LAD2based Tests for Linear Hypothesis in Censored

第 34 卷第 5 期 2004年10月

中 　国 　科 　学 　技 　术 　大 　学 　学 　报

Vol . 34 ,No. 5

JOURNAL OF UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA

Oct . 2 0 0 4

Article ID :025322778 (2004) 0520511213

LAD2based Tests for Linear Hypothesis in Censored Regression Models Ξ

CUI Wen2quan ( Department of Statistics and Finance , USTC , Hefei 230026 , China)

abstract : Zhao ( Linear hypothesis testing in censored regression models. Statistica Sinica , 2004 , 1 : 3332347) proposed some test statistics for testing linear hypothesis in a censored re2 ) model and established the asymptotic distributions of these tests gression (or censored“Tobit ” under the null hypothesis. Asymptotic distributions of the test statistics are established under local alternative hypothesis. Simulations are carried out to study the finite2sample behaviors of these tests. Key words :local alternative hypothesis , linear hypothesis testing , censored regression model , asymptotical distribution. CLC number :O212

Document code :A

AMS subject classifications ( 2000) : 62E20 , 62N03.

0 　Introduction ) model In the censored regression (or censored“Tobit ” ( Yi )

+

β + ei ) = ( x i′

+

, 　i = 1 , …, n ,

( 1)

) denotes the indicator function of a set , { xi } only Y +i = Yi I ( Yi ≥0) and x i are observed , where I ( ・

is a sequence of known p2vector , { ei } is a sequence of unobserved random errors , βis the unknown p

2vector of regression coefficients. This model is an important special case of limited dependent variable (LDV) models , for which the range of the dependent variables is restricted to a subset of the real line. Many important advances have been achieved in econometric theory related to LDV models , see , for example , Maddala [14 ] , Powell [16 ,17 ] , Pollard [15 ] , Rao and Zhao [18 ] , Chamberlain [5 ] , Chen and [6 ] Wu [7 ] , Fitzenberger [9 ] , Buchinsky and Hahn[4 ] , Chay and Honoré , Bilias , Chen and Ying[1 ] ,

Honoréand Khan[11 ] , Khan , and Powell [13 ] , Chernozhukov and Hong[8 ] and Zhao [20 ] , among others. Ξ Received date :2003201210

Foundation item :Supported by the National Natural Science Foundation of China (10171094) , Ph. D. Program Foundation of Min2 istry of Education of China and Special Foundations of the Chinese Academy of Sciences and USTC. Biogroph :Cui Wen2quan ,male ,born in 1964 ,associate professor. E2mail :wqcui @ustc. edu. cn.

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512

β) + , Powell [16 ,17 ] introduced the least absolute devia2 Based on the fact that med ( Yi + ) = ( xi′ tions (LAD) estimators of β. Simulation studies have shown these estimators to exhibit a small sample bias in the opposite direction of the least squares bias for censored data. Khan and Powell[13 ] and Chernozhukov and Hong[8 ] modified the Powell ’s estimators to adjust for this finite2sample bias. The consistency and asymptotic normality of the estimate have been studied by Powell[16 ] , Pollard [15 ] , Chen and Wu [7 ] , Rao and Zhao [18 ] , among others. Zhao [20 ] proposed some tests for testing linear hy2 pothesis and obtained the asymptotic distributions of them under the null hypothesis. In this paper , we investigate the asymptotic distributions of the test statistics under the local alternative hypotheses. In the next section , the main results are given. Section 2 presents some simulation results. The proofs of the main results are given in Appendix.

1 　Main Results Throughout this paper we denote by 0 any matrix or vector with all the elements being 0 . In the following we assume that ( C1) e1 , e2 , …are iid. random variables such that the distribution function F of e1 has zero median and the derivative f is continuous in a neighborhood of 0 with f ( 0) > 0 . ( C2) The parameter space B of βis bounded open set of Rp (with a closure Bg ) . In this paper we are interested in testing the linear hypothesis (β - β0 ) = 0 , H0 ∶H′

( 2)

where H is a known p × q matrix of rank q , and β0 is a known p ×1 vector ( 0 < q ≤ p ) .

βand S n = Throughout this paper , denote by βthe true parameter. Write μi = x i′

n

∑ 1 I (μ i=

i

> 0) x i xi′ . In the following we always assume that S n0 > 0 for some n0 and that n ≥ n0 .

We consider local alternative hypothesis (β - β0 ) = H′ H1 n ∶H′ wn

( 3)

‖S 1n/ 2 \ w n ‖ = O ( 1) 　and 　 max xi′w n = o ( 1) 　as 　n →+ ∞.

