Asymptotic normality of estimator in non-parametric model under censored samples

ASYMPTOTIC NORMALITY OF ESTIMATOR IN NON-PARAMETRIC MODEL UNDER CENSORED SAMPLES

Si-Li Niu and Qian-Ru Li

Reprinted from the

Journal of the Korean Mathematical Society Vol. 44, No. 3, May 2007

c

°2007 The Korean Mathematical Society

ASYMPTOTIC NORMALITY OF ESTIMATOR IN NON-PARAMETRIC MODEL UNDER CENSORED SAMPLES

Si-Li Niu and Qian-Ru Li

Abstract. Consider the regression model Yi= g(xi)+eifor i = 1, 2, . . . , n, where: (1) xiare fixed design points, (2) ei are independent random errors with mean zero, (3) g(·) is unknown regression function defined on [0, 1]. Under Yiare censored randomly, we discuss the asymptotic normal- ity of the weighted kernel estimators of g when the censored distribution function is known or unknown.

1. Introduction Consider the fixed design regression model

(1.1) Yi= g(xi) + ei, i = 1, 2, . . . , n,

where Y1, Y2, . . . , Yn are the observations on the fixed design points x1, x2, . . . , xn; e1, e2, . . . , en are independent random errors with mean zero; g(·) is the unknown regression function defined on [0, 1]. Without loss of generality, we assume that 0 = x0< x1< · · · < xn= 1.

Under independent samples, many researchers have studied the large sample properties of estimator of g(·). For instance, consistency and asymptotic nor- mality have been studied by Priestly and Chao [9], Benedetti [1], Georgiev and Greblicki [5] and Georgiev [4] among others. Under various dependent samples, the estimators of g(·) also have been widely studied, such as Fan [2], Roussas [10], Roussas et al. [11] and Tran et al. [12]. In particular, Prieshey and Chao [9] proposed the following weighted kernel estimator of g(x)

gn(x) = Xn

i=1

xi− xi−1

hn K(x − xi

hn )Yi, x ∈ [0, 1], (1.2)

where K(·) is a kernel function, hn is a sequence of positive constants tending to 0.

Received August 22, 2005.

2000 Mathematics Subject Classification. 62G05.

Key words and phrases. censored sample, non-parametric regression model, weighted ker- nel estimator, asymptotic normality.

c

°2007 The Korean Mathematical Society 525

In this paper, we consider the random censorship case for Yi, 1 ≤ i ≤ n, i.e.

when we observe Yi, 1 ≤ i ≤ n, we observe only the pairs (Z1, δ1), . . . , (Zn, δn):

Zi= min{Yi, Ti}, δi= I[Yi≤ Ti], i = 1, 2, . . . , n,

where {Ti, 1 ≤ i ≤ n} are censored variables, I[·] denotes the indicator func- tion. The estimator of g will be constructed by the censored data {(Zi, δi), i = 1, 2, . . . , n}.

Considering the practical background of the random censorship model, we assume that both Yi, 1 ≤ i ≤ n and Ti, 1 ≤ i ≤ n are nonnegative and independent random variables; moreover every Ti is independent of every Yi

with distribution function Fi, 1 ≤ i ≤ n. The censored variables T1, T2, . . . are assumed to be independent, identically distributed with distribution function G. Then the distribution function of Zi is

Hi(x) = P (Zi≤ x) = 1 − (1 − Fi(x))(1 − G(x)) =: 1 − Fis(x)Gs(x), where Fis= 1 − Fi, 1 ≤ i ≤ n, Gs= 1 − G. Set

F¯s= 1 n

Xn i=1

Fis, His= 1 − Hi= FisGs, ¯Hs= 1 n

Xn i=1

His.

For any distribution function F , define τF = inf{t : F (t) = 1}, F^−p(t) =

µ 1 F (t)

¶p

for all p > 0.

Throughout this paper, we assume τFi ≤ τG, i = 1, . . . , n. Note that (1.3) Eδ_iZ_iG⁻¹_s (Z_i) =

Z _τ

Fi

o

dF_i(y) Z _τG

y

yG⁻¹_s (y)dG(t) = EY_i= g(x_i).

