Appendix. Proof of Theorem 1. Define. [ ˆΛ 0(D) ˆΛ 0(t) ˆΛ (t) ˆΛ. (0) t. X 0 n(t) = D t. and. 0(t) ˆΛ 0(0) g(t(d t)), 0 < t < D, t.

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1 Appendix Proof of Theorem. Define [ ˆΛ (D) X n (t) = ˆΛ (t) D t ˆΛ (t) ˆΛ () g(t(d t)), t < t < D X n(t) = [ ˆΛ (D) ˆΛ (t) D t ˆΛ (t) ˆΛ () g(t(d t)), < t < D, t where ˆΛ (t) = log[exp( ˆΛ(t)) + ˆp/ˆp, ˆΛ(t) is the Nelson-Aalen estimator of Λ(t) defined in (3.8), ˆp is defined in (3.), ˆΛ (t) = log[exp( ˆΛ(t)) + p/p. Note that Λ () = ˆΛ () = Λ() =. Then X(t) = t q (D t) q tλ (D) DΛ (t). For any given small ε >, let c (,minx(τ) X(τ ε),x(τ) X(τ +ε)), which depends on ε,τ,τ 2,θ q. Then X(τ) X(t) > c whenever t τ > ε. Noticing that X n (t) attains its maximum at ˆτ n, for sufficiently large n, we have Pr ˆτ n τ > ε PrX(τ) X(ˆτ n ) > c PrX(τ) X(ˆτ n ) + X n (ˆτ n ) X n (τ) > c Pr X n (ˆτ n ) X(ˆτ n ) + X n (τ n ) X(τ) > c Pr sup X n (t) X(t) > c τ <t<τ 2 2 Pr sup X n (t) Xn (t) > c + Pr sup Xn τ <t<τ 2 (t) X(t) > c τ <t<τ 2 Pr Dτ p (D τ 2 ) p sup Ũn(t) +τ p 2 (D τ 2) p Ũn(D) > c τ <t<τ 2 + Pr sup X n (t) Xn(t) > c, (A.) τ <t<τ 2 where U n(t) = ˆΛ (t) Λ (t). Consequently, there exist c 2 > c 3 >, depending on c,τ,τ 2,D q, such that Pr ˆτ n τ > ε Pr sup Un (t) > c 2 + Pr Un (D) > c 3 τ <t<τ 2 + Pr sup X n (t) Xn(t) > c τ <t<τ 2 = I + II + III. (A.2)

2 From the definition of Λ (t) we find that Un(t) = [ log exp( ˆΛ(t)) + p log p = exp( η(t)) exp( η(t)) + p ˆΛ(t) Λ(t), p [exp( Λ(t)) + p (A.3) where η(t) is a number between ˆΛ(t) Λ(t). Thus exp( η(t)) lies on the segment between Ŝ(t) = ˆF n (t) S(t) = F(t) = pf (t). Under the i.i.d. censoring model, according to Wang (987), sup t [,τf ˆF n (t) F(t) almost surely for τ F τ G. Thus for any α < pf (D) sufficiently large n, exp( η(t)) > [ F(D) α = [ pf (D) α = φ(d) provided that τ F > D. It follows from (A.3) that U n(t) φ(d) + p ˆΛ(t) Λ(t) = φ(d) + p U n(t), where U n (t) = ˆΛ(t) Λ(t) is a martingale. By (A.2) (A.3) there exists c > depending on c,c 2,τ,τ 2,D, q, p F, such that We know that I Pr sup U n (t) > c,τ 2 Y (n) + PrY (n) < τ 2 τ <t<τ 2 n Pr sup U n (t Y (n) ) > c + PrY i < τ 2 τ <t<τ 2 i= [ 2 c 2 E sup U n (t Y (n) ) + (PrY < τ 2 ) 2. (A.) τ <t<τ 2 ˆΛ(t Y (n) ) Λ(t Y (n) ) = t Y(n) is a mean zero, square integrable martingale, E[ˆΛ(t Y (n) ) Λ(t Y (n) ) 2 = E t Y(n) n H i (s) dm n (s) (A.5) i= n H i (s) λ(s)ds, (A.6) where M n (t) = n i= N i(t) t n i= H i(s)λ(s)ds is the basic martingale. 2 i=

