Joint Analysis of Survival Time and Longitudinal Categorical Outcomes

Abstract

In biomedical or public health research, it is common for both survival time and longitudinal categorical outcomes to be collected for a subject, along with the subject’s characteristics or risk factors. Investigators are often interested in finding important variables for predicting both survival time and longitudinal outcomes which could be correlated within the same subject. Existing approaches for such joint analyses deal with continuous longitudinal outcomes. New statistical methods need to be developed for categorical longitudinal outcomes. We propose to simultaneously model the survival time with a stratified Cox proportional hazards model and the longitudinal categorical outcomes with a generalized linear mixed model. Random effects are introduced to account for the dependence between survival time and longitudinal outcomes due to unobserved factors. The Expectation–Maximization (EM) algorithm is used to derive the point estimates for the model parameters, and the observed information matrix is adopted to estimate their asymptotic variances. Asymptotic properties for our proposed maximum likelihood estimators are established using the theory of empirical processes. The method is demonstrated to perform well in finite samples via simulation studies. We illustrate our approach with data from the Carolina Head and Neck Cancer Study (CHANCE) and compare the results based on our simultaneous analysis and the separately conducted analyses using the generalized linear mixed model and the Cox proportional hazards model. Our proposed method identifies more predictors than by separate analyses.

Appendices

Appendix A: EM Algorithms

1.1 A.1 EM Algorithm—Binary Longitudinal Data and Survival Time

(1) E-step: For binary longitudinal outcomes and survival time, we calculate the conditional expectation of q(b _i ) for subject i with S _i =s given the observations and the current estimate $(\boldsymbol{\theta}^{(k)}, \varLambda_{s}^{(k)})$ for some known function q(⋅). The conditional expectation denoted by $E [ q(\boldsymbol{b}_{i}) | \boldsymbol{\theta}^{(k)}, \varLambda_{s}^{(k)} ]$ can be expressed as the following:

Given the current estimate $(\boldsymbol{\theta}^{(k)}, \varLambda_{s}^{(k)})$,

(6)

where

(7)

$( \boldsymbol{\Sigma}_{b}^{(k)} )^{\frac{1}{2}}$ is an unique non-negative square root of $\boldsymbol{\Sigma}_{b}^{(k)}$ (i.e. $( \boldsymbol{\Sigma}_{b}^{(k)} )^{\frac{1}{2}} \times ( \boldsymbol{\Sigma}_{b}^{(k)} )^{\frac{1}{2}} = \boldsymbol {\varSigma}_{b}^{(k)}$), and z _G follows a multivariate Gaussian distribution with mean zero.

(2) M-step: Since the parameter ϕ is set to 1 for logistic distribution, we estimate only β in the longitudinal process. β ^(k+1) solves the conditional expectation of complete data log-likelihood score equation, using one-step Newton–Raphson iteration,

$\boldsymbol{\Sigma}_{b} ^{(k+1)}$, ψ ^(k+1),γ ^(k+1) and $\varLambda_{s}^{(k+1)}$ have the same expressions as in Sect. 2.2.

1.2 A.2 EM Algorithm—Poisson Longitudinal Data and Survival Time

(1) E-step: For Poisson longitudinal outcomes and survival time, given the current estimate $(\boldsymbol{\theta}^{(k)}, \varLambda_{s}^{(k)})$, the conditional expectation denoted by $E [ q(\boldsymbol{b}_{i}) | \boldsymbol{\theta}^{(k)}, \varLambda_{s}^{(k)} ]$ can be expressed as in (6) with R(z _G ) defined in (7):

and z _G follows a multivariate Gaussian distribution with mean zero.

(2) M-step: Since the parameter ϕ is set to 1 for Poisson distribution, we estimate only β in the longitudinal process. β ^(k+1) solves the conditional expectation of complete data log-likelihood score equation, using one-step Newton–Raphson iteration,

$\boldsymbol{\Sigma}_{b} ^{(k+1)}$, ψ ^(k+1),γ ^(k+1) and $\varLambda_{s}^{(k+1)}$ have the same expressions as in Sect. 2.2.

