)β⊤X,∑k=0∞Λk(t)(E‖U‖e(k+1)β⊤U)2·EXe(k+1)β⊤Wk!Mk,β3≤∑k=0∞ak+1(β)Λk(t)e(k+1)β⊤Xk!<∞.

Therefore, condition (6) yields (7), and P(W,t) is a solution to the third equation. □

The condition of Theorem2is fulfilled in each of the following cases:

(a) the measurement error U is bounded,

(b) U is normally distributed with zero mean and variance-covariance matrixσU2Im, withσU>00$]]>, and

(c) U has independent componentsU(i)which are shifted Poisson random variables, i.e.U(i)=U˜(i)−μi, whereU˜(i)∼Pois(μi),i=1,…,m.

(a) Let ‖U‖≤K. Then
E‖U‖2e(k+1)β⊤UMk,β≤K2,
and (6) holds true.

(b) For a normally distributed vector U with components U(i), we have EetU(i)=exp(t2σU22). Differentiation twice in t gives
EU(i)2e(k+1)βiU(i)=(1+(k+1)2βi2σU2)σU2exp((k+1)2βi2σU22),
and
EU(i)2e(k+1)β⊤UMk,β=(1+(k+1)2βi2σU2)σU2.
Thus,
E‖U‖2e(k+1)β⊤UMk,β=∑i=1m(1+(k+1)2βi2σU2)σU2.
Then (6) holds true.

(c) We have MU(i)(t):=EetU(i)=exp(μi(et−1)−μit). Differentiation twice in t gives
MU(i)″(t)=EU(i)2eU(i)t=μi2(et−1)2MU(i)(t)+μietMU(i)(t),EU(i)2e(k+1)β⊤UMk,β=μi2(e(k+1)βi−1)2+μie(k+1)βi≤const·e2(k+1)·|βi|,
where the factor ‘const’ does not depend of k. Thus,
E‖U‖2e(k+1)β⊤UMk,β≤const·∑i=1me2(k+1)·|βi|,
and condition (6) holds. This completes the proof. □

Now, we can construct estimators of a(t), b(t) and p(t) for t∈[0,τ]. Take Λˆ(t):=∫0tλˆn(2)(s)ds as a consistent estimator of Λ(t), t∈[0,τ]. Indeed, the consistency of λˆn(2)(·) implies
supt∈[0,τ]|Λˆ(t)−Λ(t)|→0
a.s. as n→∞.

For any fixed (λ,β)∈ΘR and for all t∈[0,τ], a sequence
1n∑i=1nB(Wi,t;λ,β)
converges to b(t;λ,β) a.s. due to SLLN. The sequence is equicontinuous a.s. on the compact set ΘR, and the limiting function is continuous on ΘR. The latter three statements ensure that the sequence converges to b uniformly on ΘR. Thus,
bˆ(t)=1n∑i=1nB(Wi;λˆn(2),βˆn(2),Λˆ)→b(t;λ0,β0,Λ),t∈[0,τ],
a.s. as n→∞.

In a similar way for all t∈[0,τ],
aˆ(t)=1n∑i=1nA(Wi;λˆn(2),βˆn(2),Λˆ)→a(t;λ0,β0,Λ)
a.s. and
pˆ(t)=1n∑i=1nP(Wi;λˆn(2),βˆn(2),Λˆ)→p(t;λ0,β0,Λ)
a.s. Then
Tˆ(t)Kˆ(t)=(pˆ(t)−aˆ(t)aˆ⊤(t)bˆ(t))λˆn(2)(t)
is a consistent estimator of T(t)K(t), t∈[0,τ].

The Kaplan–Meier estimator of the survival function of censor C is defined as
GˆC(u)=∏j=1n(N(Yj)N(Yj)+1)Δ˜jIYj≤uifu≤Y(n);0,otherwise,
where Δ˜j:=1−Δj, N(u):=♯{Yi>u,i=1,…,n}u,\hspace{0.2778em}i=1,\dots ,n\}$]]>, and Y(n) is the largest order statistic.

We state the convergence of the Kaplan–Meier estimator. Remember that Y=min{T,C}. Let GY(t) be the survival function of Y.

it holdsmin{GY(S),1−GY(S)}≥δ,for some fixed0<S<∞and0<δ<12.