( 4)

where w n is a p2vector satisfying 1 ≤i ≤n

The local alternative we considered is almost the same as the one in the complete data case[21 ] . Zhao [20 ] proposed some test criteria and obtained the asymptotic distributions of the test statistics under the null hypothesis. In this paper , we are interested in the asymptotic distribution of the test statistics under the local alternative hypothesis. To test hypothesis ( 2) , Zhao [20 ] suggested the following test criterion n

Mn =

inf β)

( bH′

∑

=0 i =1 0

n

( xi b) + - Yi + - inf b

∑ ( x b)

+

i

- Yi + ,

( 5)

i =1

(β - β0 ) = 0 , and assumed where infima are taken over all b ∈g B with and without the restriction H′

to be reached at βn3 and β ^ n respectively. Also he studied the following Wald type test criterion W n = (β ^ n - β0) ′ H ( H′ S -n 1 H)

-1

H′(β ^ n - β0)

and Rao′ s score type test criterion © 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

( 6)

第 5 期 　　　　　　LAD2based Tests for Linear Hypothesis in Censored Regression Models

513

3 -1 3 R n = ξ (βn ) ′ S n ξ (βn ) n

ξ ( b) =

( 7)

n

∑I ( x ′b > 0) sgn ( x ′b i

∑I ( x ′b > 0) sgn ( x ′b -

+ Yi ) xi =

i

i

i =1

i

Yi ) xi .

( 8)

i =1

To obtain the limiting distribution of M n , W n and R n under H1 n , further assume that n

∑ 1 ‖x ‖2 I ( ‖x

( C3) For any σ> 0 ,there exists a finite α> 0 such that

i

i=

i

λ( S n) for ‖ > α) < σ

) denotes the smallest eigenvalue of a matrix. large n , where λ( ・ n

∑ 1 ‖x

( C4) For any σ > 0 , there exists δ > 0 such that

i=

i

λ( S n ) for ‖2 I ( μi ≤δ) < σ

large n . ( C5) λ( S n ) ( log n) - 2 → ∞ as n → ∞ . Remark 1. Our conditions ( C1) ～ ( C5) are the same as the ones in Zhao [20 ] except that ( C1) is slightly strengthened. Write xin = S -n 1/ 2 xi , Hn = S -n 1/ 2 H ( H′ S -n 1 H)

- 1/ 2

n

∑ 1 I (μ

. Then

i

i=

> 0) x in xin′= Ip , and

Hn′ Hn = Iq , where Ip and Iq are the identity matrices of order p and q respectively. We have the fol2

lowing main theorem.

　　Theorem 1. Suppose that ( C1) ～ ( C5) are satisfied and that ( 3) and ( 4) hold , then we have n

4 f ( 0) M n = ‖

∑I (μ > 0) sgn ( e ) H ′x i

i

n

1/ 2

2

in

+ 2 f ( 0) Hn′ S n w n ‖ + op ( 1) ,

in

+ 2 f ( 0) Hn′ S n w n ‖ + op ( 1) ,

( 10)

in

+ 2 f ( 0) Hn′ S 1n/ 2w n ‖2 + op ( 1) .

( 11)

( 9)

i =1 n

2 ( 0) W n = ‖ 4f 　

∑I (μ > 0) sgn ( e ) H ′x i

i

n

1/ 2

2

i =1 n

Rn = ‖

∑I (μ > 0) sgn ( e ) H ′x i

i

n

i =1

2 (0) Wn and Rn have the same noncentral chi2square limiting distribution Consequently , 4 f (0) Mn , 4 f 　

χ2q ,ν with q degree of freedom and noncentrality parameterν, the limit of 4f 2　(0) wn′S1n/ 2 Hn Hn′S 1n/ 2 wn . For the proof of Theorem 1 , see Appendix. Remark 2. If S 1n/ 2w n →0 then Hn′ S 1n/ 2w n →0 and hence the limiting distribution under the local alternative is the central chi2square distribution with q df . But the inverse is not true since for any w n = S -n 1/ 2 Knηwith η ∈ Rp - q , Hn′ S 1n/ 2w n = 0 while S 1n/ 2w n = Knη does not necessarily to converge to

0 . In a word , S 1n/ 2w n →0 is a sufficient but not necessary condition for the limiting chi2square distribu2 tion under local alternative to be central . In order for the results of the Theorem 1 to be useful in testing the hypothesis H0 against H1 n , some consistent estimates of S n and f ( 0) should be obtained. Here we present the estimators in Zhao [20 ] . Use ^