Appendix B: Proofs for Theorems

In Appendices B.1 and B.2, we sketch the proofs for Theorems 1 and 2. All detailed technical proofs are available from the authors. From (3) in Sect. 2.2, we can have the observed log-likelihood function, l _f (θ,Λ;Y,V)=log{L _f (θ,Λ;Y,V)}. We obtain and use the following modified object function, by replacing λ _s (V _i ) with Λ _s {V _i } in l _f (θ,Λ;Y,V) where Λ _s {V _i } is the jump size of Λ _s (t) at the observed time V _i with Δ _i =1,

(8)

and $(\widehat{\boldsymbol{\theta}},\widehat{\boldsymbol{\Lambda }})$ maximizes l _n (θ,Λ) over the space $\{ (\boldsymbol{\theta}, \boldsymbol{\Lambda}) : \boldsymbol{\theta}\in\varTheta, \boldsymbol {\varLambda}\in \mathbb{W}\times\mathbb{W}\times \cdots\times\mathbb{W}\}$, where $\mathbb{W}$ consists of all the right-continuous step functions only; that is, $\boldsymbol{\Lambda}= (\varLambda_{1}, \ldots, \varLambda_{S})^{T}, s=1, \ldots, S, \varLambda_{s} \in \mathbb{W}$. For the proofs of both Theorems 1 and 2, the modified object function is used in place of the observed log-likelihood function.

2.1 B.1 Proof of Consistency—Theorem 1

Consistency can be proved by verifying the following three steps: First, we show the maximum likelihood estimate $(\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{\Lambda}})$ exists. Second, we show that, with probability one, $\widehat{\varLambda}_{s} (\tau)$, s=1,…,S, are bounded as n→∞. Third, if the second step is true, by Helly’s selection theorem [38], we can choose a subsequence of $\widehat{\varLambda}_{s}(t)$ such that $\widehat{\varLambda}_{s}(t)$ weakly converges to some right-continuous monotone function $\varLambda^{*}_{s}(t)$ with probability one. For any subsequence, we can find a further subsequence, still denoted as $\widehat{\boldsymbol{\theta}}$, such that $\widehat{\boldsymbol {\theta}} \rightarrow\boldsymbol{\theta}^{*}$. Thus, in the third step, we show θ ^∗=θ ₀ and $\varLambda_{s}^{*} = \boldsymbol{\Lambda}_{s0}$, s=1,…,S. Once the three steps are completed, we can conclude that, with probability one, $\widehat{\boldsymbol{\theta}}$ converges to θ ₀ and $\widehat{\boldsymbol{\Lambda}}_{s}(t)$ converges to Λ _s0(t) in [0,τ], s=1,…,S. Moreover, since Λ _s0(t) is right-continuous in [0,τ], the latter can be strengthened to uniform convergence; that is, $\sup_{t \in[0, \tau]} \| \widehat{\boldsymbol{\Lambda}} (t) - \boldsymbol{\Lambda}_{0} (t) \| \rightarrow0$ a.s. Then, the proof of Theorem 1 will be done.

In the first step, since θ belongs to a compact set Θ by Assumption (A1), it is sufficient to show that Λ _s {V _i } with Δ _i =1 is finite. For each subject i with Δ _i =1, after simple algebra, we have that, from (8),

If Λ _s {V _i }→∞ for some i with Δ _i =1, then l _n (θ,Λ)→−∞, which is contradictory to that l _n (θ,Λ) is bounded. Therefore, we conclude that Λ _s {⋅} must be finite. By the conclusion and assumption (A1), the maximum likelihood estimate $(\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{\Lambda}})$ exists.