Therefore, we may assume that {δiZiG⁻¹_s (Zi), 1 ≤ i ≤ n} obey the following model:

(1.4) δiZiG⁻¹_s (Zi) = g(xi) + e⁰_i, i = 1, 2, . . . , n,

where {e⁰_i, 1 ≤ i ≤ n} are independent random errors with mean zero. Hence, when G is known, taking advantage of model (1.4) and comparing with (1.2), the weighted kernel estimator of g(x) is defined as follows:

g_n⁽¹⁾(x) = Xn i=1

x_i− x_i−1

hn K(x − x_i

hn )δiZiG⁻¹_s (Zi), x ∈ [0, 1].

Also, we assume that τ(n) =: max{τFi : 1 ≤ i ≤ n} ≤ τG ≤ +∞ for all n ≥ 1, and that τ0=: limnτ(n)exists.

When G is unknown, the estimate of g(x) is defined by g_n⁽²⁾(x) =

Xn i=1

xi− xi−1

hn K(x − xi

hn )δiZiGˆ⁻¹_ns(Zi)I[Zi≤ Mn], x ∈ [0, 1],

where Mn is a sequence of constants, which are less than τ_(n) and monoto- nously increase to τ0, ˆGnsdenotes the Kaplan-Meier estimator of the distribu- tion function G, i.e.,

Gˆns(t) =



 Q_n

j=1

µ

1+N⁺(Zj) 2+N⁺(Zj)

¶_{I[δj=0, Zj≤t]}

, t ≤ Z_(n),

0, t > Z_(n)

(cf. Kaplan and Meier [6]), Z_(n)= max{Z1, Z2, . . . , Zn}, N⁺(Zj) =P_n

i=1I[Zi

> Zj], 1 ≤ j ≤ n.

In the sequel, let C, C1 and C2, . . . will represent positive constants whose value may change from one place to another.

For model (1.1), the consistent rates of estimators of g have also been studied by some authors under censored assumptions, see Xue [14] and Wang [13]. In particular, Wang [13] discussed the weak consistency and consistent rates of the estimators g⁽¹⁾n and g⁽²⁾n of g. His results are as follows.

Theorem 1.1. Let p >√

2, 0 < α < 1, and let K(u) be a continuous prob- ability density kernel. Set ˆK(u) = |u|^αK(u) withR_∞

−∞K(u)du < ∞. Supposeˆ that

(A.1) there exists some M0≥ 0 such that ˆK(u) is non-increasing when u > M0, and nondecreasing when u < −M0,

(A.2) max1≤i≤n(xi− xi−1) ≤ C/n for all n ≥ 1,

(A.3) g(x) satisfies Lipschitz condition of order α on [0, 1].

If max1≤i≤nE[Y_i^pG^1−p_s (Yi)] ≤ C, and one of the following conditions holds.

(i)√

2 < p ≤ 2, n^2/p−ph^1−p_n → 0; (ii) p > 2, n^−p/2h^1−p_n → 0.

Then g⁽¹⁾n (x)−→ g(x) for all x ∈ [0, 1].^P Theorem 1.2. Let p > ^1−β+

√(1−β)²+8

2 for some 0 ≤ β ≤ 1. Suppose that (A.1)-(A.3) in Theorem 1.1 are satisfied. If max1≤i≤nE[Y_i^pG^1−p_s (Yi)] ≤ C, and one of the following conditions holds.

(a) 0 < β ≤ 1, 1 − β +p

(1 − β)²+ 8

2 < p ≤ 2, X∞ n=1

n^2/p−p−βh^1−p_n < ∞;

(b) 0 ≤ β ≤ 1, p > 2, X∞ n=1

n^−2/p−βh^1−p_n < ∞.

Then, for any fixed x ∈ [0, 1], we have X∞

n=1

n^−βP (|g⁽¹⁾_n (x) − g(x)| > ²) < ∞.

