Copula-based Cox models for dependent current status data with a cure fraction

Proof of Lemma 1:

Define two parameters θ ( 1 ) = ( β ( 1 ) ⊤ , γ ( 1 ) ⊤ , η ( 1 ) ⊤ , Λ T ( 1 ) , Λ C ( 1 ) ) and θ ( 2 ) = ( β ( 2 ) ⊤ , γ ( 2 ) ⊤ , η ( 2 ) ⊤ , Λ T ( 2 ) , Λ C ( 2 ) ) , where θ ⁽¹⁾, θ ⁽²⁾ ∈ Θ. Under the regularity conditions (A1), (A2), and (A4), by applying the Taylor expansion, we obtain

Additionally, based on the calculation [40] (p.94), we can derive

The proof of Lemma 1 is complete.

Lemma 2

(Uniform Convergence) Suppose conditions (A1), (A2), and (A4) hold. Then, we have

almost surely, where P n l ( θ , O ) = 1 n ∑ i = 1 n l ( θ , O i ) and Pl(θ, O) = ∫l(θ, O) dP(O).

Proof of Lemma 2:

We note that when the regularity conditions (A1), (A2) and (A4) are satisfied, |l(θ, X)| is bounded. Therefore, without loss of generality, sup _θ∈Θ|l(θ, O)| ≤ 1, so we have P l 2 ( θ , O ) ≤ P sup θ ∈ Θ | l ( θ , O ) | 2 ≤ 1 . Let a n = n − 1 / 2 + ϕ 1 ( log ⁡ n ) 1 / 2 , where ν/2 < ϕ ₁ < 1/2. Thus, the sequence a _n is a non-increasing positive sequence. For any given ϵ > 0, let ϵ _n = ϵa _n . Then, for any θ ∈ Θ _n and sufficiently large n, we obtain

Let P _n ° denote a measure that places mass ± n ⁻¹ at each observation sample {O ₁, O ₂, …, O _n }, with the random ± independently determined from O _i .

Therefore, based on the above inequality (B1) and Pollard’s paper [41], we can obtain the following inequality:

Given the sample observation data H = { O 1 , O 2 , … , O n } , for any θ ∈ Θ _n , choose θ ⁽¹⁾, …, θ ^(κ), where κ = N ( ϵ , L n , L 1 ( P n ) ) . Suppose the parameter θ* represents the value of θ ^(j) where l(θ, O) achieves its optimal value. Then, there exists.

Thus, we have.

According to the definition of the covering number N ( ϵ n / 2 , L n , L 1 ( P n ) ) , for each θ ^(j), there exists θ ̃ ( j ) ∈ Θ n such that P n | l ( θ ̃ ( j ) , O ) − l ( θ ( j ) , O ) | < ϵ n / 2 . Therefore, we can obtain the following inequality:

Therefore, based on Hoeffding’s inequality [41], we can obtain the conclusion:

Based on inequalities (B2), (B3), and (B4), we can obtain the following inequality:

From inequality (B5), it can be seen that the right-hand side of inequality (B5) does not depend on the observation data H . Therefore, by taking the expectation over the observation data H = { O 1 , O 2 , … , O n } , we obtain.

Combining inequality (B6) and the symmetry of the inequalities, we get the following inequality:

Therefore, we can conclude that ∑ n = 1 ∞ P sup θ ∈ Θ n | P l ( θ , O i ) − P l ( θ , O i ) | > 8 ε n < ∞ . By the Borel-Cantelli Lemma, we have sup θ ∈ Θ n | P n l ( θ , O ) − P l ( θ , O ) | → 0 almost surely. The proof of Lemma 2 is complete.

Proof of Theorem 1:

Based on the conclusions of Lemma 1 and Lemma 2, it is known that the covering number of the class L n = { l ( θ , O ) : θ ∈ Θ n } satisfies N ( ϵ , L n , L 1 ( P n ) ) ≤ K M p + 2 q M n 2 ( m + 1 ) ϵ − ( p + 2 ( q + m + 1 ) ) , and that

For a given ϵ > 0, we define the following:

Therefore, we can prove that.

If the parameter θ ̌ ∈ K ϵ , we can obtain.

