Consistent test for parametric models with right-censored data using projections

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Model 3G

pp. 1–14 (col. fig: nil)

Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

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Consistent test for parametric models with right-censored data using projections✩ Zhihua Sun a,b, *, Xue Ye a , Liuquan Sun c a b c

University of Chinese Academy of Sciences, Beijing, China Key Laboratory of Big Data Mining and Knowledge Management of CAS, Beijing, China Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

article

info

Article history: Received 8 February 2017 Received in revised form 7 September 2017 Accepted 13 September 2017 Available online xxxx Keywords: Consistent test Curse-of-dimensionality-free Empirical process Projection Right-censored data

a b s t r a c t In the literature, there are several methods to test the adequacy of parametric models with right-censored data. However, these methods will lose effect when the predictors are medium-high dimensional. In this study, a projection-based test method is built, which acts as if the predictors were scalar even if they are multidimensional. The proposed test is shown to be consistent and can detect the alternative hypothesis converging to the null hypothesis at the rate n−r with 0 ≤ r ≤ 1/2. Also, it is free from the choices of the subjective parameters such as bandwidth, kernel and weighting function. A wild bootstrap method is developed to determine the critical value of the test, which is shown to be robust to the model conditional heteroskedasticity. Simulation studies and real data analyses are conducted to validate the finite sample behavior of the proposed method. © 2017 Published by Elsevier B.V.

1. Introduction In recent years, there has been an upsurge of study on statistic models, especially on semiparametric models. However, parametric models are still valuable tools because they have prominent advantages such as the high precision, the good interpretability, the desiring prediction capability and the easy accessibility via existing procedures in statistic software. Li and Racine (2007) concluded that the correctly specified parametric models are usually a ‘‘first-best’’ solution for statistic inference. But if the parametric models are misspecified, the statistical analysis results will be erroneous. Therefore, it is significant to develop a formal testing procedure for parametric models. The adequacy check of parametric models has attracted a lot of attention since (Bierens, 1982) first proposed a consistent test. Escanciano (2006) divided the existing methods into two categories: local approaches and integrated ones. For the local approaches, see Dette (1999), Härdle et al. (1998), Horowitz and Härdle (1994), Eubank and Hart (1992), Härdle and Mammen (1993), and Zheng (1996), among others. For the integrated method, we can refer to Bierens (1982), Bierens and Ploberger (1997), Stute et al. (1998), Stute and Zhu (2002), Escanciano (2006), Sun and Wang (2009) and the references within. In medical, biologic and economic research fields, one may find that the response variable is often rightly censored because of the end of the study or the loss of follow-up. Several existing approaches aim for checking the adequacy of parametric models with right-censored data. For example, Stute et al. (2000) extended the method of Stute et al. (1998) ✩ The research was supported by the National Natural Science Foundation of China (Grant Nos. 11231010, 11690015, 11571340, U1430103), the National Center for Mathematics and Interdisciplinary Sciences and the Open Project of Key Laboratory of Big Data Mining and Knowledge Management, CAS. Corresponding author. E-mail address: [email protected] (Z. Sun).

*

https://doi.org/10.1016/j.csda.2017.09.005 0167-9473/© 2017 Published by Elsevier B.V.

Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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to deal with the right-censored data. Pardo-Fernández et al. (2007) constructed a testing method based on the distribution difference of the estimated residuals of the parametric and nonparametric models. Lopez and Patilea (2009) employed a U-process to build a test statistic. All these methods perform well in obtaining reasonable empirical sizes and high empirical powers for a low dimensional parametric model. However, they will lose effect when the predictors are medium-high dimensional. The methods of Pardo-Fernández et al. (2007) and Lopez and Patilea (2009) applied the local smoothing method, which causes the testing methods suffering from ‘‘curse of dimensionality’’. For the integrated method of Stute et al. (2000), the test statistic based on the indicator function tends to degenerate to zero when the covariates are medium-high dimensional. In this paper, we develop a testing method free from the curse of dimensionality for parametric models with rightcensored data. A projection-based testing statistic is constructed by applying a linear indicator weighting function. We show that even if the covariates are multivariant, the proposed testing method acts as if they were scalar. Furthermore, we carry through extensive simulation studies to validate that the proposed method performs over the existing tests in terms of the empirical sizes and powers when the predictors are medium or high dimensional. Besides the powerful advantage to deal with medium or high dimensional data, the proposed method has the following merits: it is consistent; it can detect the alternative hypothesis converging to the null hypothesis at the rate n−r with 0 ≤ r ≤ 1/2; it is free from the subjective parameters such as bandwidth, kernel and weighting function. To determine the critical value, a wild bootstrap method is proposed, which is shown to be robust to the conditional heteroskedasticity. The rest of this paper is organized as follows. We describe the testing problem and an estimating procedure in Section 2. In Section 3, we propose the testing method and study its asymptotic properties. The analysis of the power is conducted in Section 4. In Section 5, we consider the calculation of the testing statistic and develop a bootstrap method to calculate the critical value. Simulation studies and two real data analyses are conducted in Sections 6 and 7, respectively. The proofs of the main results are collected in Appendix.

23

2. Existing estimation of the null hypothetical model

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

24 25

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

41

Let X be a p-dimensional predictor and Y be a scalar response. Consider the regression model: Y = m(X ) + e, where m(x) = E(Y |X = x) and e is the error variable with E(e|X ) = 0. In this study, we consider the null hypothesis: H0 : P {m(X ) = g(X , β )} = 1 for some β, where g is a known function and β is an unknown parameter. The alternative hypothesis can be written as: H1 : P {E(Y |X ) = g(X , β )} < 1 for all β ∈ Rp . In presence of random right censoring, the response Y is not always available. Let C denote the censoring time variable with the distribution G(·) and δ = I(Y ≤ C ). Then, instead of Y , the variable Z = Y ∧ C is observed. We first describe an estimating procedure of the null hypothetical model. To deal with randomly-censored data, the inverse probability weighting method is an effective way. There are two means to employ the inverse probability: One aims for adjusting the response variable; and the other aims for adjusting the objective least square function or equivalently for adjusting the estimating equation. The former is called the synthetic data (SD) method and the latter is called the weighted least squares (WLS) method. It is well known that the estimating procedure is very critical to the effect of the model checking method. Lopez and Patilea (2009) showed that the testing method based on the WLS estimating procedure outperforms the test based on the SD method. The WLS method calibrates the model error directly, which is appropriate for the model checking problem. Therefore, we apply the WLS method to estimate the null hypothetical model. Assume that we have an i.i.d. sample {(Zi , δi , Xi ), i = 1, 2, . . . , n} from (Z , δ, X ). The WLS method defines an estimator of β , denoted by βˆ n , which minimizes the following weighted least squared objective function: Mn (β ) =

n ∑ i=1

δi (Zi − g(Xi , β ))2 , ˆ i −) 1 − G(Z

∑n

43

ˆ where G(z) = 1 − j:Zj ≤z (1 − 1/ referred to in Koul et al. (1981).

44

3. Testing methods

45

3.1. Existing testing methods

42

46 47 48

∏

k=1 I(Zj

(2.2)

ˆ ≤ Zk ))1−δj is the Kaplan–Meier estimator of G(z). More details on G(z) can be

Based on the estimated null hypothetical model, an estimated model error can be constructed: eˆ i = nWin (Zi − g(Xi , βˆ n )) ˆ i −))). Observe that the null hypothesis (2.1) is equivalent to E(eI(X < x)) = 0 for any x with with Win = δi /(n(1 − G(Z e = Y − g(X , β ). An empirical-process based test statistic can be defined as:

∫ 49

(2.1)

Tn,e =

[Rn,e (x)]2 Fn (dx),

Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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where the estimated empirical processes Rn,e (x) = n−1/2 i=1 eˆ i I(Xi < x), and Fn (·) is the empirical distribution function of X based on the data {Xi , i = 1, 2, . . . , n}. The null hypothesis (2.1) is also tantamount to E(eE(e|X = x)) = 0 for any x. By estimating E(eE(e|X = x)) via the local smoothing method, a test U-statistic can be constructed:

∑n

Tn,u = (n(n −

1)hpn Vn )−1

∑

eˆ i eˆ j K ((Xi − Xj )/hn ),

1 2 3 4

5

i̸ =j

where hn is a bandwidth sequence, K (·) is a kernel function, and Vn2

= 2(n(n −

1)hpn )−1

∑

e2i e2j K 2 ((Xi

ˆ ˆ

6

− Xj )/hn ).