Theorem 1.3. Suppose that (A.1)-(A.3) in Theorem 1.1 are satisfied, and that (A.4) both Fi (1 ≤ i ≤ n), and G are continuous distribution functions,

(A.5) τ0< τG,

(A.6) lim infn→∞Fs(Mn) log n ≥ b for some b > 0.

If {Fi, i ≥ 1} are equi-continuous at τ0 andP_∞

n=1n^−p/2h^1−p_n < ∞, then X∞

n=1

P (|g_n⁽²⁾(x) − g(x)| > ²) < ∞ for all ² > 0 and x ∈ [0, 1].

In this paper, we will establish the asymptotic normality of the estimators g⁽¹⁾n and gn⁽²⁾ of g under some suitable conditions.

The paper is organized as follows. In Section 2, we introduce the assumptions used throughout the paper and give main results. Proofs will be provided in Sections 3.

2. Main results

In order to state the main results, we first list the following some assump- tions.

(B.1) Let K(u) be a continuous probability density kernel with 0 <R_+∞

−∞ K²(u)du < ∞; there exists some M0≥ 0 such that K(u) is non-increasing when u > M0 and nondecreasing when u < −M0. (B.2) For all n ≥ 1, ^C_n¹ ≤ xi− xi−1≤^C_n², 1 ≤ i ≤ n.

(B.3) For 0 < θ < 1, lim infn→∞n^θF¯s(Mn) ≥ b for some b > 0;

limn→∞hn^p2F¯_s^1−2p(Mn) = 0 for some p ≥ 2.

Theorem 2.1. Suppose that (B.1) and (B.2) are satisfied and nhn → ∞.

Set ZiG = δiZiG⁻¹_s (Zi). If limC→∞sup_iEZ_iG² I(ZiG > C) = 0, and σ_i² =:

Var(ZiG) ≥ δ², 1 ≤ i ≤ n for some δ > 0, then g⁽¹⁾n (x) − Eg⁽¹⁾n (x)

q

Var(gn⁽¹⁾(x))

→dN (0, 1), x ∈ [0, 1].

Theorem 2.2. Suppose that (A.4)-(A.5) and (B.1)-(B.3) are satisfied. If σ²_i =:

Var(ZiG) ≥ δ², 1 ≤ i ≤ n for some δ > 0 and√

nhnsup_i[Fi(τ0)−Fi(Mn)] → 0, then

g⁽²⁾n (x) − Eg⁽¹⁾n (x) q

Var(gn⁽¹⁾(x))

→dN (0, 1), x ∈ [0, 1].

3. Proofs of main results

Lemma 3.1. ([13, Lemma 3.1]) Suppose that ¯K(·) is a continuous function withR_∞

−∞K(u)du < +∞, and there exists some M¯ 0≥ 0 such that ¯K(u) is non- increasing when u > M0and non-decreasing when u < −M0. If max1≤i≤n(xi− x_i−1) ≤ C/n for all n and lim_n→∞nh_n= ∞, then

h⁻¹_n Xn i=1

(xi− xi−1) ¯K

µx − xi

hn

¶

→ Z _+∞

−∞

K(u)du.¯

Lemma 3.2. ([7]) For any r > 0,

E[(1 + N⁺(Zi))^−r|(Zi, δi)] ≤ n^−r[ ¯Hs(Zi) − n⁻¹]^−r.

Lemma 3.3. Suppose that (A.4)-(A.5) and (B.1)-(B.3) are satisfied. If σ_i²=:

Var(ZiG) ≥ δ², 1 ≤ i ≤ n for some δ > 0, then sup_0≤t≤Mn| log ˆGns(t) − log Gs(t)|

q

Var(gn⁽¹⁾(x))

→p0.