Sn =

n

β ^ ∑ 1 I( x′ i

i=

n

> 0) xi x i′

( 12)

as an estimator of S n . To estimate f ( 0) , take h = h n > 0 such that h n → 0 and use n

f n^( 0) = 　

h

n

1 β β β ^ > 0) I( x′ ^ > 0) I ( x ′ ^ ∑I ( x ′ ∑ -

i

i =1

n

i

n

i

n

β < Yi + ≤ xi′ ^ n + h)

i =1

© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

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中国科学技术大学学报 　　　　　　　　　　　　　　第 34 卷

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^ as an estimator of f ( 0) . Under conditions ( C1) ～ ( C5) , Zhao [20 ] showed that S n is a consistent esti2 ^

mate of S n in that S -n 1/ 2S nS -n 1/ 2 → Ip in probability as n →+ ∞. Let d n = max ‖xin ‖. The following condition is assumed throughout the article 1 ≤i ≤n

( C6) 　 h n → 0 , 　 d n / h n →0 , 　 h n lim

n →∞

n

∑ 1 I( i=

n

∑1 i=

β > 0) → ∞, I ( xi′

n

β > 0) ∑ 1 I( x′

β ≤δn ) / x i′

i=

i

= 0 for any δn → 0 .

　　Zhao [20 ] showed that f ^( 0) is a consistent estimate of f ( 0) if ( C1) ～ ( C6) hold. Now define ^-1 ^-1 -1 3 3 ^ = (β (β ( 14) W ^ n - β0) ′ H ( H′ S n H) H′ ^ n - β0 ) , ^R n = ξ(βn ) ′ S n ξ (βn ) , n 　

and use ^ and ^R 4 f ^( 0) M n , 4 f ^( 0) 2 W n n 　

( 15)

　

as test statistics for testing the hypothesis H0 . For the distributions of the statistics under the local al2 ternative , we have Corollary 1. Assume that ( C1) ～ ( C6) are satisfied and ( 3) and ( 4) hold. Then ^ + o ( 1) = ^R + o ( 1) = 4 f ^( 0) M n = 4 f ^( 0) 2 W n p n p 　

　

n

∑I (μ i

> 0) sgn ( ei - xi′ w n ) Hn′ xin

2

d 2 + op ( 1) →χq , v ,

( 16)

i =1

2 1/ 2 ( 0) w n′S 1n/ 2 Hn Hn′ where νis the limit of 4 f 　 S n wn .

For fixed n , the estimate f ^( 0) of f ( 0) is sensitive to the choice of h n . So it is important to 　

choose appropriate h n . In the complete data case , h n can be estimated with a cross2validation estima2 tor [10 ] . In principle , when censoring happens , h n can be similarly estimated. However , the corre2 sponding objective function is non2convex and there is no appropriate minimization algorithm. This dif2 ficulty also arises when estimating the variance2covariance matrix of the Powell estimator. Some resam2 pling methods have been proposed to avoid estimating f ( 0) in literature (Bilias et al [1 ] and references therein) . We are pursuing developing a bootstrap method to avoid estimating f ( 0) .

2 　A Simulation Study In this section we carry out some simulations to study the finite sample behavior of the test statis2 tics under local alternative hypothesis. To save computer time , we use f ^( 0) as the estimator of f ( 0) 　

with h n fixed at 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 . We use the iterative linear programming algorithm (Buchinsky[3 ] ) to find solutions β ^ n and βn3 to ( 5) . The algorithm is described as follows. Without loss of generality , we minimize the objective func2 j) j- 1 β tion without constraint . In the jth iteration solve for β ^ n using the observations for which x i′ ^n > (

(

)

0 . Stop when the sets of observations in two consecutive iterations are the same. It is seen that drop2 ( ) ( ) β β ping the observations for which x i′ ^ nj - 1 ≤ 0 is of no consequence since y +i - ( x i′ ^ nj - 1 ) + = y +i ,

which does not depend onβ ^ nj - 1 . If the number of iterations is finite , then a global minimum is guar2 (

)

anteed (Buchinsky[2 ] ) . Linear programming algorithm we used here is the interior2point method imple2 © 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

第 5 期 　　　　　　LAD2based Tests for Linear Hypothesis in Censored Regression Models

515

mented by MATLAB ( Zhang[19 ] ) . We consider the model in Zhao [20 ] . The model assumes that p is fixed to be 2 , e is a Laplace random variable with probability distribution function