Combining inequalities (B8) and (B9), we get inf K ϵ P M ( θ , O ) ≤ ξ n + P M ( θ 0 , O ) , where ξ _n = ξ ₁ n + ξ ₂ n.According to condition (A3), we have.

Therefore, we have ξ _n ≥ δ _ϵ , which implies that { θ ̌ ∈ K ϵ } ⊆ { ξ n ≥ δ ϵ } . According to the law of large numbers and the conclusion of Lemma 2, we have ξ _1n → 0 and ξ _2n → 0 almost surely. Hence, ⋃ k = 1 ∞ ⋂ n = k ∞ { θ ̌ n ∈ K ϵ } ⊆ ⋃ k = 1 ∞ ⋂ n = k ∞ { ξ n ≥ δ ϵ } , which establishes that d(θ, θ ₀) → 0. The proof of Theorem 1 is complete.

Proof of Theorem 2:

Next, we will use Theorem 3.4.1 [40] to discuss the convergence rate of θ ̂ n . First, by using Theorem 1.6.2 [42], we know that there exist Bernstein polynomials Λ _Tn0 and Λ _Cn0 , such that ‖ Λ T 0 − Λ Tn0 ‖ ∞ = O ( n − r ν 2 ) and ‖ Λ C 0 − Λ Cn0 ‖ ∞ = O ( n − r ν 2 ) . For any ρ > 0, define the class F ρ = { l ( θ , O ) − l ( θ n 0 , O ) : θ ∈ Θ n , d ( θ , θ n 0 ) ≤ ρ } , where θ n 0 = ( β n 0 , γ n 0 , η n 0 , Λ Tn0 , Λ Cn0 ) . Through the calculation [43](p. 597), we get log N [ ] ( ϵ , F ρ , ‖ ⋅ ‖ ) ≤ K N ⁡ log ( ρ / ϵ ) , N = 2 ( m + 1 ) , where N [ ] ( ϵ , F ρ , ‖ ⋅ ‖ ) represents the bracket number of the function class F with respect to the metric or semi-metric d. Furthermore, for any l ( θ n 0 , O ) − l ( θ , O ) ∈ F ρ , we have ‖ l ( θ n 0 , O ) − l ( θ , O ) ‖ 2 2 ≤ K ρ 2 . Thus, by Lemma 3.4.2 [40], we can obtain

where H ρ ( ϵ , F ρ , ‖ ⋅ ‖ 2 ) = ∫ 0 ρ 1 + log N [ ] ( ϵ , F ρ , ‖ ⋅ ‖ 2 ) 1 / 2 d ϵ . We can see that X n ( ρ ) ρ is a decreasing function of ρ, and satisfies r n 2 X n 1 r n = r n N 1 2 + r n 2 N n 1 2 < 2 n 1 2 , where r n = N − 1 2 n 1 2 = n ( − ν + 1 ) / 2 , 0 < ν < 0.5 . By Theorem 3.2.5 [40], we can obtain n ( 1 − ν ) / 2 d ( θ ̂ , θ n 0 ) = O P ( 1 ) . Also, by Theorem 1.6.2 [42], we get d ( θ n 0 , θ 0 ) = O p ( n − r ν 2 ) . Finally, we obtain the convergence rate d ( θ ̂ , θ 0 ) = O p n − ( 1 − ν ) / 2 + n − r ν / 2 , which implies that ‖ Λ ̂ T n − Λ T 0 ‖ 2 + ‖ Λ ̂ C n − Λ C 0 ‖ 2 = O q n − ( 1 − ν ) / 2 + n − r / 2 .

Proof of Theorem 3:

Define the linear expansion of Θ − θ ₀ as V, where θ 0 = β 0 ⊤ , γ 0 ⊤ , η 0 ⊤ , Λ T 0 , Λ C 0 is the true value of θ, and let Θ₀ represent the true parameter space. Let l(θ; O) denote the log-likelihood function, and define δ _n = n ^−rν/2 + n ^{−(1−ν)/2}. For any θ ∈ {θ ∈ Θ₀: d(θ, θ ₀) = O(δ _n )}, define the first and second directional derivatives of l(θ; O) in the direction v ∈ V as follows:

We define the Fisher inner product on the space V as ⟨ v , v ̃ ⟩ = P l ̇ ( θ ; O ) [ v ] | l ̇ ( θ ; O ) [ v ̃ ] , and for v ∈ V, the Fisher norm is ‖v‖2 = ⟨v, v⟩. Let V ̄ denote the closed linear expansion of the space V under the Fisher norm, and it can be proven that ( V ̄ , ‖ ⋅ ‖ ) is a Hilbert space.