7

i̸ =j

The test methods based on Tn,e and Tn,u are investigated by Stute et al. (2000) and Lopez and Patilea (2009). Both testing methods are consistent and perform well when there are only several predictors, as shown by the simulation results in Section 6. Note that both test methods are persecuted by the ‘‘curse of dimensionality’’. With the increase of the dimension of X , the application of the indicator function would make Tn,e degenerate to zero, and it is difficult to carry through the local estimation of E(e|X = x) in Tn,u . As a reviewer pointed out, Pardo-Fernández et al. (2007) also considered the same parametric mean regression model checking problem with the right-censored response variable. The method in Pardo-Fernández et al. (2007) is based on the difference of the error distributions under the parametric and nonparametric models. In terms of the dimension of the covariate, the method in Pardo-Fernández et al. (2007) only deals with one-dimensional covariates, while Stute et al. (2000) and Lopez and Patilea (2009), in principle, deal with multidimensional covariates. In this work, we also aim for developing a test method suitable for models with multidimensional covariates. 3.2. Projection-based testing method

8 9 10 11 12 13 14 15 16 17 18

19

The null hypothesis (2.1) is also equivalent to E(eW (X , x)) = 0, ∀x ∈ Rp , for some special weighting function W (X , x). Bierens (1982), Escanciano (2006) and Ma et al. (2014) presented the primitive conditions on the weighting function to make the equivalence hold. The following five weighting functions are often recommended: the exponential weighting function exp(ixτ X ) where i = (−1)1/2 is the imaginary unit; the linear indicator (LI) weighting function I(X τ θ ≤ u) for any vector θ ∈ Rp and any real number u; the logistic weighting function {1 + exp(xτ X )}−1 ; the simple indicator (SI) weighting function I(X ≤ x); and the trigonometric weighting function cos(xτ X ) + sin(xτ X ). Furthermore, Bierens and Ploberger (1997) showed that all these weighting functions lead to asymptotic admissible tests. Hence it is of no significance to find a theoretically optimal weighting function. We employ the LI weighting function to construct a test statistic. The similar idea has been applied by Stute and Zhu (2002) and Xia et al. (2004). They took the projecting parameter θ to be the estimated regression coefficient. These testing methods suffer from the inconsistency. Escanciano (2006) first considered the projecting parameter as a random variable. The application of the LI weighting function I(X τ θ ≤ u) makes the multivariate predictors act as a scalar variable. Therefore, the tests based on the LI weighting function can avoid the ‘‘curse of dimensionality’’. By considering the projecting parameter as a random variable with the continuous distribution function, the test will be consistent as shown in the next section. Based on the LI weighting function, we construct an estimated empirical process:

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

n

Rn (u, θ ) = n−1/2

∑

eˆ i I(Xiτ θ ≤ u),

(3.1)

35

i=1

where θ is the projection parameter. We assume that θ is a random variable with density fθ (·). Then the projection-based test statistic can be constructed as follows:

∫ Tn =

[Rn (u, θ )]2 fθ (θ )Fnθ (du)dθ,

36 37

38

where Fnθ (·) is the empirical distribution function of X τ θ . Via integration, the test statistic Tn is unrelated to θ and u.