Proof. For every n ≥ 1 and t ∈ [0, Mn], define βj(t) = I[δj = 0, Zj ≤ t], then by the Taylor expansion we have

log ˆGns(t) = Xn j=1

I[δj = 0, Zj≤ t] log(1 + N⁺(Zj) 2 + N⁺(Zj))

= Xn j=1

βj(t) log(1 − 1 2 + N⁺(Zj))

= − Xn j=1

βj(t) X∞

l=1

1

l(2 + N⁺(Zj))^−l. Hence

log ˆGns(t) − log Gs(t)

= [−1 n

Xn j=1

βj(t) ¯H_s⁻¹(Zj) − log Gs(t)] − Xn j=1

βj(t) X∞ l=2

1

l(2 + N⁺(Zj))^−l

−1 n

Xn j=1

βj(t)[n(2 + N⁺(Zj))⁻¹− ¯H_s⁻¹(Zj)]

=: Ln1+ Ln2+ Ln3.

Since {δiZiG⁻¹_s (Zi), 1 ≤ i ≤ n} are independent random variables, by assump- tion (B.2) and Lemma 3.1, we have

Var(g⁽¹⁾_n (x)) = Var[

Xn i=1

(xi− xi−1

hn )K(x − xi

hn )δiZiG⁻¹_s (Zi)]

= Xn i=1

(xi− xi−1

hn )²K²(x − xi

hn )σ²_i

≥ Cδ² nhn

Xn i=1

xi− xi−1

hn K²(x − xi

hn )

≥ C

nhn

. (3.5)

Hence, for any ² > 0 and t ∈ [0, Mn],

P (| log ˆGns(t) − log Gs(t)|

q

Var(g⁽¹⁾n (x))

> ²)

≤ P (| log ˆGns(t) − log Gs(t)| > C²

√nhn

)

≤ P (|Ln1| > C² 3√

nhn

) + P (|Ln2| > C² 3√

nhn

) +P (|Ln3| > C²

3√ nhn

).

(3.6)

Next, we divide three steps to estimate the three terms in (3.6) respectively.

Step 1. Note that

−Eβj(t) ¯H_s⁻¹(Zj)

= − Z _t

o

( Z _τFj

u

1

H¯s(u)dFj(y))dG(u)

= − Z _t

0

1 − Fj(u)

1 n

P_n

i=1Fis(u)Gs(u)dG(u)

= −n Z _t

0

F_js(u) Gs(u)P_n

i=1Fis(u)dG(u), therefore,

−1 n

Xn j=1

E[βj(t) ¯H_s⁻¹(Zj)] = − Z _t

0

1

Gs(u)dG(u) = log Gs(t).

Thus, Ln1(t) = _n¹P_n

j=1[−βj(t) ¯H_s⁻¹(Zj) + Eβj(t) ¯H_s⁻¹(Zj)] and we obtain from (B.3) and the Dharmadhikar-Jogdeo inequality (Praksas Rao [8]) that for p ≥ 2

P (|Ln1| > C² 3√

nhn

)

≤ Cn^−p2hn^p2E|

Xn j=1

[−βj(t) ¯H_s⁻¹(Zj) + Eβj(t) ¯H_s⁻¹(Zj)]|^p

≤ Cn^−p2hn^p2n^p2−1 Xn j=1

E|βj(t) ¯H_s⁻¹(Zj) − Eβj(t) ¯H_s⁻¹(Zj)|^p

≤ Cn^−p2hn^p2n^p2−1 Xn j=1

[E|βj(t) ¯H_s⁻¹(Zj)|^p+ |Eβj(t) ¯H_s⁻¹(Zj)|^p]

≤ Cn⁻¹hn^p2

Xn j=1

Eβj(t) ¯H_s^−p(Zj)

= Cn⁻¹hn^p2 · n Z _t

0 1 n

P_n

j=1Fjs(u) G^ps(u)(_n¹P_n

i=1Fis(u))^pdG(u)

≤ Chn^p2F¯_s^1−p(Mn) → 0.