1 e 2c

x c

, x 1 a { 0 , 1} valued random variable

1 , x 2 a standard normal random variable. Set β0 = ( 0 , 1) ′and 2 consider testing H0 ∶H′(β - β0 ) = 0 with H = ( 1 , 0) ′ . Let c = 1 and covariates be fixed in all the

with P ( x 1 = 0) = P ( x 1 = 1) =

samples. Under the local alternative β = β0 + w n with w n = ( 1 , 1) ′ / n , 1 000 samples are gener2 ated for sample size n = 50 , 100 , 200 . For n = 50 , 100 , 200 , empirical quantiles of the statistics a2 gainst that of their limiting noncentral chi2square distributions χ2 distribution with q degree of freedon 2 1/ 2 1/ 2 ( 0) w n′ and noncentrality parameter 4 f 　 S n Hn Hn′ S n w n are plotted in Figures 1～3 , respectively. In

^ the figures , the dashed , dotted and dash2dot lines are for statistics T1 = 4 f ^( 0) M n , T2 = 4 f ^2 ( 0) W n 　

　

and T3 = ^R n , respectively. The solid line is a diagonal line indicating the limiting distribution. The figures show that the distributions of the test statistics get micreasingly closer to the limiting one as the sample size increases. It is seen that the distribution is a little biased for small or moderate sample size , which is not surprising since Powell ’s estimators do have a small sample bias.

Fig 1. 　Quantile2quantile plot for n = 50

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516

中国科学技术大学学报 　　　　　　　　　　　　　　第 34 卷

Fig 2. 　Quantile2quantile plot for n = 100

Figure 3. 　Quantile2quantile plot for n = 200

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第 5 期 　　　　　　LAD2based Tests for Linear Hypothesis in Censored Regression Models

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3 　Appendix Proof of Theorem 1. Let K be p × ( p 1/ 2

S n K ( K′ S n K)

- 1/ 2

q) such that H′ K = 0 . Write Kn =

q) matrix of rank ( p -

. We have Kn′ Kn = Ip - q , 　 Hn′ Hn = Iq , Hn′ Kn = 0 , and hence Hn Hn′+ Kn Kn′= Ip .

( a . 1)

　　To prove the theorem , we need two important lemmas , corresponding to Lemma 3. 1 , 3. 2 of p β - xi′ Zhao [20 ] . Let μ gi = μi - xi′w n = xi′ w n , γ = b - β + w n for b ∈ R . Write n

μ gi + - Yi ) =

∑

Gn (γ) =

+ ( ( x i′ b) - Yi -

i =1

n

∑(

(μ γ) + - Yi gi + xi′

μ gi + - Yi ) .

i =1

( a . 2)

It ’s easily seen that

γ ^ n ≡arg min Gn (γ) = β ^ n - β + wn . 　　 Lemma A. 1. Assume that ( C1) ～ ( C5) are satisfied , then

( a . 3)

n

∑I (μ

2 f ( 0) S 1n/ 2γ ^n =

i

> 0) sgn ( ei - x i′ w n ) xin + op ( 1) ,

( a . 4)

i =1

Gn (γ ^ n ) = - f ( 0) γ ^ n′ Sγ n^ n + op ( 1) = n

- ( 4 f ( 0) )

∑I (μ

-1

i

> 0) sgn ( ei - x i′ w n ) xin

2

+ op ( 1) .

( a . 5)

i =1

　　The proof is somewhat lengthy and is postponed. Assume that H′( b - β0 ) = 0 and H′(β - β0) = H′ w n , then by the definition of K , there exists p- q η ∈ R such that b = β - w n + Kη. Write Gn3 (η) =

n

+ ( xi′ b) - Yi - μ g +i - Yi

∑

i =1

n

=

∑

μ ( gi + xi′ η) + - Yi - μ K g +i - Yi

.

i =1

( a . 6)

It ’s easily seen that 3 βn3 = β - w n + K^ ηn , where η ^ n ≡arg min G n (η) . 　　Using Lemma A. 1 , similar to Zhao [20 ] , one can show that Lemma A. 2. Assume that ( C1) ～ ( C5) are satisfied and ( 3) and ( 4) hold , then

( a . 7)

n

1/ 2 2 f ( 0) ( K′ S n K) η ^n =

∑I (μ

> 0) sgn ( ei - xi′ w n ) ( K′ S n K)

i

- 1/ 2

K′ x i + op ( 1) =

i =1 n

∑I (μ

> 0) sgn ( ei - xi′ w n ) Kn′ xin + op ( 1) ,

i

( a . 8)

i =1

( K′ G n3 (η ^ n ) = - f ( 0)η ^ n′ S n K)η ^ n + op ( 1) = n

- ( 4 f ( 0) )

-1

∑I (μ i

> 0) sgn ( ei - xi′ w n ) Kn′ x in

2

+ op ( 1 ) .