39

3.3. The asymptotic results

40

By applying the techniques of the kernel regression method and the theory of empirical process, we can obtain the following asymptotic properties for the test statistic Tn under the null hypothetical model. Theorem 1. Suppose that Conditions (C1)–(C5) in Appendix hold. Under the null hypothesis (2.1), Rn (u, θ ) converges in distribution to R(u, θ ) in the Skorokhod space D[−∞, ∞]× D[0, 1]p , where R(u, θ ) is a centered Gaussian process. The covariance function of R(u, θ ) is [1 ]

[1]

Cov{R(u1 , θ1 ), R(u2 , θ2 )} = E(H(u,θ ) (Z , δ, X ; u1 , θ1 )H(u,θ ) (Z , δ, X ; u2 , θ2 )),

(3.2)

Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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where [1]

2

H(u,θ ) (Z , δ, X ; u, θ ) = E [H(u,θ ) (Z , δ, X ; Z1 , δ1 , X1 ; u, θ )|Z1 , δ1 , X1 ],

3

and H(u,θ ) (Z , δ, X ; Z1 , δ1 , X1 ; u, θ ) is listed in Appendix A. Furthermore, we have

∫

L

4

5 6 7 8 9

10

11 12

13 14 15

16 17

where Fθ (u) is the conditional distribution of X τ θ given θ . Here we take θ as a random variable with a density function f (θ ) satisfying Condition (C5). The continuity of the density function f (θ ) ensures that infinite values of the projection parameter θ are considered. When there is no empirical information on f (θ ), we can assume that θ follows the uniform distribution on the unit sphere. Because all the directions on the unit sphere are considered, the proposed test is consistent to all the alternatives. 4. Analysis of the power To investigate the sensitivity of the proposed test, we analyze its power by considering the alternative hypothetical models: H1n : Y = g(X , β ) + dn D(X ) + η,

19 20 21

(4.1)

with E [η|X ] = 0, some constant dn and some bounded measurable function D(·). If dn = 1, Eq. (4.1) is the global alternative hypothetical models. When n1/2 dn −→ 1, Eq. (4.1) is Pitman alternative hypothetical models. Theorem 2. Under Conditions (C1)–(C5) in Appendix, for the alternative (4.1), if nγ dn −→ a with 0 ≤ γ < 1/2 and some constant a > 0, then Tn converges to ∞; if n1/2 dn −→ 1, then

∫

L

18

[R(u, θ )]2 f (θ )Fθ (du)dθ,

Tn −→

[R(u, θ ) + Ω ]2 f (θ )Fθ (du)dθ,

Tn −→

(4.2)

˙ , β ) is the derivative of g with respect to β , where R(u, θ ) is defined in Theorem 1, g(x ˙ , β )I(X τ θ ≤ u)]A−1 E [D(X )g(X ˙ , β )], Ω = E [D(X )I(X τ θ ≤ u)] − E [g(X ˙ , β )g(X ˙ , β )τ ]. and A = E [g(X

25

For the global alternative hypothetical models with dn = 1, Tn converges to ∞. Therefore, the test Tn is consistent. Furthermore, the test Tn has asymptotic power 1 for the local alternative hypothetical models close to the null hypothetical model with the rate n−r , 0 < r < 1/2. For the Pitman alternative hypothetical models with r = 1/2, the proposed test has the non-trivial power.

26

5. Realization of the test

22 23 24

27 28 29

30

In this section, we consider two problems related to the realization of the proposed test: the computation of the test statistic and the determination of the critical value. By simple algebraic manipulations, we have Tn = n−2

n n n ∑ ∑ ∑

eˆ i eˆ j Aijk ,

i=1 j=1 k=1 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

with Aijk = I(Xiτ θ ≤ Xkτ θ )I(Xjτ θ ≤ Xkτ θ )f (θ )dθ. If we know the density of θ , we would be able to calculate Tn . When θ is uniformly distributed on the unit sphere, inspired by Escanciano (2006), we have