In the second step, we define $\widehat{\zeta}_{s} = \log\widehat{\varLambda}_{s} (\tau)$ and rescale $\widehat{\varLambda}_{s}$ by the factor $e^{\widehat{\zeta}_{s}}$. Then, we let $\widetilde{\varLambda}_{s}$ denote the rescaled function; that is, $\widetilde{\varLambda}_{s} (t) = \widehat{\varLambda}_{s} (t) / \widehat{\varLambda}_{s} (\tau) = \widehat{\varLambda}_{s} (t) e^{-\widehat{\zeta}_{s} }$. thus, $\widetilde{\varLambda}_{s} (\tau) =1$. To prove this second step, it is sufficient to show that $\widehat{\zeta}_{s}$ is bounded. After some algebra in (8), we obtain that, for any $\boldsymbol{\Lambda}\in\mathbb{W}\times \mathbb{W} \cdots\times\mathbb{W}$,

where

and

Thus, since $0 \le n^{-1} l_{n} (\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{\Lambda}}) - n^{-1} l_{n} (\widehat{\boldsymbol {\theta}}, \widetilde{\boldsymbol{\Lambda}})$ where $\widehat{\boldsymbol {\varLambda}} = e^{\widehat{\boldsymbol{\xi}}} \circ\widetilde{\boldsymbol {\varLambda}}$, it follows that

(9)

According to the assumption (A2), there exist some positive constants C1, C2 and C3 such that $| Q_{1i} ( t, \boldsymbol{b}_{0}, \widehat{\boldsymbol{\theta}})| \le C_{1} \|\boldsymbol {b}_{0}\|+C_{2}\|\boldsymbol{Y}_{i} \| + C_{3}$. By denoting b ₀ as a vector of variables following a standard multivariate normal distribution, from concavity of the logarithm function, in the third term of (9),

where C ₄ and C ₅ are positive constants. Since it is easily verified that

$$\operatorname{E}_{\boldsymbol{b}_0} \Biggl[ \sum_{ j= 1}^{n_i} \frac{B(\widehat{\boldsymbol{\beta}}; \boldsymbol{b}_0)}{A(D_{i} (t_{j} ; \widehat{\phi} ) )} + e^{ C_1 \|\boldsymbol{b}_0\|+C_2\|\boldsymbol{Y}_{ i} \| + C_3 } \Biggr] < \infty, $$

by the strong law of large numbers and the assumption (A4), the third term of (9) can be bounded by a constant C ₆. i.e. $- \frac{1}{n} \sum_{i=1}^{n} H_{i} \le\frac{1}{n} \sum_{i=1}^{n} (e^{C_{2}\|\boldsymbol{Y}_{i} \| + C_{4}} + C_{5}) \triangleq C_{6}$. Then, (9) becomes

(10)

where C ₇ is a constant. On the other hand, since, for any Γ≥0 and x>0, Γlog(1+x/Γ)≤Γx/Γ=x, we have that e ^−x≤(1+x/Γ)^−Γ. Therefore, with $Q_{1i} (t, \boldsymbol{b}_{0}, \widehat{\boldsymbol{\theta}} ) \ge-C_{1} \|\boldsymbol{b}_{0}\| - C_{2} \| \boldsymbol{Y}_{i} \| -C_{3}$, (10) gives that

(11)

where C ₈(Γ) is a deterministic function of Γ. For the sth stratum, (11) is

By the strong law of large numbers, $\sum_{i=1}^{n} I(V_{i}=\tau) I(S_{i}=s) / n \longrightarrow P (V_{i} = \tau, S_{i} = s) > 0$. Then, we can choose Γ large enough such that $\sum_{i=1}^{n} \Delta_{i} I(S_{i}=s)/ n \le(\varGamma/2n) \sum_{i=1}^{n} I(V_{i}=\tau) I(S_{i}=s)$. Thus, we obtain $0 \le C_{7} + C_{8} (\varGamma) - \frac{\varGamma }{2n}\times \sum_{i=1}^{n} I(V_{i}=\tau) I(S_{i}=s) \widehat{\zeta}_{s}$ . In other words,

If we denote B _s0=exp{2(C ₇+C ₈(Γ))/(ΓP(V _i =τ,S _i =s))}, we conclude that $\widehat{\varLambda}_{s} (\tau) \le B_{s0}, s=1, \ldots, S$. Note that the above arguments hold for every sample in the probability space except a set with zero probability. Therefore, we have shown that, with probability one, $\widehat{\varLambda}_{s} (\tau)$ is bounded for any sample size n.