Step 2. We observe that

|Ln2| ≤ Xn j=1

βj(t) X∞ l=2

(2 + N⁺(Zj))^−l

= Xn j=1

βj(t)

1 (2+N⁺(Zj))²

1 −_2+N¹+(Zj)

≤ Xn j=1

β_j(t) 1 (1 + N⁺(Zj))². Hence, by applying the Minkowski inequality we have

P (|Ln2| ≥ C² 3√

nhn

)

≤ P (|

Xn j=1

βj(t) 1

(1 + N⁺(Zj))²| > C² 3√

nhn

)

≤ Cn^p2hn^p2E|

Xn j=1

βj(t) 1

(1 + N⁺(Zj))²|^p

≤ Cn^p−1n^p2hn^p2

Xn j=1

Eβj(t)(1 + N⁺(Zj))^−2p

= Cn^3p2−1hn^p2

Xn j=1

E{βj(t)E[(1 + N⁺(Zj))^−2p|(Zj, δj)]}.

(3.7)

Lemma 3.2 yields

(3.8) E[(1 + N⁺(Zj))^−2p|(Zj, δj)] ≤ n^−2p[ ¯Hs(Zj) − n⁻¹]^−2p.

The assumptions (A.5) and (B.3) imply that for sufficiently large n and all t ∈ [0, Mn],

n ¯Hs(t) − 1 ≥ n ¯Hs(Mn) − 1 ≥ Gs(τ0)n ¯Fs(Mn) − 1

= Gs(τ0)n^1−θn^θF¯s(Mn) − 1 ≥ bGs(τ0)n^1−θ− 1 → ∞, further we have

(3.9) [ ¯Hs(t) − n⁻¹]⁻¹= 1

H¯s(t)[1 + 1

n ¯Hs(t) − 1] ≤ C ¯H_s⁻¹(t).

From (3.7)-(3.9) and (B.3), we find P (|Ln2| ≥ C²

3√ nhn

)

≤ Cn^3p2−1hn^p2n^−2p Xn j=1

Eβj(t) ¯H_s^−2p(Zj)

= Cn^−p2−1hn^p2

Xn j=1

EI[Yj> Tj, Yj∧ Tj≤ t] ¯H_s^−2p(Yj∧ Tj)

≤ Cn^−p2−1hn^p2

Xn j=1

Z _t

o

Z _τ

Fj

u

1

H¯s^2p(u)dFj(y)dG(u)

≤ Cn^−p2−1hn^p2

Z _t

0

P_n

j=1Fjs(u) G^2ps (u)(_n¹P_n

i=1Fis(u))^2pdG(u)

≤ Cn^−p2hn^p2F¯_s^1−2p(Mn) → 0.

Step 3. Note that _2+N⁻¹+(Zj)≤ _(1+N+¹(Zj))² −_1+N¹+(Zj), so

L_n3 ≤ 1 n

Xn j=1

β_j(t)[ n

(1 + N⁺(Zj))² − n

1 + N⁺(Zj)] + 1 n

Xn j=1

β_j(t) ¯H_s⁻¹(Z_j)

= Xn j=1

βj(t)

(1 + N⁺(Zj))² −1 n

Xn j=1

βj(t)( n

1 + N⁺(Zj)− ¯H_s⁻¹(Zj))

=: Ln31− Ln32. (3.10)

From Step 2 we have

(3.11) P

µ

|Ln31| ≥ C² 6√

nhn

¶

→ 0.

As to Ln32 we have P

µ

|Ln32| ≥ C² 6√

nh_n

¶

= P (|1 n

Xn j=1

βj(t)( n

1 + N⁺(Zj)− 1

H¯s(Zj))| ≥ C² 6√

nhn

)

≤ Cn^p2hn^p2n^−pE|

Xn j=1

βj(t)( n

1 + N⁺(Zj)− 1 H¯s(Zj))|^p

≤ Cn^−p2hn^p2n^p−1 Xn j=1

Eβj(t)| n

1 + N⁺(Zj)− 1 H¯s(Zj)|^p

= Cn^p2−1hn^p2

Xn j=1

E{βj(t)E[| n

1 + N⁺(Zj)− 1

H¯s(Zj)|^p|(Zj, δj)]}

(3.12)

and

E

·

| n

1 + N⁺(Zj)− 1

H¯s(Zj)|^p|(Zj, δj)

¸

= E

·

|1 + N⁺(Zj) − n ¯Hs(Zj)

H¯s(Zj)(1 + N⁺(Zj)) |^p|(Zj, δj)

¸

≤ H¯_s^−p(Zj)E¹²[(1 + N⁺(Zj))^−2p|(Zj, δj)]

·E¹²[|1 + N⁺(Zj) − n ¯Hs(Zj)|^2p|(Zj, δj)].