3 In view of (A.5) (A.6) the martingale inequality (Hall Heyde, 98, p.5), the first term on the right h side of (A.), denoted by I, satisfies I 2c 2 E[U n(τ 2 Y (n) ) 2 = 2c 2 E τ2 Y (n) I(s Y (n) )H n (s)λ(s)ds. Thus I converges to zero as n since τ 2 Y (n) H n (s)λ(s)ds almost surely. Next, by (A.2), II Pr U n (D) > c 3,D Y (n) + PrY (n) < D. Similarly, II converges to zero as n. In order to prove III, we rewrite X n (t) X n(t) as X n (t) = t q (D t) t[ˆλ q (D) ˆΛ (t) (D t)ˆλ (t) (A.7) Xn (t) = tq (D t) t [ˆΛ q (D) ˆΛ (t) (D t)ˆλ (t). (A.8) By (A.), (A.7) (A.8), III Pr sup X n (t) Xn(t) > c τ <t<τ 2,D < Y (n) + PrY (n) D Pr 2 sup ˆΛ (t) ˆΛ (t) τ q 2 (D τ 2) q > c <t<d 8 + Pr sup ˆΛ (t) Λ (t) τ q 2 (D τ 2 ) q > c + PrY D n τ <t<τ 2 8 = I 2 + I 3 + (PrY D) n, say. It is easy to see that I 2 Pr log ˆp logp + sup <t<d log exp( ˆΛ(t)) + ˆp exp( ˆΛ(t)) + p > c. (A.9) 8 Since ˆp converges to p in probability under conditions of Maller Zhou (996, p.67), logx is a continuous function for x >, sup <t<d ˆΛ(t) Λ(t) almost surely (cf. Anderson et al., 993, p.93), (A.9) shows that I 2 converges to zero as n. Similarly, I 3 as n. This completes the proof of Theorem. In order to show the asymptotic properties in Section, we need some Lemmas. We first state some conditions from Hu (998), which correspond to Conditions of 3,,, 2, 5 of Huang (996), respectively. Note that o p () in the following representations 3

4 indicates convergence to zero in outer probability in case that the term involved is not Borel measurable. Condition. (Stochastic Equicontinuity Condition) n(p n P ) l µ (ˆµ, ˆν) n(p n P ) l µ (µ,ν ) + n ˆµ µ = o p (), where ˆµ µ = o p () ˆν ν = o p (). Condition 2. np n lµ (µ,ν ) = O p (). For i.i.d. observations, Condition 2 holds automatically if P l2 µ (µ,ν ) < by the central limit theorem. Condition 3. (Smoothness Condition) For (µ,ν) D n, P lµ (µ,ν) P lµ (µ,ν ) P lµµ (µ,ν )(µ µ ) P lµν (µ,ν )(ν ν ) = o( µ µ ) + o( ν ν ), where D n = (µ,ν) : µ µ η n, ν ν cn /2 for some constant c. Condition. np lµν (µ,ν ) ˆν ν = O p (). When ˆν is a n-consistent, this condition holds automatically. Condition 5. under the true probability P, [ (Pn P ) l µ (µ,ν ) d n Λ = ˆν ν [ Λ where Λ N (,Σ) with Σ being a positive definite matrix. Λ 2, (A.) The following Lemmas are due to Hu (998), which also correspond to Theorem 6. in Huang (996) for the semiparametric model with a infinite-dimensional parameter space. Lemma. (Consistency) Suppose that µ is the unique solution to P lµ (µ,ν ) = ˆν is an estimator of ν such that ˆν ν = o p (). If P n lµ (µ,ν) P lµ (µ,ν ) sup µ Θ, ν ν η n + P n lµ (µ,ν) + P lµ (µ,ν ) = o p () for every sequence η n, then the ˆµ almost surely solving P n lµ (ˆµ, ˆν) = o p () converges in outer probability to µ.

5 Proof. See Theorem 3.. of Hu (998). Lemma 2. Suppose that the class of functions ψ(µ,ν) : µ µ < γ, ν ν < γ is P -Donsker for some γ >, that P ψ(µ,ν X) ψ(µ,ν X) 2, as µ µ ν ν. If ˆµ p µ ˆν ν p, then n(p n P )(ψ(ˆµ, ˆν) ψ(µ,ν )) = o p (). Proof. See Lemma 3.. of Hu (998). We should note that the conditions of Lemma 2 imply Condition. But they give a set of simple sufficient conditions for Condition, so we will verify the conditions of Lemma 2 in the proof of Theorem below. Lemma 3. (Rate of Convergence) Suppose that ˆµ satisfies P n lµ (ˆµ, ˆν) = o p (n /2 ) is a consistent estimator of µ, which is the unique point for which P lµ (µ,ν ) =, ˆν is an estimator of ν satisfying ˆν ν = O p (n /2 ). Then under Conditions -, n(ˆµ µ ) = O p (). Proof. See Theorem 3..3 of Hu (998). Lemma. (Normality) Suppose that µ is the unique solution to P lµ (µ,ν ) = ˆν is an estimator of ν satisfying ˆν ν = O p (). Then under Conditions 3-5, n(ˆµ µ ) d ( P lµµ (µ,ν )) N (,V ), where V = V ar(λ + P lµν (µ,ν )Λ 2 ). Proof. See Corollary 3..2 of Hu (998). Lemma 5. For l β (µ,ν X) l θ (µ,ν X)defined in (.3) (.), if µ µ η n ν ν cn /2, then P l µ (µ,ν)) l µ (µ,ν )) 2 = o p (). Proof. We only show that P l β (µ,ν)) l β (µ,ν )) 2 = o p (), when µ µ η n ν ν cn /2, as the proof for l θ is similar. Denote A(µ,ν,y) = ( δ)( p)y p + pexp( βy)) B(µ,ν,y) = δθ β(β + θ) + ( δ)( p)y p + pexp( βy θ(y τ)). 5