Then a.s. for alln≥2,sup1≤i≤n,Yi≤S|Gˆn(Yi)−GC(Yi)|=O(lnnn).

In our model, the lifetime T has a continuous survival function, and if we assume that the same holds true for the censor C, then the first condition of Theorem 4 is satisfied. Next, it holds GY(t)=GT(t)GC(t) and due to condition (v) for all small enough positive ε there exists 0<δ<12 such that
δ≤GT(τ−ε)GC(τ−ε)≤1−δ.
Therefore, the second condition holds as well, with S=τ−ε.

Relation (9) is equivalent to the following: there exists a random variable CS(ω) such that a.s. for all n≥2,
sup0≤u≤S|GˆC(u)−GC(u)|≤CS(ω)lnnn.

Let
Mˆ=∫0Y(n)Tˆ(u)Kˆ(u)GˆC(u)du.
We have
‖Mˆ−M‖=||∫0Y(n)(Tˆ(u)Kˆ(u)GˆC(u)−T(u)K(u)GC(u))du++∫Y(n)τT(u)K(u)GC(u)du||≤sup0≤u≤τ‖Tˆ(u)Kˆ(u)−T(u)K(u)‖∫0Y(n)GˆC(u)du+∫0Y(n)‖T(u)K(u)‖·|GˆC(u)−GC(u)|du+GC(Y(n))∫Y(n)τ‖T(u)K(u)‖du.
Due to the above-stated consistency of Tˆ(·)Kˆ(·) and since GˆC is bounded by 1, the first summand in (10) converges to zero a.s. as n→∞.

Consider the second summand. Let S=τ−ε for some fixed ε>00$]]>. There are two possibilities: Y(n)≤S and S<Y(n)≤τ. In the first case,
∫0Y(n)‖T(u)K(u)‖·|GˆC(u)−GC(u)|du≤const·sup0≤u≤S|GˆC(u)−GC(u)|.
In the second case,
∫0Y(n)‖T(u)K(u)‖·|GˆC(u)−GC(u)|du≤const(sup0≤u≤S|GˆC(u)−GC(u)|+∫SY(n)|GˆC(u)−GC(u)|du)≤const(sup0≤u≤S|GˆC(u)−GC(u)|+Y(n)−S).

It holds that Y(n)→τa.s. Utilizing Theorem 4, we first tend n→∞ and then ε→0 and obtain convergence of the second summand of (10) to 0 a.s. as n→∞.

The convergence of Y(n) yields the convergence of the third summand. Finally,
‖Mˆ−M‖→0a.s. asn→∞.

Because Eζi=0, it holds Σβ=4·Eζ1ζ1⊤. Therefore, we take
Σˆβ=4n∑i=1nζiˆζˆi⊤,withζˆi=−Δiaˆ(Yi)bˆ(Yi)+exp(βˆn(2)TWi)MU(βˆn(2))∫0Yiaˆ(u)Kˆ(u)du+∂q∂β(Yi,Δi,Wi,βˆn(2),λˆn(2)),
as an estimator of Σβ. We have
Σˆβ→Σβa.s. asn→∞.
Then
Mˆ−1ΣˆβMˆ−1→M−1ΣβM−1a.s.,
and eventuallyMˆ−1ΣˆβMˆ−1>00$]]>. Convergences (5) and (11) yield
n(Mˆ−1ΣˆβMˆ−1)−1/2(βˆn(2)−β0)→dN(0,Im).
Thus,
‖n(Mˆ−1ΣˆβMˆ−1)−1/2(βˆn(2)−β0)‖2=n(βˆn(2)−β0)⊤(Mˆ−1ΣˆβMˆ−1)−1(βˆn(2)−β0)→dχm2.

Given a confidence probability 1−α, the asymptotic confidence ellipsoid for β is the set
En={z∈Rm|(z−βˆn(2))⊤(Mˆ−1ΣˆβMˆ−1)−1(z−βˆn(2))≤1n(χm2)α}.
Here (χm2)α is the upper quantile of χm2 distribution.

Confidence intervals for the baseline hazard rate

Theorem 1 implies the following statement.