(3.13)

By Lemma 3.2, for 1 ≤ j ≤ n,

E[(1 + N⁺(Zj))^−2p|(Zj, δj)] ≤ n^−2p[ ¯Hs(Zj) − n⁻¹]^−2p

≤ Cn^−2pH¯_s^−2p(Zj).

(3.14)

On applying the Dharmadhikar-Jogdeo inequality (Praksas Rao [8]) we have E[|1 + N⁺(Zj) − n ¯Hs(Zj)|^2p|(Zj, δj)]

= E[|1 + Xn l=1

(I(Zl> Zj) − Hls(Zj))|^2p|(Zj, δj)]

≤ C{1 + n^p−1 Xn

l=1

E[|I(Zl> Zj) − Hls(Zj)|^2p|(Zj, δj)]}

≤ Cn^p. (3.15)

Hence, (3.12)-(3.15) and (B.3) yield that

P (|Ln32| ≥ C² 6√

nhn

) ≤ Cn⁻¹hn^p2

Xn j=1

Eβj(t) ¯H_s^−2p(Zj)

= Cn⁻¹hn^p2

Xn j=1

Z _t

o

Z _τFj

u

1

H¯s^2p(u)dFj(y)dG(u)

≤ Cn⁻¹hn^p2

Z _t

0

P_n

j=1Fjs(u) G^2ps (u)(_n¹P_n

i=1Fis(u))^2pdG(u)

≤ Chn^p2F¯_s^1−2p(Mn) → 0.

(3.16)

Then, (3.10), (3.11) and (3.16) follow that P (|Ln3| ≥ ₃^√C²_nh

n) → 0.

(3.6) and Steps 1-3 yield that for all t ∈ [0, Mn],

(3.17) | log ˆGns(t) − log Gs(t)|

q

Var(gn⁽¹⁾(x))

→p0.

Finally, by the continuity of G and (3.17), utilizing the method used in F¨oldes and Rejt¨o [3] (or see Wang [13]), we can obtain that

sup_0≤t≤Mn| log ˆGns(t) − log Gs(t)|

q

Var(gn⁽¹⁾(x))

→p0

for sufficiently large n. ¤

Proof of Theorem 2.1. Set ξni=^xi−x_hⁱ⁻¹

n K(^x−x_h ⁱ

n )ZiG. Thengn⁽¹⁾(x) =P_n

i=1ξni. From the assumption of independence on Yiand Ti, it suffices to verify Linder- berg condition: for any selected x ∈ [0, 1] and any ² > 0,

A_n(ε) = 1 s²_n

Xn i=1

E|ξ_ni− Eξ_ni|²I(|ξ_ni− Eξ_ni| ≥ εs_n) → 0,

where s²_n = Var(gn⁽¹⁾(x)). Note that limC→∞sup_iEZ_iG² I(|ZiG| > C) = 0, namely

C→∞lim sup

i

Z

|δiZiG⁻¹s (Zi)|>C

|δiZiG⁻¹_s (Zi)|²dP = 0, which follows that

sup

i E|δiZiG⁻¹_s (Zi)|²< ∞ and sup

i |g(xi)| ≤ sup

i E|δiZiG⁻¹_s (Zi)| < ∞.