( a . 9)

i =1

　　Lemma A. 3. For any constant C > 0 ,as n → ∞, n

sup p ‖ζ‖≤C ,ζ∈R

∑ I (μ i

ζ - ei ) + sgn ( ei ) xin - 2 f ( 0) ζ →0 in pro. , > 0) { sgn ( x in′

i =1

( a . 10) © 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

中国科学技术大学学报 　　　　　　　　　　　　　　第 34 卷

518

　　The proof can be found in the proof of Lemma 3. 4 in Zhao[20 ] . Now we are in a position to prove the theorem. By ( 5) , ( a. 1) , Lemma A. 1～A. 3 , we have 4 f (0) M n = 4 f (0) ( G n3 (η ^ n) - Gn (γ ^ n) ) = n

∑I (μ i

2

> 0) sgn ( ei - xi′ w n) xin

n

∑I (μ

-

i =1

i

2

+ op (1) =

i =1

n

∑I (μ

> 0) sgn ( ei - xi′ w n) Kn′ xin

i

2

> 0) sgn ( ei - xi′ w n ) Hn′ x in + op ( 1) =

i =1

n

∑ I (μ

Hn′

i

1/ 2 > 0) sgn ( ei - x in′ S n w n ) xin + op ( 1) =

i =1

n

Hn′

∑ I (μ i

1/ 2 > 0) sgn ( ei ) xin - 2 f ( 0) S n w n

+ op ( 1) ,

( a . 11)

i =1

which proves ( 9) . The proof of ( 10) is similar to that in Zhao [20 ] and is omitted. Define n

Vn ≡

∑I (μ i

> 0) sgn ( ei - xi′ w n ) Hn′ xin =

i =1 n

∑I (μ i

1/ 2 > 0) sgn ( ei ) Hn′ xin - 2 f ( 0) Hn′ S n w n + op ( 1) .

( a . 12)

i =1

By ( C3) and ( C5) we have d n ≡ max ‖x in ‖ →0 , noticing that Hn′ Hn = Iq , we have 1 ≤i ≤n

1/ 2

EV n = - 2 f ( 0) Hn′ S n w n ( 1 + o ( 1) ) ,

( a . 13)

by the Lindeberg theorem , it follows that V n is asymptotically normally distributed d 1/ 2 V n + 2 f ( 0) Hn′ S n w n →N ( 0 , Iq) .

( a . 14)

The last assertion of the theorem follows immediately if ( 11) holds. To prove ( 11) , write 1/ 2 φ ηn - w n ) , ^ n ≡ S n ( K^

( a . 15)

3

βn = xi′ β - xi′ ηn = μi + xin′ φ then by ( a. 7) and ( 7) , we have xi′ w n + x i′K^ ^ n and =

n

∑1 i=

ξ(βn3 )

- 1/ 2 Sn

φ φ I (μi + xin′ ^ n > 0) sgn ( x in′ ^ n - ei ) x in .

By the same arguments as in Zhao [20 ] , ( a. 15) , ( a. 8) and a similar version of ( a. 13) , we have

ξ(βn3 ) = -

S -n 1/ 2

n

∑ I (μ i

> 0) sgn ( ei ) xin + 2 f ( 0) φ ^ n + op ( 1) =

i =1 n

-

∑ I (μ i

1/ 2

> 0) sgn ( ei ) xin - 2 f ( 0) S n w n +

i =1 n

　 ∑ I (μi > 0) sgn ( ei - xi′ w n ) Kn Kn′ xin + op ( 1) = i =1

n

∑

-

i =1

n

I (μi > 0) sgn ( ei ) xin -

∑ I (μ i

> 0) E[ sgn ( ei - xi′ w n) ] xin +

i =1

n

　 ∑ I (μi > 0) sgn ( ei - xi′ w n ) Kn Kn′ xin + op ( 1) . i =1

It is easily seen that © 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

( a . 16)

第 5 期 　　　　　　LAD2based Tests for Linear Hypothesis in Censored Regression Models n

519

n

∑

I (μi > 0) { sgn ( ei - xi′ wn) - sgn ( ei ) } xin -

i =1

∑ I (μ i

> 0) E[ sgn ( ei - xi′ wn) ] xin = op (1)

i =1

( a . 17)

since the expectation of the left hand is 0 and the covariance matrix of the left hand converges to 0 . Now ( 11) follows by ( a. 16) , ( 1. 17) and ( a. 1) . Theorem 1 is proved. Proof of Lemma A. 1. We will follow the line of Rao and Zhao [18 ] to prove Lemma A. 1. Here2 after we mean by C a positive constant independent of n , which may take different values in different places. Let n

Vn =

∑

I (μi > 0) sgn ( ei - xi′ w n ) xin . 　　γ gn ≡

i =1

- 1/ 2

S n Vn 2 f ( 0)

( a . 18)

Similar to ( a. 14) , we have d 1/ 2 γn →0 　 in probability V n + 2 f ( 0) S n w n →N ( 0 , Ip) , 　thus 　g

By ( C5) and ( 4) , it follows ‖w n ‖ →0 . Notice ( a. 3) , by Lemma 2. 2 in Rao and Zhao

( a . 19) [18 ]

, we

have that γ ^ n →0 in probability . Thus , there exists a sequence : n ∈ ( 0 , 1) , : n →0 such that

γn ‖ + ‖γ P ( ‖g ^ n ‖ ≤: n ) →1 　 as n → ∞.