In the third step, for convenience, we omit the index i. Then, for the number of observed longitudinal measurements per subject, we use n _N instead of the n _i without subscript i since we denoted sample size as n. Use O to abbreviate the observed statistics $(\boldsymbol{Y}, \boldsymbol{X}, \widetilde {\boldsymbol{X}} , V, \Delta, n_{N}, s )$ and $\{ \boldsymbol{Z}(t), \widetilde{\boldsymbol {Z}} (t), 0 \le t \le V \}$ for a subject, and define

and a class , where B _s0 is the constant given in the second step and $\mathbb{W}$ contains all nondecreasing functions in [0,τ]. Employing empirical process formulation, the class can be proved to be as P-Donsker by Theorems 2.5.6 and 2.7.5 in [39]. In stratum s, denote m _s as the number of subjects, and V _sk and Δ _sk as the observed time and censoring indicator for the kth subject, respectively. By differentiating (8) with respect to Λ _s {V _sk }, we obtain

$$ \widehat{\varLambda}_s \{ V_{sk} \} = \frac{\Delta_{sk} }{ m_s \mathbf{P}_{m_s} \{ I(V_s \ge v) Q (v, \boldsymbol{O}; \widehat {\boldsymbol{\theta}}, \widehat{\varLambda}_s) \} |_{v=V_{sk}}}. $$

We also construct $\bar{\varLambda}_{s}(t)$, another step function with the jump size $\bar{\varLambda}_{s} \{ V_{sk} \}$, given by

$$ \bar{\varLambda}_s \{ V_{sk} \} = \frac{\Delta_{sk} }{ m_s \mathbf{P}_{m_s} \{ I(V_s \ge v) Q (v, \boldsymbol{O}; \boldsymbol {\theta}_0, \varLambda_{s0} ) \} |_{v=V_{sk}} }. $$

Through the arguments using empirical process and relevant properties of P-Donsker and Glivenko–Cantelli class, we can prove that $\bar{\varLambda}_{s} (t)$ uniformly converges to Λ _s0(t) in [0,τ]. Next, by the bounded convergence theorem, the fact that $\widehat{\boldsymbol {\theta}}$ converges to θ ^∗ and $\widehat{\varLambda}_{s}$ weakly converges to $\varLambda_{s}^{*}$ , and the Arzela–Ascoli theorem, we can prove that $\widehat{\varLambda}_{s} (t)$ uniformly converges to $\varLambda_{s}^{*} (t)$. Then, from $n^{-1} l_{n} (\widehat{\boldsymbol {\theta}}, \widehat{\boldsymbol{\Lambda}} ) - n^{-1} l_{n} (\boldsymbol{\theta}_{0}, \bar{\boldsymbol{\Lambda}}) \ge0$, using the properties of Glivenko–Cantelli class and Kullback–Leibler information, the following holds, with probability one,

$$ \bigl(\lambda_s^* (V_s) \bigr)^{\Delta_s} \int_{\boldsymbol{b}} G \bigl(\boldsymbol{b}, \boldsymbol{O}, \boldsymbol{\theta}^*, \varLambda_s^* \bigr) \,d \boldsymbol{b} = \bigl(\lambda_{s0} (V_s) \bigr)^{\Delta_s} \int_{\boldsymbol{b}} G (\boldsymbol{b}, \boldsymbol{O}, \boldsymbol{\theta}_0, \varLambda _{s0} ) \,d\boldsymbol{b}. $$