(3.18)

Therefore, from (B.1), (B.2) and (3.5) we have

An(ε) = 1 s²_n

Xn i=1

E|ξni− Eξni|²I(|ξni− Eξni| ≥ εsn)

= 1

s²_n Xn i=1

E[δiZiG⁻¹_s (Zi) − g(xi)]²[xi− xi−1

hn

K(x − xi

hn

)]²

·I(|(δiZiG⁻¹_s (Zi) − g(xi))xi− xi−1

hn

K(x − xi

hn

)| ≥ εsn)

≤ C Xn i=1

E(δiZiG⁻¹_s (Zi) − g(xi))²[xi− xi−1

hn K(x − xi

hn )]

·I(|δiZiG⁻¹_s (Zi) − g(xi)| ≥ εsn

|^xi−x_hⁱ⁻¹

n K(^x−x_h ⁱ

n )|)

≤ C Xn i=1

E(δiZiG⁻¹_s (Zi) − g(xi))²[xi− xi−1

hn K(x − xi

hn )]

·I(|δiZiG⁻¹_s (Zi) − g(xi)| ≥ C²p nhn).

(3.19)

From Lemma 3.1

(3.20)

Xn i=1

xi− xi−1

hn K(x − xi

hn ) → Z _+∞

−∞

K(u)du = 1.

We observe that

E(δiZiG⁻¹_s (Zi) − g(xi))²I(|δiZiG⁻¹_s (Zi) − g(xi)| ≥ C²p nhn)

≤ C{E(δiZiG⁻¹_s (Zi))²I(|δiZiG⁻¹_s (Zi) − g(xi)| ≥ C²p nhn) +g²(xi)EI(|δiZiG⁻¹_s (Zi) − g(xi)| ≥ C²p

nhn)}

=: E1+ E2. (3.21)

(3.18) and nhn → ∞ yield that

E1 ≤ CE(δiZiG⁻¹_s (Zi))²I(|δiZiG⁻¹_s (Zi)| ≥ C²p

nhn− |g(xi)|)

→ 0.

(3.22)

On applying the Chebyshev inequality, from (3.18) and nhn→ ∞ we have

E2 ≤ g²(xi)E(δiZiG⁻¹_s (Zi) − g(xi))² C²²²nhn

≤ g²(xi)[E(δiZiG⁻¹_s (Zi))²+ g²(xi)]

Cnhn → 0.

(3.23)

Hence, we complete the proof of Theorem 2.1 by (3.19)-(3.23). ¤

Proof of Theorem 2.2. Put ani(x) = ^xi−x_hⁱ⁻¹

n K(^x−x_h ⁱ

n ). For any selected x ∈ [0, 1], we have

g⁽²⁾_n (x) − Eg⁽¹⁾_n (x)

= [ Xn i=1

aniδiZiGˆ⁻¹_ns(Zi)I(Zi ≤ Mn) − Xn i=1

aniZiGI(Zi≤ Mn)]

+[

Xn i=1

aniZiG− Eg⁽¹⁾_n (x)] − E Xn i=1

aniZiGI(Zi> Mn)

−[

Xn i=1

a_niZ_iGI(Z_i> M_n) − E Xn i=1

a_niZ_iGI(Z_i> M_n)]

=: B1+ B2+ B3+ B4. (3.24)

Note that, from (A.4) and (A.5) we have sup

i EZ_iG² I(Zi≥ Mn)

= sup

i

Z _τ

Fi

Mn

G⁻²_s (y)y²dFi(y) Z _τG

y

dG(t)

= sup

i

Z _τ

Fi

Mn

y²

1 − G(y)dFi(y)

≤ τ₀² 1 − G(τ0)sup

i [Fi(τ0) − Fi(Mn)] → 0 as n → ∞.

(3.25)

Therefore

C→∞lim sup

i EZ_iG² I(ZiG> C) = 0.

Next, we will analyse the four parts in (3.24) respectively as follows.

(i) Obviously, B2= g⁽¹⁾n (x) − Eg⁽¹⁾n (x). Hence, from Theorem 2.1 we have B2

q

Var(gn⁽¹⁾(x))

→dN (0, 1).