Let0<ε<τ. Assume that the censor C has a bounded pdf on[0,τ−ε]. Under conditions (i) – (x), for any Lipschitz continuous function f on[0,τ]with support on[0,τ−ε],n∫0τ−ε(λˆn(2)−λ0)(u)f(u)du→dN(0,σφ2(f)),whereσφ2(f)=σφ2withφ=(φλ,φβ),φβ=−A−1m(φλ)andφλis a unique solution inC[0,τ]to the Fredholm integral equationφλ(u)K(u)−a⊤(u)A−1m(φλ)=f(u)GC(u),u∈[0,τ].Here we setf(τ)GC(τ)=0. Notice that1GCis Lipschitz continuous on[0,τ−ε].

We show that asymptotic variance σφ2 is positive and construct its consistent estimator.

A random variable ξ is called nonatomic if P(ξ=x0)=0, for all x0∈R.

Suppose that assumptions of Corollary1are satisfied. Additionally assume the following:

m(φλ)≠0, forλ=λ0andβ=β0.

For all nonzeroz∈Rm, at least one of random variablesz⊤Xandz⊤Uis nonatatomic.

Thenσφ2(f)≠0.

We prove by contradiction. For brevity we drop zero index writing φλ=φλ0, φβ=φβ0 and omit arguments where there is no confusion. In particular, we write MU instead of MU(β0) and σφ2 instead of σφ2(f).

Denote η=ξ(C,0,W). From (4) we get
MU2·η=∫0C(αWφλ(u)+γWλ0(u))du,
with
αW:=−MU·exp(β0⊤W),γW:=−φβ⊤(MU·W−E(Ueβ0⊤U).)

Suppose that σφ2=0. This yields ξ=0 a.s. Then
η=ξ·I(Δ=0)=0a.s.
It holds P(Δ=0)>00$]]> and according to (x), C>00$]]> a.s. Thus, in order to get a contradiction it is enough to prove that
P(η=0|C>0)=0.0)=0.\]]]>
Since C and W are independent, it holds
P(η=0|C>0)=E[πx|x=C|C>0],0)=\mathsf{E}[\pi _{x}|_{x=C}\hspace{2.5pt}|\hspace{2.5pt}C>0],\]]]>
where for x∈(0,τ],
πx:=P(∫0x(αWφλ(u)+γWλ0(u))du=0)=P(MU∫0xφλ(u)du+φβ⊤(MU·W−E(Ueβ0⊤U))∫0xλ0(u)du=0)=P(φβ⊤W=vx).
Here vx is a nonrandom real number. In the latter equality we use assumption (vii) to guarantee that ∫0xλ0(u)du>00$]]>.

Further, φβ=−A−1m(φλ)≠0 because according to (xi) m(φλ)≠0. Using independence of X and U together with assumption (xii), we conclude that for all nonzero z∈Rm, z⊤W=z⊤X+z⊤U is nonatomic. Then φβ⊤W is nonatomic as well and πx=0.

Thus, P(η=0|C>0)=00)=0$]]> which proves (13). Therefore, σφ2(f)≠0. □

Now, we can construct an estimator for the asymptotic variance σφ2. Rewrite
A=E[XX⊤eβ0⊤X∫0Yλ0(u)du]=∫0τλ0(u)p(u)GC(u)du.
Let
Aˆ=∫0Y(n)λˆn(2)(u)pˆ(u)GˆC(u)du.
Results of Section 3 yield that Aˆ is a consistent estimator of A. Denote
mˆ(φλ)=∫0Y(n)φλ(u)aˆ(u)GˆC(u)du
and define φˆλ as a solution in L2[0,τ] to the Fredholm integral equation with a degenerate kernel
φλ(u)Kˆ(u)−aˆ⊤Tˆ(u)Aˆ−1mˆ(φλ)=f(u)GˆC(u),u∈[0,τ].Eventually, a solution is unique because the limiting equation (12) has a unique solution. The function φˆλ can be assumed right-continuous and it converges a.s. to φλ from (12) in the supremum norm. Therefore,
φˆβ=−Aˆ−1mˆ(φˆλ)
is a consistent estimator of φβ.

Finally, we construct an estimator of σφ2. Put
σˆφ2=4n−1∑i=1n(ξˆi−ξ¯)2,
with
ξˆi:=Δi·φˆλ(Yi)λˆn(2)(Yi)−exp(βˆn(2)TWi)MU(βˆn(2))∫0Yiφˆλ(u)du+Δi·φˆβ⊤Wi−φˆβ⊤MU(βˆn(2))Wi−EUeβˆn(2)TUMU(βˆn(2))2exp(βˆn(2)TWi)∫0Yiλˆn(2)(u)du
and
ξ¯:=1n∑i=1nξˆi.