∫

Aijk = Cq ⏐π − arccos (Xi − Xk )′ (Xj − Xk )/(|Xi − Xk | |Xj − Xk |) ⏐ ,

⏐

{

}⏐

where Cq = π q/2−1 /(Γ (q/2) + 1), q = p + 1 and Γ (·) is the gamma function. To obtain the critical value of Tn , following Wu (1986), Stute et al. (1998) and Escanciano (2006), we apply the wild bootstrap method to mimic the null distribution of the proposed test statistic and to determine the critical value. Let {ξi }ni=1 be a sequence of independent and identically distributed random √ variables with mean 0 and variance √ 1. For instance, ξi could be chosen from a two-point distribution that has values (1 ∓ 5)/2 with the probabilities (5 ± 5)/10. The testing process can be performed as follows: Step 1: Compute the test statistic Tn ; Step 2: Generate random variables {ξi }ni=1 and let Yi∗ = g(Xi , βˆ n ) + ξi eˆ i . Compute Tn using the bootstrap sample (Xi , Yi∗ )ni=1 . Denote the bootstrap test statistic by Tn∗ ; Step 3: Repeat B times of Step 2. For given level α , take the critical value by calculating the 1 − α quantile of the bootstrap test statistics. We can find that the wild bootstrap method does not depend on the distributional information of the model error. Therefore, it is robust to the conditional heteroscedasticity of the model error. Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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Table 6.1 The values of µ of the censoring distribution for Settings I–IV. Censoring rate