Next, we define the linear function of θ as γ ( θ ) = h 1 ⊤ β + h 2 ⊤ γ + h 3 ⊤ η , where h = h 1 ⊤ , h 2 ⊤ , h 3 ⊤ ⊤ is an arbitrary vector of dimension p + 2q, and ‖h‖ ≤ 1. Additionally, let ϑ = β ⊤ , γ ⊤ , η ⊤ ⊤ , and for any v = v ϑ , φ T , φ C ∈ V , we define.

We can easily derive that γ ( θ ) − γ ( θ 0 ) = γ ̇ ( θ 0 ) [ θ − θ 0 ] . Also, by the Riesz representation theorem, for all v ∈ V ̄ , there exists v * ∈ V ̄ such that γ ̂ ( θ 0 ) [ v ] = ⟨ v * , v ⟩ , and ‖ v * ‖ 2 = ‖ γ ̂ ( θ 0 ) ‖ . Since h ⊤ ( ( β ̂ − β 0 ) ⊤ , ( γ ̂ − γ 0 ) ⊤ , ( η ̂ − η 0 ) ⊤ ) = γ ( θ ̂ n ) − γ ( θ 0 ) = γ ̇ ( θ 0 ) [ θ ̂ n − θ 0 ] = ⟨ θ ̂ n − θ 0 , v * ⟩ , to prove Theorem 3 by the Cramr-Wold theorem, we need to show

in distribution. First, we need to prove that n θ ̂ n − θ 0 , v * → N ( 0 , ‖ v * ‖ 2 ) in distribution, and then prove that ‖v*‖2 = h⊤Σh.

First, by condition (A4) and Theorem 1.6.2 [42], we know that for any v* ∈ Θ0, there exists ∏nv* ∈ Θn such that ∏ n v * − v * = o ( 1 ) and δn‖Πnv* − v*‖ = o(n−1/2). At the same time, we define r [ θ − θ 0 , O ] = l ( θ ; O ) − l ( θ 0 , O ) − l ̇ ( θ , O ) [ θ − θ 0 ] , and from the definition of θ ̂ , we can obtain.

For I ₁, by condition (A1), condition (A2), Chebyshev’s inequality, and ‖Πnv* − v*‖ = o(1), we can obtain I ₁ = ϵnop(n−1/2). For I ₂, we have

where θ ̃ is between θ ̂ n and θ ̂ n ± ϵ n ∏ n v * . From Theorem 2.8.4 [40], we know that l ̇ ( θ , O ) Π n v * − l ̇ ( θ 0 , O ) Π n v * : ‖ θ − θ 0 ‖ = O ( δ n ) is a Donsker class. Therefore, by Lemma 2.3.12 [40], we get I 2 = ϵ n × o p n − 1 / 2 .

For I ₃, we have

where θ ̃ lies between θ0 and θ, and the last inequality follows from the Taylor expansion and conditions (A1) and (A2). Therefore,

where the last inequality in the above derivation is due to the Cauchy-Schwarz inequality, along with δ n ∏ n v * − v * = o ( n − 1 / 2 ) and ∏ n v * 2 → ‖ v * ‖ 2 . Additionally, combining P l ̇ ( θ 0 , O ) [ v * ] = 0 , we obtain.

Therefore, we obtain n ⟨ θ ̂ n − θ 0 , v * ⟩ = n ( P n − P ) l ̇ ( θ 0 , O ) [ v * ] + o p ( 1 ) → N ( 0 , ‖ v * ‖ 2 ) , where the asymptotic normality is guaranteed by the central limit theorem, with asymptotic variance ‖ v * ‖ 2 = ‖ l ̇ ( θ 0 ; O ) [ v * ] ‖ 2 . This implies n 1 / 2 ( γ ( θ ̂ n ) − γ ( θ 0 ) ) = n 1 / 2 ⟨ θ ̂ n − θ 0 , v * ⟩ + o p ( 1 ) → N ( 0 , ‖ v * ‖ 2 ) .