Lemma 2 and the consistency of auxiliary estimators yield the following consistency result.

Assume that condition (6) together with conditions (i) – (xii) are fulfilled and censor C has a continuous survival function. Thenσφ2>00$]]>andσˆφ2→σφ2a.s. asn→∞.

For fixed ε>00$]]>, consider an integral functional of the baseline hazard rate, If(λ0)=∫0τ−ελ0(u)f(u)du. Corollary 1 gives
n(If(λˆn(2))−If(λ0))σφ→dN(0,1),
which together with (14) yields
n(If(λˆn(2))−If(λ0))σˆφ→dN(0,1).
Let
In=[If(λˆn(2))−zα/2σˆφn,If(λˆn(2))+zα/2σˆφn],
where zα/2 is the upper quantile of normal law. Then In is the asymptotic confidence interval for If(λ0).

Computation of auxiliary estimators

In Section 3, we constructed estimators in a form of absolutely convergent series expansions. E.g., in Theorem 2 (a) we derived an expansion of such kind for t∈[0,τ]:
B(W,t)=∑k=0∞Bk(W,t),EB(W,t)=b(t)
and
1n∑i=1nB(Wi,t)→b(t),
a.s. as n→∞. Now, we show that we can truncate the series.

Let {Nn:n≥1} be a strictly increasing sequence of nonrandom positive integers. Fix t for the moment and omit this argument t. Consider the head of series B(Wi),
BNi(Wi):=∑k=0NiBk(Wi).
Fix j≥1, then for n≥j it holds:
1n∑i=jn|B(Wi)−BNi(Wi)|≤1n∑i=jn∑k=Ni+1∞|Bk(Wi)|≤1n∑i=jn∑k=Nj+1∞|Bk(Wi)|,lim supn→∞1n∑i=jn|B(Wi)−BNi(Wi)|≤lim supn→∞1n∑i=jn∑k=Nj+1∞|Bk(Wi)|=E∑k=Nj+1∞|Bk(W1)|.
The latter expression tends to zero as j→∞. Therefore, almost surely
limj→∞lim supn→∞1n∑i=jn|B(Wi)−BNi(Wi)|=0.
We conclude that
1n∑i=1nBNi(Wi)→EB(W1)=b(t)
a.s. as n→∞. Moreover, with probability one the convergence is uniform in (λ,β) belonging to a compact set. Therefore, it is enough to truncate the series B(W,t) by some large numbers, which makes feasible the computation of estimators from Section 3.

Conclusion

At the end of Section 3, we constructed asymptotic confidence intervals for integral functionals of the baseline hazard rate λ0(·), and at the end of Section 4, we constructed an asymptotic confidence region for the regression parameter β. We imposed some restrictions on the error distribution. In particular, we handled the following cases: (a) the measurement error is bounded, (b) it is normally distributed, and (c) it has independent components which are shifted Poisson random variables. Based on truncated series, we showed a way to compute auxiliary estimates which are used in construction of the confidence sets.

In future we intend to elaborate a method to construct confidence regions in case of heavy-tailed measurement errors.

ReferencesAugustin, T.: An exact corrected log-likelihood function for Cox’s proportional hazards model under measurement error and some extensions. Chimisov, C., Kukush, A.: Asymptotic normality of corrected estimator in Cox proportional hazards model with measurement error. Cox, D.R.: Regression models and life tables (with discussion). Földes, A., Rejtö, L.: Strong uniform consistency for nonparametric survival curve estimators from randomly censored data. Kukush, A., Baran, S., Fazekas, I., Usoltseva, E.: Simultaneous estimation of baseline hazard rate and regression parameters in Cox proportional hazards model with measurement error. Kukush, A., Chernova, O.: Consistent estimation in Cox proportional hazards model with measurement errors and unbounded parameter set. arXiv preprint arXiv:1703.10940 (2017). MR3666874Kukush, A., Chernova, O.: Consistent estimation in Cox proportional hazards model with measurement errors and unbounded parameter set (Ukrainian). Stefanski, L.A.: Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models.