(12)

Our proof will be completed if we can show θ ^∗=θ ₀ and $\varLambda_{s}^{*} = \varLambda_{s0}$ from (12). To show that β ^∗=β ₀, ϕ ^∗=ϕ ₀ and $\boldsymbol {\varSigma}_{b}^{*} = \boldsymbol{\Sigma}_{b0}$, we let Δ _s =0 and V _s =0 in (12). By the comparison of the coefficients of Y ^T Y and Y in the exponential part and the constant term out of the exponential part and assumption (A5), we obtain ϕ ^∗=ϕ ₀, β ^∗=β ₀ and $\boldsymbol {\varSigma}_{b}^{*} = \boldsymbol{\Sigma}_{b0}$. To show that ψ ^∗=ψ ₀, γ ^∗=γ ₀ and $\varLambda_{s}^{*} = \varLambda_{s0}$, we let Δ _s =0 in (12). By the similar arguments done for β ^∗=β ₀, ϕ ^∗=ϕ ₀ and $\boldsymbol{\Sigma}_{b}^{*} = \boldsymbol{\Sigma}_{b0}$, both sides of (12) with Δ _s =0 are expressed as the expected values with respect to the random effects b following a multivariate normal distribution with mean $\boldsymbol{\Sigma}_{b0}^{1/2} ( \sum_{j=1}^{n_{N}} Y_{j} \widetilde{\boldsymbol{X}}_{j} / A(D (t_{j} ; \phi_{0}) ) )^{T}$ and covariance Σ _b0. By the fact that for any fixed $\widetilde{\boldsymbol{X}}$, treating $\widetilde{\boldsymbol {X}}^{T} \boldsymbol{Y}$ as a parameter in the normal family, b is the complete statistic for $\widetilde{\boldsymbol{X}}^{T} \boldsymbol{Y}$, and the assumptions (A2) and (A5), we have ψ ^∗=ψ ₀, γ ^∗=γ ₀ and $\varLambda_{s}^{*} = \varLambda_{s0}$. Therefore, the proof of Theorem 1 is completed.

2.2 B.2 Proof of Asymptotic Normality—Theorem 2

Asymptotic distribution for the proposed estimators can be shown if we can verify the conditions of Theorem 3.3.1 in [39]. Then, it will be shown that the distribution is Gaussian. For completeness, we use Theorem 4 in [25] which restated the Theorem 3.3.1 of [39].

Theorem 4

[25]

Let U _n and U be random maps and a fixed map, respectively, from ξ to a Banach space such that:

• (a) $\sqrt{n} (U_{n} - U) ( \widehat{\xi}_{n} ) - \sqrt{n} (U_{n} - U) ( \xi_{0} ) = o_{P}^{*} (1+ \sqrt{n} \| \widehat{\xi}_{n} - \xi_{0} \| )$.

• (b) The sequence $\sqrt{n} (U_{n} - U) ( \xi_{0} )$ converges in distribution to a tight random element Z.

• (c) the function ξ→U(ξ) is Fréchet differentiable at ξ ₀ with a continuously invertible derivative $\nabla U_{\xi_{0}}$ (on its range).

• (d) $U_{\xi_{0}}$ and $\widehat{\xi}_{n}$ satisfies $U_{n} ( \widehat {\xi }_{n} ) = o_{P}^{*} (n^{-1/2})$ and converges in outer probability to ξ ₀.

Then $\sqrt{n} ( \widehat{\xi}_{n} - \xi_{0} ) \Rightarrow\nabla U_{\xi_{0}}^{-1} Z$.