( a . 20)

By ( C5) we can take αn → ∞ such that

α2n: n →0 　 and 　α-n 2λ( S n ) / log2 n → ∞.

( a . 21)

Write n

∑1 ∑1

S n1 =

I (μi > 0 , ‖xi ‖ > αn ) xi xi′,

i= n

S n0 =

I (μi > 0 , ‖xi ‖ ≤αn ) xi xi′ .

i=

Define n

Hn (γ) =

∑

i =1 n

(μ γ) g + x′

n

hni (γ) ≡

∑∫ μ g i

+

i

( sgn ( v - Yi ) + sgn ( ei - xi′ wn ) ) d v =

+

i =1

i

γ) + - Yi gi + xi′ ( (μ

n

μ gi + - Yi ) +

∑

γ) ∑( (μg + x ′ i

i =1 n

-μ gi +) sgn ( ei - xi′ wn ) ≡

n

∑

i =1

( 1) h ni (γ) +

∑h 2

( ) ni

(γ) ,

( a . 22)

i =1

n

Hn0 (γ) =

+

i

i =1

n

∑

I ( ‖xi ‖ ≤αn ) h ni (γ) , 　Hn1 (γ) =

i =1

∑I ( ‖x

i

‖ > αn ) h ni (γ) .

( a . 23)

i =1

By ( C3) and ( C4) , as n → ∞, n

∑I ( ‖x

i

‖ > αn ) ‖xi ‖2 = o (λ( S n ) ) ,

( a . 24)

i =1 n

∑I ( ‖μ ‖ ≤α: ) ‖x i

n n

i

‖2 = o (λ( S n ) ) .

( a. 25)

i =1

Similar to Pollard[15 ] , pages 62～63 , the sets , for k = 1 ,2 , ( ( k) γ ‖, …, h (( k)nn) (γ) / ‖γ ‖) ′∈ R n , 0 ≠γ ∈ BgT} H ( k) n , w) = { ( h n1 (γ) / ‖

© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

( a. 26)

中国科学技术大学学报 　　　　　　　　　　　　　　第 34 卷

520

have finite pseudodimensions determined by p. By the maximal inequality for manageable processes , ( noting that h nik) (γ) / ‖γ ‖ ≤ ‖x i ‖, by ( a. 24) we have n

-1 Hn1 (γ) - EHn1 (γ) ) 2 ≤ C E sup ( ‖γ ‖ ‖γ‖≤: n

∑I ( ‖x

i

‖ > αn ) ‖xi ‖2 = o (λ( S n ) ) .

i =1

( a . 27)

By ( C1) and ( 4) , there exists aΔ > 0 such that F ( v + x i′ w n ) - F ( xi′ w n) = ( f ( 0) + o ( 1) ) v , v < Δ when n is large , which implies that

uniformly for

γ) 2 　for any γ. 0 ≤ E h ni (γ) ≤ C ( xi′

( a. 28)

γ>- μ To show this , we take the case when μ gi > 0 and xi′ gi as an example. One has E h ni (γ) =

μ γ g+ x ′

∫

i

E[ sgn ( v - Yi ) + sgn ( ei - xi′ wn ]d v =

i

μ γ gi + x i′

∫ μ gi

[ 2 F ( v - μi ) - 1 + 1 - 2 F ( x i′ w n) ]d v =

γ x′

∫ 2 [ F ( u + μg - μ) i

i

0

γ x′

∫ vd v = f (0) (1 + o (1) ) ( x ′γ)

2 f ( 0) ( 1 + o ( 1) )

i

2

i

0

i

F ( x i′ w n) ]d u =

γ < Δ( a. 29) γ) 2 , 　 if xi′ , 1 ≤ C ( xi′

C γ ≥Δ. So ( a. 28) follows immediately. By x′ ( γ) 2 Δ xi′ , 　if i ( a. 28) and ( a. 24) , for 0 < ‖γ ‖ ≤: n ,

γ ≤ and E h ni (γ) ≤ C xi′

n

∑I ( ‖x

EHn1 (γ) / ‖γ ‖2 ≤ C

i

‖ > αn ) ‖xi ‖2 = o (λ( S n) ) ,

( a . 30)

i =1 n

γ′ S n1γ/ ‖γ ‖2 ≤ C

∑I ( ‖x

i

‖ > αn ) ‖xi ‖2 = o (λ( S n ) ) .

