A Proofs of asymptotic results

To establish the asymptotic properties of the proposed estimators, we need the following regularity conditions: (C1)P(C≤τ)>0, and N(τ) is bounded almost surely; (C2) the covariate Z is bounded; (C3) k∗(t) is right continuous with left-hand limits, and has bounded total variation on [0, τ]; (C4) the limit of matrix Dˆ(τ,β), D(τ, β), is non-singular.

Proof of Theorem 1. Using expression (5) and the Volterra equation [16], it can be written that

Then, by the standard counting process martingale theory, (11) implies that kˆ0(t;β∗) converges almost surely to k∗(t) in t ∈ [0, τ]. Denote minus the first derivative of U(τ, β) with respect to β by I(τ, β), where it can be obtained that

By the uniform strong law of large numbers [22], it follows that n−1I(τ,β) converges almost surely to a non random function D(τ, β) uniformly in β, where

with µ(t) and μ˜(t) are the limit in probability of Zˉ(t) and Z˜(t), respectively. Because n−1U(τ,β∗) converges to 0 almost surely, and D(τ, β) is non-singular by condition (C4), there must exist a small neighborhood of β∗ in which n−1I(τ,β) is non-singular. Hence it follows from the inverse function theorem [23] that within a small neighborhood of β∗, there exists a unique solution βˆ to U(τ, β) = 0 for all large n. Since this neighborhood of β∗ can be arbitrarily small, the preceding proof also implies that βˆ is strongly consistent. It then follows from the uniform convergence of kˆ0(t;β∗) to k0(t;β∗) that kˆ0(t) ≡kˆ0(t;βˆ)→k0(t;β∗) ≡k∗(t) almost surely uniformly in t ∈ [0, τ].

Proof of Theorem 2(i). Consider a decomposition of n−1/2U(τ,β∗;kˆ0(t;β∗)) as

By the representation of kˆ0(t;β∗)−k∗(t) in (11) and interchanging the order of integration, the first term in this decomposition is −n−1/2∑i=1n∫0τZ˜(t)k∗(t)dMi(t). Then it can be obtained that

Therefore, by the uniform strong law of large numbers and using the Lemma of [24], n−1/2U(τ,β∗;kˆ0(t;β∗)) can be decomposed as a sum of independent and identically distributed terms

where ξi=∫0τ{Zi−μ(t)−μ˜(t)}k∗(t)dMi(t), and the op(1) is uniform in t. Utilizing the multivariate central limit theorem, n−1/2U(τ,β∗;kˆ0(t;β∗)) converges in distribution to zero-mean normal distribution whose variance-covariance matrix Σ=E{ξi⊗2} can be consistently estimated by Σˆ=n−1∑i=1nξˆi⊗2 as defined in Theorem 2(i).

Proof of Theorem 2(ii). A Taylor series expansion of the score function (8) around βˆ gives

where β∗∗ is on the line segment between β∗ and βˆ. From the uniform convergence of n−1I(τ,β) to a non-singular matrix D(τ, β) (refer to condition (C4)) along with the consistency of βˆ and representation (12), asymptotic approximation for n1/2(βˆ−β∗) can be displayed by

Thus, it follows that n1/2(βˆ−β∗) is asymptotically normal with mean zero and covariance matrix D(τ,β∗)−1ΣD(τ,β∗)−1, which can be consistently estimated by Dˆ(τ,βˆ)−1ΣˆDˆ(τ,βˆ)−1 as defined in Theorem 2(ii).

Proof of Theorem 3. To show the weak convergence of n1/2{kˆ0(t)−k∗(t)}, we first note that

It follows from (11) that n1/2kˆ0(t;β∗)−k∗(t)=n−1/2∑i=1nζi(t)+op(1), where ζi(t)=−q(t;β∗)−1∫0tq(s;β∗){k∗(s)/y(s)}dMi(s), with q(t; β) and y(t) are the limit in probability of Q(t; β) and n−1Y(t), respectively. Taking the Taylor expansion of kˆ0(t,βˆ), together with the consistency of βˆ and the uniform strong law of large numbers, we have n1/2{kˆ0(t,βˆ)−kˆ0(t,β∗)}=−B(t;β∗)n1/2(βˆ−β∗)+op(1), where B(t;β)=q(t;β)−1∫0tq(s;β)E{Yi(s)ZiTexp(−ZiTβ)}/y(s)ds, denotes the limit in probability of −∂kˆ0(t;β)/∂β. Therefore, it follows from (13) that uniformly in t ∈ [0, τ],

where ϕi(t)=ζi(t)−B(t;β∗)D(τ,β∗)−1ξi are independent and identically distributed zero-mean random variables for each t. By the multivariate central limit theorem, n1/2{kˆ0(t)−k∗(t)} converges in finite-dimensional distribution to a zero-mean Gaussian process for 0≤t≤τ. The processes {ζi(t);i=1,…,n} can be written as sums or products of monotone functions of t because any function of bounded variation can be expressed as the difference of two increasing functions. Therefore, they are manageable and using the functional central limit theorem [22], the first term on the right-hand side of (14) is tight. The second term is also tight because n−1/2∑i=1nξi converges in distribution and B(t,β∗) is a deterministic function. Thus n1/2kˆ0(t)−k∗(t) is tight and converges weakly to a zero-mean Gaussian process whose covariance function Γ(s,t)=E{ϕi(s)ϕi(t)} can be consistently estimated by Γˆ(s,t) defined in Theorem 3.

Proof of (9) in Section (3). Taking the Taylor series expansion of U(t,βˆ) at β∗ yields that

where I(t, β) is minus the first derivative of U(t, β) with respect to β. Then it can be written that n1/2(βˆ−β∗)=[n−1I(τ,βˆ)]−1n−1/2U(τ,β∗)+op(1). Therefore, we have

It follows from eq. (12) that n−1/2U(τ,β∗) can be written in terms of a sum of independent and identically distributed zero-mean decompositions. Hence using the resampling approach as in [18], distribution of n−1/2U(τ,β∗) is asymptotically equivalent to the process n−1/2∑i=1nξiˆΩi, where Ω1,…,Ωn are independent standard normally distributed random variables. Therefore, from the uniform convergence of n−1I(τ,β) to D(τ, β) and resampling approach as in [18], the distribution of U(t,βˆ) can be approximated by Uˆ(t,βˆ)=∑i=1nξˆi(t)−Dˆ(t,βˆ)Dˆ(τ,βˆ)−1ξˆiΩi, where Dˆ(t,βˆ)=−n−1∑i=1n∫0t{Zi−Zˉ(s)−Z˜(s)}Yi(s)ZTiexp(−ZTiβˆ)ds, and ξˆi(t)=∫0t{Zi−Zˉ(s)−Z˜(s)}kˆ0(s)dMˆi(s).