In our situation, the parameter ξ _s =(θ,Λ _s )∈Ξ={(θ,Λ _s ):∥θ−θ ₀∥+sup _t∈[0,τ]|Λ _s (t)−Λ _s0(t)|≤δ,s=1,…,S} for a fixed small constant δ. We define , where ∥h ₂∥ _V is the total variation of h ₂ in [0,τ] defined as

$$ \sup_{0=t_0 \le t_2 \le\cdots\le t_k= \tau} \sum_{j=1}^{k} \big|h_2 (t_j) - h_2 (t_{j-1}) \big|, $$

and also define that, for stratum s,

where l _θ (θ,Λ _s ) is the first derivative of the log-likelihood function from one single subject belonging to stratum s, denoted by l(O;θ,Λ _s ), with respect to θ, and $l_{\varLambda_{s}} (\boldsymbol {\theta}, \varLambda_{s} )$ is the derivative of l(O;θ,Λ _sε ) at ε=0, where $\varLambda_{s \varepsilon} (t) = \int_{0}^{t} (1+\varepsilon h_{2} (u) ) \,d \varLambda_{s0} (u)$. Therefore, both $U_{m_{s}}$ and U _s map from Ξ to , and $\sqrt{m_{s}} \{ U_{m_{s}} (\xi_{s}) - U_{s} (\xi_{s}) \}$ is an empirical process in the space .

Denoting $\mathbf{h}_{1} = (\mathbf{h}_{1}^{\beta T}, \mathbf{h}_{1}^{\phi T}, \mathbf{h}_{1}^{b T} , \mathbf{h}_{1}^{\psi T} , \mathbf{h}_{1}^{\gamma T} )^{T}$ corresponding to $\boldsymbol{\theta}= (\boldsymbol{\beta}^{T}, \boldsymbol{\phi}^{T}, \operatorname{Vech}(\boldsymbol{\Sigma}_{b})^{T},\boldsymbol{\psi}^{T},\boldsymbol{\gamma}^{T})^{T}$, for any , the class

can be shown as P-Donsker. From the P-Donsker property, it is also implied that

as ∥θ−θ ₀∥+sup _t∈[0,τ]|Λ _s (t)−Λ _s0(t)|→0. Then, for the conditions (a)–(d) in Theorem 4 of [25], we have that: (a) follows from Lemma 3.3.5 (p. 311) of [39]; (b) holds as a result of P-Donsker property and the convergence is defined in the metric space by the Donsker theorem in [39]; (d) is true because $(\widehat{\boldsymbol{\theta}} , \widehat {\varLambda}_{s})$ maximizes $\mathbf{P}_{m_{s}} l(\boldsymbol{O}; \boldsymbol{\theta}, \varLambda_{s})$, (θ ₀,Λ _s0) maximizes P l(O;θ,Λ _s ), and $(\widehat{\boldsymbol {\theta}}, \widehat{\varLambda}_{s})$ converges to (θ ₀,Λ _s0) from Theorem 1;

The first half of condition (c), that the function ξ→U(ξ) is Fréchet differentiable at ξ ₀, is proved by showing there exists a bounded linear operator for the function.

Thus, it only remains to prove the second half of condition (c), that the derivative $\nabla U_{\xi_{0}}$ is continuously invertible on its range . From the proof of the Fréchet differentiability of U(ξ) at ξ ₀, we have that for any (θ ₁,Λ _s1) and (θ ₁,Λ _s1) in Ξ,

(13)

where both Ω ₁ and Ω ₂ are bounded linear operators on , and Ω=(Ω ₁,Ω ₂) maps to R ^d×BV[0,τ], where BV[0,τ] contains all the functions with finite total variation in [0,τ]. The explicit expressions of Ω ₁ and Ω ₂ can be obtained from P-Donsker property and the derivation of $\nabla U_{\xi_{0}}$ by definition. Thus, $\nabla U_{\xi_{0}}$ is a linear operator from to itself. We note that to prove that $\nabla U_{\xi_{0}}$ is continuously invertible is equivalent to showing that Ω is invertible. Then, by Theorem 4.25 of [30], for the proof of invertibility of Ω, it is sufficient to verify that Ω is one to one: if Ω[h ₁,h ₂]=0, then, by choosing θ ₁−θ ₂=ε ^∗ h ₁ and Λ _s1−Λ _s2=ε ^∗∫h ₂  dΛ _s0 in (13) for a small constant ε ^∗, we obtain