6 Then l β (µ,ν)) l β (µ,ν )) = [A(µ,ν,y)I(y τ) A(µ,ν,y)I(y τ ) Thus it suffices to show + [B(µ,ν,y)I(y > τ) B(µ,ν,y)I(y > τ ) + [δ/β δ/β. P A(µ,ν,y)I(y τ) A(µ,ν,y)I(y τ ) 2 = o p () P B(µ,ν,y)I(y > τ) B(µ,ν,y)I(y > τ ) 2 = o p (). (A.) (A.2) Note that A(µ,ν,y) is continuous for (µ,ν) C C η, A(µ,ν,y)I(y τ) A(µ,ν,y)I(y τ ) 2 = [A(µ,ν,y)I(y τ) A(µ,ν,y)I(y τ ) + [A(µ,ν,y)I(y τ ) A(µ,ν,y)I(y τ ) 2 = A 2 (µ,ν,y)i 2 (τ < y τ) + [A(µ,ν,y) A(µ,ν,y) 2 I 2 (y τ ), P (I 2 (τ < y τ)) = p[f (τ) F (τ ) as τ τ. Thus (A.) is proved. The proof of (A.2) is similar. Proof of Theorem 2. To prove the consistency of the pseudo estimator ˆµ, we mainly need sup µ C, ν ν η n P n lµ (µ,ν) P lµ (µ,ν ) = o p () for every sequence η n. Then the consistency of ˆµ follows from Lemma. Since P n lµ (µ,ν) P lµ (µ,ν ) (P n P ) l µ (µ,ν) + P ( l µ (µ,ν) l µ (µ,ν )), by (.) the second term obviously tends to zero when ν ν η n, it suffices to show that the class of functions F η l µ (µ,ν) : µ C R 2, ν ν η is a VC-class for some η >, where C is defined in (.). This implies that the uniform strong law of large numbers holds, i.e., sup f Fη (P n P )f p (see Van der Vaart Wellner, 996, Chap , for details). Let F η = I (,τ (y) : τ τ η. Then the VC-index of the class of functions F η is 2 by Example 2.6. of Van der Vaart Wellner (996). Thus the class of functions ( δ)( p)yi(y τ) p + pexp( βy) 6 : β > A,v C η

7 is Donsker by Lemma Example 2..8 of Van der Vaart Wellner (996), because ( δ)( p)/( p + pexp( βy)) is bounded. Let F 2η = I (τ, ) (y) : τ τ η, we apply Lemma of Van der Vaart Wellner (996) to show that F 2η is VC-class. Thus the class of functions δθ β(β + θ) I(y > τ) : µ C, τ τ η is Donsker since δθ/β(β + θ) is bounded. It is similar to show that the other classes of functions are also Donsker. Thus the class of functions of F η is VC-class by applying Example 2..7 Theorem 2..6 of Van der Vaart Wellner (996). Finally, by Lemma, ˆµ is consistent. Proof of Theorem 3. We first verify the stochastic equicontinuity condition: n(p n P )[ l µ (ˆµ, ˆν) l µ (µ,ν ) = o p (). (A.3) Let F γ = l µ (µ,ν) l µ (µ,ν ) : µ µ γ, ν ν γ. Simular to the proof of Theorem we can show that F γ is a VC-class. Thus (A.3) follows from Lemma 2 together with Lemma 5. Next, the smoothness Condition 3 holds by (.5) Lemma 5, P n lµ (µ,ν ) converges in distribution to a normal rom variable by the central limit theorem. Thus n ˆµ µ = O p () by Lemma 3. Proof of Theorem. By the consistency of ˆp ˆτ together with Slutsky s theorem the central limit theorem, we can show that (A.) holds with normally distributed Λ with mean zero positive variance. Hence by Lemma, n(ˆµ µ ) is asymptotically normal with mean variance P lµµ (µ,ν ) 2 V. 7