( a . 31)

i =1

By ( a. 27) , ( a. 30) and ( a. 31) , as n → ∞, Hn1 (γ) - f ( 0) γ′ S n1γ = op ( ‖S n γ ‖ + ‖S n γ ‖ ) 　uniformly for ‖γ ‖ ≤: n . 1/ 2

1/ 2

2

( a. 32)

For ‖xi ‖ ≤αn and ‖γ ‖ ≤: n , we consider the following cases : ( Ⅰ) When μ γ>- μ gi > 0 and xi′ gi , hni (γ)

xγ ′

=

∫[sgn ( u + x ′w - e ) + sgn( x ′w - e ]d u i

i

0

By ( C1) , ( 4) and

n

i

i

n

i

γ I ( ei - xi′ γ ). wn ≤ xi′ ≤2 xi′

γ ≤αn: n →0 , it follows , uniformly for γ ≤: n , that xi′

γ) = γ) 2 P ( ei - xi′ w n ≤ xi′ Var [ h ni (γ) ] ≤E h2ni (γ) ≤4 ( xi′ γ) 2 ( F ( xi′ 4 ( x i′ wn + γ xi′

8 ( f ( 0) + o ( 1) ) γ x′

∫ ( F ( u + x ′w ) -

Eh ni (γ) = 2

i

i

0

n

γ ) - F ( xi′ x i′ wn 3

γ )) = x i′

γ 3, ≤ C xi′ γ) 2 ( 1 + o ( 1) ) . F ( xi′ w n ) d u = f ( 0) ( xi′

γ ≤- μ 　　( Ⅱ) When μ gi > 0 and xi′ gi , then h ni (γ) =

-μ gi

∫[ sgn ( u + x ′w 0

i

n

γ - ei ) - sgn ( xi′ w n - ei ) ] d u , Var [ h ni (γ) ] ≤ C xi′

© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

3

第 5 期 　　　　　　LAD2based Tests for Linear Hypothesis in Censored Regression Models -μ gi

∫[ F ( u + x ′w ) -

0 ≤E h ni (γ) = 2

i

0

γ x′

∫ [ F ( u + x ′w ) i

0

F ( xi′ w n) ]d u ≤

γ) F ( x i′ w n ) ] d u ≤ C ( x i′

i

2

n

n

521

2

I ( 0 < μi ≤αn: n )

γ>- μ 　　( Ⅲ) Similarly , when μ gi ≤0 and xi′ gi , γ Var [ h ni (γ) ] ≤C x i′

3

I (μi ≤0 , μi ≤αn: n) ,

γ) 2 I (μi ≤0 , μi ≤αn: n ) 0 ≤ E h ni (γ) ≤C ( x i′ Summarizing ( Ⅰ) ～ ( Ⅲ) , by ( a. 25) and ( a. 23) , we obtain n

E Hn0 (γ) =

∑I ( ‖x ‖ ≤α ) E h i

γ) =

ni (

n

i =1

n

n

∑

f (0) (1 + o (1))

γ) 2 + O I (μi > 0 , ‖xi ‖ ≤αn) ( xi′

i =1

∑I ( μ i

γ) 2) = ≤αn: n ( xi′

i =1

f ( 0) γ′ S nγ( 1 + o ( 1) ) 　uniformly for ‖γ ‖ ≤: n as n → ∞,

( a. 33)

and n

　　　　Var Hn0 (γ) = ∑I ( ‖xi ‖≤αn ) Var h ni (γ) ≤ i =1

n

Cαn ‖γ ‖

n

∑

γ) 2 + I (μi > 0 , ‖xi ‖ ≤αn) ( xi′

i =1

∑I (

μi ≤αn: n) ( xi′ γ) 2 ≤

i =1

Cαn ‖γ ‖γ′ S nγ.

( a. 34)

Using ( a. 21) , ( a. 33) and ( a. 34) , similar to Rao and Zhao

[18 ]

we can show that , with proba2

bility one for the a large n , Hn0 (γ) = f ( 0) γ′ S nγ + o ( ‖S n γ ‖ + ‖S n γ ‖ ) + O ( 1/ n ) 　uniformly for ‖γ ‖ ≤: n . 1/ 2

1/ 2

2

( a. 35)

Write n

Z n (γ) =

n

∑

z ni (γ) ≡

i =1

∑I ( ‖x

i

γ) + - μ γ] × ‖ ≤αn ) [ (μ gi + xi′ gi + - I (μi > 0) xi′

i =1

[ sgn ( ei - xi′ w n ) - ( 1 - 2 F ( x i′ w n ) ) ].