Since $\nabla U_{\xi_{0}} ( \mathbf{h}_{1}, \int h_{2} \,d \varLambda_{s0}) [\mathbf{h}_{1}, h_{2}]$ is the negative information matrix in the submodel (θ ₀+ε h ₁,Λ _s0+ε∫h ₂  dΛ _s0), the score function along this submodel is $l_{\theta} (\boldsymbol{\theta}_{0}, \varLambda_{s0} )^{T} \mathbf{h}_{1} + l_{\varLambda_{s}} (\boldsymbol{\theta}_{0}, \varLambda_{s0} ) [h_{2}]= 0$; that is, with probability one, the numerator of the score function

(14)

where A′(D(t _j ;ϕ ₀)) and C′(Y _j ;D(t _j ;ϕ ₀)) are the derivatives of A(D(t _j ;ϕ)) and C(Y _j ;D(t _j ;ϕ)) with respect to ϕ evaluated at ϕ ₀ and B′(β ₀;b) is the derivative of B(β;b) with respect to β evaluated at β ₀. The proof of invertibility of Ω will be completed if we can show h ₁=0 and h ₂(t)=0 from (14).

To show h ₁=0, particularly we let Δ _s =0 and V _s =0 in (14). Examining the coefficient for Y and the constant terms without Y and using assumption (A5) and the similar arguments done in Appendix B.1 give $\mathrm{h}_{1}^{\phi}=0$, $\mathbf{h}_{1}^{\beta}=0$ and D _b =0. On the other hand, letting Δ _s =0 in (14) with assumptions (A2) and (A5) and the similar arguments done in Appendix B.1 lead to $\mathbf{h}_{1}^{\psi} = 0$, $\mathbf{h}_{1}^{\gamma} = 0$ and h ₂(t)=0. Hence, the proof of condition (c) is completed.

Since the conditions (a)–(d) have been proved, Theorem 3.3.1 of [39] concludes that $\sqrt{m_{s}} ( \widehat{\boldsymbol{\theta}} - \boldsymbol{\theta}_{0}, \widehat {\varLambda}_{s} - \varLambda_{s0} )$ weakly converges to a tight random element in . Then, we have

where o _P (1) is a random variable which converges to zero in probability in , and, from (13),

By denoting $(\mathbf{h}_{1}^{*}, h_{2}^{*}) = \varOmega^{-1} (\mathbf{h}_{1}, h_{2})$, we have $(\mathbf{h}_{1}, h_{2}) = \varOmega(\mathbf{h}_{1}^{*}, h_{2}^{*})$, and by replacing (h ₁,h ₂) with $(\mathbf{h}_{1}^{*}, h_{2}^{*})$ in the above two equations, we obtain

(15)

We can see that the first term on the right-hand side in (15) is $\sqrt{m_{s}} \{ U_{m_{s}} (\boldsymbol {\theta}_{0}, \varLambda_{s0}) - U_{s} (\boldsymbol{\theta}_{0}, \varLambda_{s0}) \}$, which is an empirical process in the space . Furthermore, it is already shown that is P-Donsker. Therefore, $\sqrt{m_{s}}( \widehat{\boldsymbol{\theta}} - \boldsymbol {\theta}_{0}, \widehat{\varLambda}_{s} - \varLambda_{s0} )$ weakly converges to a Gaussian process in .

Choose h ₂=0 in (15) with Proposition 3.3.1 in [4] concludes that $\widehat{\boldsymbol{\theta}}$ is an efficient estimator for θ ₀. Therefore, Theorem 2 is proved.