(ii) Note that

pnhn max

1≤i≤nEZiGI(Zi> Mn)

= p

nhn max

1≤i≤nEYiG⁻¹_s (Yi)I(Yi> Mn)

= p

nhn max

1≤i≤n

Z _τFi

Mn

G⁻¹_s (y)ydFi(y) Z _τG

y

dG(t)

= p

nhn max

1≤i≤n

Z _τFi

Mn

y(1 − G(y)) Gs(y) dFi(y)

≤ τ0

pnhn max

1≤i≤n[Fi(τ0) − Fi(Mn)] → 0, which, together with Lemma 3.1, follows that

|B3| q

Var(gn⁽¹⁾(x))

≤ Cp nhn

Xn i=1

(xi− xi−1

hn )K(x − xi

hn )EZiGI(Zi> Mn) → 0.

(iii) We observe that P

µ

|B₄| Áq

Var(gn⁽¹⁾(x)) ≥ ε

¶

≤ P (|B₄| ≥ Cε

√nhn

)

≤ CnhnE[

Xn i=1

(aniZiGI(Zi> Mn) − EaniZiGI(Zi> Mn))]²

≤ Cnhn

Xn i=1

E|aniZiGI(Zi> Mn) − EaniZiGI(Zi> Mn)|²

≤ C Xn i=1

xi− xi−1

hn K²(x − xi

hn )EZ_iG² I(Zi> Mn).

(3.26)

By Lemma 3.1 we have (3.27)

Xn i=1

xi− xi−1

hn K²(x − xi

hn ) → Z

K²(u)du < ∞.

Hence, (3.25)-(3.27) follow that P µ

|B4| Áq

Var(g⁽¹⁾n (x)) ≥ ε

¶

→ 0.

(iv) Note that Mn < τ0 < τG. By Lemma 3.1, for sufficiently large n we have

P

µ |B1| q

Var(g⁽¹⁾n (x))

≥ ε

¶

≤ P µ1

sn

Xn i=1

x_i− x_i−1

hn K(x − x_i

hn )|Ziδi( ˆG⁻¹_ns(Zi)

−G⁻¹_s (Zi))|I(Zj≤ Mn) > ε

¶

≤ P µτ0

s_n sup

0≤t≤Mn

| ˆG⁻¹_ns(t) − G⁻¹_s (t)|

Xn i=1

xi− xi−1

h_n K(x − xi

h_n ) > ε

¶

≤ P µ1

sn sup

0≤t≤Mn

| exp{log ˆG⁻¹_ns(t)} − exp{log G⁻¹_s (t)}| > ε 2τ0

¶

= P µ1

sn sup

0≤t≤Mn

| exp{θ(log ˆG⁻¹_ns(t) − log G⁻¹_s (t))}

·(log ˆG⁻¹_ns(t) − log G⁻¹_s (t))| > ε 2τ0

¶

≤ P µ1

snexp{ sup

0≤t≤Mn

| log ˆGns(t) − log Gs(t)|}

· sup

0≤t≤Mn

| log ˆGns(t) − log Gs(t)| > ε 2τ0

¶

≤ P µ

exp{sup_0≤t≤Mn| log ˆGns(t) − log Gs(t)|

sn }

· 1 sn sup

0≤t≤Mn

| log ˆGns(t) − log Gs(t)| > ε 2τ0

¶

= P µ

e^∆n· ∆n > ε 2τ0

, ∆n> 1

¶ + P

µ

e^∆n· ∆n > ε 2τ0

, ∆n≤ 1

¶

≤ P (∆n > 1) + P (∆n> ε/2τ0e),

where 0 < θ < 1, ∆n = ^{sup0≤t≤Mn}√^{| log ˆGns(t)−log Gs(t)|}

Var(g⁽¹⁾n (x)) . Therefore, according to Lemma 3.3 we have

P

µ |B₁| q

Var(g⁽¹⁾n (x))

≥ ε

¶

→ 0.

On combining with (3.24) and (i)-(iv), Theorem 2.2 is proved. ¤ Acknowledgements. This research was supported by the National Natural Science Foundation of China (No. 10571136).

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