( a. 36)

n

Z1 n (γ) =

∑ I ( ‖x

i

γ) + - μ ‖ ≤αn ) [ (μ gi + xi′ gi + ][ 1 - 2 F ( xi′ wn) ]

( a. 37)

γ) + - μ xi′ g+i ] sgn ( ei - xi′ wn)

( a. 38)

i =1 n

Z2 n (γ) =

∑[ (μg + i

i =1

As before , by using the maximal inequality for manageable processes , noting that E Z n (γ) = 0 and that : ‖γ ‖- 1 z ni (γ)

≤ C ‖xi ‖ I (μi ≤αn: n ) 　for ‖γ ‖ ≤: n . We obtain n

-1 Z n (γ) ) 2 ≤ C E sup ( ‖γ ‖ ‖γ‖≤: n

∑I (

μi ≤αn: n ) ‖xi ‖2 .

( a. 39)

i =1

Since αn: n →0 , by ( a. 25) and ( a. 39) , Z n (γ)

= op ( ‖S 1n/ 2γ ‖) 　 as n → ∞ uniformly for ‖γ ‖ ≤: n .

By ( a. 21) and ( a. 25) © 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

( a. 40)

中国科学技术大学学报 　　　　　　　　　　　　　　第 34 卷

522 n

Z1 n (γ)

≤o (1)

∑

n

∑ I ( ‖μ ‖ ≤α: ) ‖x

γ) 2 ≤ o (1) I ( ‖μi ‖ ≤αn: n) ( xi′

i

i =1

n n

i

‖2 ‖γ‖2 =

i =1

o (γ′ S nγ) 　　　uniformly for 　　‖γ ‖ ≤: n , 　if n → ∞.

( a. 41)

It follows by ( a. 36) , ( a. 40) ～ ( a. 41) and ( a. 19) that Z2 n (γ) = V n′ S n γ + op ( ‖S n γ ‖) + op ( ‖S n γ ‖ ) . 1/ 2

1/ 2

1/ 2

2

( a. 42)

Now by ( a. 2) , ( a. 22) , ( a. 23) , ( a. 32) , ( a. 35) , ( a. 38) and ( a. 42) Gn (γ) = - V n′ S n γ + f ( 0) γ′ S nγ + op ( 1 + γ′ S nγ) uniformly for ‖γ ‖ ≤: n . 1/ 2

( a. 43)

By ( a. 18) , ( a. 20) and ( a43) , we have Gn (γ ^ n ) = - 2 f ( 0) γ gn′ Sγ n ^ + op ( 1 + γ S nγ) + op ( 1) = 2 2 f (0) ( 1 + op (1) ) ‖S 1n/ γ ^ n - S 1n/ γ gn (1 + op (1) ) ‖2 - f (0) (1 + op (1) )γ gn′ Sγ ngn + op ( 1) ,

( a. 44) Gn (γ g) = f (0) (1 + op (1) ) ‖S n γ gn ( op (1) ) ‖ - f (0) (1 + op (1) )γ gn′ Sγ ngn + op (1) . 1/ 2

2

( a. 45)

Since S n γ g = Op ( 1) ( refer to ( a. 18 ) and ( a. 19 ) ) , by ( a. 44 ) and ( a. 45 ) , and noting that 1/ 2

Gn (γ ^ n ) - Gn (γ gn ) ≤0 , one obtains f (0) (1 + op (1) ) ‖S 1n/ 2 (γ ^n - γ gn) + op (1) ‖2 + op (1) ≤0. It

follows that 1/ 2 S n (γ ^n - γ gn ) →0 　in probability as n → ∞.

( a. 46)

Now Lemma A. 1 follows by ( a. 19) , ( a. 44) and ( a. 46) .

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删失回归模型中基于 LAD 的线性假设检验研究 崔文泉 ( 中国科学技术大学统计与金融系 ,安徽合肥 ,230026)

摘要 : 在 Zhao L C. Linear hypothesis testing in censored regression models 基础上讨论了删失回归 模型的线性假设检验问题 ,在局部备择假设成立时 ,给出了基于最小绝对偏差所构造的检验 统计量的渐近分布 ,并对检验统计量的表现进行了统计模拟 . 关键词 : 局部备择假设 ,线性假设检验 ,删失回归模型 ,渐近分布 .

© 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.