20%

d

Setting

40%

I

II

III

IV

I

II

III

IV

0.0 0.5 1.0 1.5

54.6 59.6 61.0 65.9

4.8 5.2 5.6 5.9

62.7 64.2 67.6 73.3

225.5 231.5 234.6 236.3

22.9 24.4 26.6 28.0

1.7 1.9 2.0 2.1

26.9 28.2 29.9 30.5

97.4 98.1 100.6 101.2

6. Simulations To investigate the finite sample behaviors of the proposed method, we carry out extensive simulation studies. The projection parameter θ is assumed to follow the uniform distribution on the unit sphere in the space Rp . Then the proposed test can be implemented according to the method in Section 5. We consider four models with 2-, 4-, 7- and 13-dimensional predictors described in the following: Setting I: Y = β1 X1 + β2 X2 + d{sin(|X1 X2 |) + 1.5} + ε, where X1 , X2 ∼ U(0, 2π ), (β1 , β2 ) = (1, 3). And the model error ε = ϕ + 0.5770 where ϕ follows standard extreme value distribution. We can validate that the model error ε has zero expectation. ∑4 ′ Setting II: Y = exp(X τ β ) + d{0.1( i=1 Xi2 ) − 0.6} + ε, where X1 , X2 , X3 , X4 ∼ N4 (0.114 , Σ ), Σ = (σjj′ ), σjj′ = 0.1|j−j | , β = (−0.5, 0, 0.5, 1)τ and ε ∼ N(0, 0.52 ). Setting III: Y = X τ β + d exp{0.01(X τ β ) + 0.1} + ε, where X1 , X2 , X3 ∼ N3 (13 , I), X4 , X5 , X6 , X7 ∼ U(0, 2π ), β = (−1, −0.5, 0, 0.5, 1, 1.5, 2)τ and ε ∼ ∑ N(0, 0.52 ). ′ 13 τ Setting IV: Y = X β + d{0.1 cos( i=1 |Xi |) + 1.5} + ε, where X1 , X2 , . . . , X6 ∼ N6 (16 , Σ ), Σ = (σjj′ ), σjj′ = 0.1|j−j | , τ 2 X7 , X8 , . . . , X13 ∼ U(0, 2π ), β = (−1.5, −1, −0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5) and ε ∼ N(0, 0.5 ). Here 1j is a j × 1 vector with all components one. We take the sample sizes n = 100 and 200 into account. For Setting IV with 13-dimensional predictor, the data are quite sparse. Five hundred replications are conducted for each simulation experiment. For the bootstrap procedure, the number of replication B is 300. The censoring variable C follows an exponential distribution with the mean µ. By choosing different values of µ, the censoring rates are 20% or 40%. The values of µ are listed in Table 6.1. For all the simulated models, we calculate the empirical sizes and powers by choosing d ∈ {0, 0.5, 1, 1.5}. When d = 0, the null hypothesis holds. The choice of d = 0.5, 1, 1.5 means that different alternative models are considered. The bigger the b is, the further the alternative models are from the null hypothetical model. The detailed simulated results are presented in Tables 6.2–6.6. We also report the simulation results for the existing testing methods in Stute et al. (2000) and Lopez and Patilea (2009). The method of Pardo-Fernández et al. (2007) focuses on the one-dimensional case only. Since we aim for developing model diagnosis method for the parametric models with multidimensional covariates, we do not make comparison with the method of Pardo-Fernández et al. (2007). For the U-process test in Lopez and Patilea (2009), we consider two choices of the kernel functions: Epanechnikov and Gaussian kernel functions. The bandwidth is chosen according to the rule of thumb and the necessity of undersmoothing: 2.34std(X )n−1/3 and 1.06std(X )n−1/3 respectively for Epanechnikov and Gaussian kernel functions. Here by choosing different kernel functions with bounded and unbounded supports, we rule out the influence of the kernel functions on the dimensionality problem because two kernel functions yield similar results. Because of the ‘‘curse of dimensionality’’, the existing methods in Stute et al. (2000) and Lopez and Patilea (2009) sometimes lose effect. We compute the empirical sizes and powers from the un-degenerated simulation results. We also record the failure times of the tests in Tables 6.2–6.6. For the U-process test in Lopez and Patilea (2009), two kernel functions yield the same failure times, which is recorded by F u . The projection-based test does not degenerate in all situations. From the simulated results of the empirical sizes in Table 6.2, we find that all the four tests have acceptable empirical ep ga sizes in Setting I. However, the empirical sizes of the existing methods Tn,u , Tn,u and Tn,e are getting worse from Settings II to IV. In particular, for Settings III and IV, the empirical sizes are small and far from the ideal values. The phenomenon is actually an exhibition of the ‘‘curse of dimensionality’’. Because of the sparsity of data, in some simulated experiments, even if the test statistics do not degenerate to zero, some addends corresponding to some subjects in the testing statistics tend ep to be zero. Note that the existing testing statistics are some summations for all the subjects. Thus the testing statistics Tn,u , ga Tn,u and Tn,e tend to be smaller and the null hypothesis cannot be rejected easily than it should be. So the empirical sizes are always smaller than the theoretical levels. We also find that the proposed test Tn is free of the curse of dimensionality and the empirical sizes are very close to the test levels for all four settings. ep ga Tables 6.3–6.6 present the empirical powers of the four tests Tn , Tn,u , Tn,u and Tn,e . As shown in Table 6.3, when the predictor is 2-dimensional, all tests perform well though there are slight differences among these tests. But for Setting II with 4-dimensional predictor, the proposed test outperforms the other tests for all the situations in terms of the empirical powers. For setting III with 7-dimensional predictor, the U-process tests sometimes lose effect; the empirical process test with the SI weighting function has poor empirical powers, but the proposed test still keeps high empirical powers. For the Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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Table 6.2 Empirical sizes of different testing methods for Settings I–IV under different censoring rates and sample sizes. % Cens.

20%

40%

α:

0.05

n

Tn

T n, u

T n, u

T n, e

Tn

T n, u

T n, u

Tn,e

I

100 200

0.042 0.054

0.040 0.046

0.042 0.046

0.042 0.052

0.106 0.102

0.092 0.096

0.106 0.102

0.092 0.096

0 0

0 0

II

100 200

0.048 0.052

0.028 0.042

0.028 0.036

0.042 0.064

0.080 0.094

0.072 0.096

0.062 0.092

0.078 0.098

0 0

0 0

III

100 200

0.062 0.048

0.000 0.000

0.000 0.000

0.022 0.022

0.112 0.096

0.000 0.000

0.006 0.016

0.050 0.066

305 321

0 0

IV

100 200

0.060 0.054

0.000 0.000

0.000 0.000

0.000 0.000

0.128 0.104

0.000 0.000

0.000 0.000

0.008 0.004

499 500

206 25

I

100 200

0.060 0.056

0.038 0.042

0.040 0.058

0.040 0.056

0.114 0.094

0.110 0.090

0.090 0.106

0.108 0.096

0 0

0 0

II

100 200

0.078 0.044

0.018 0.012

0.020 0.018

0.070 0.060

0.118 0.082

0.060 0.062

0.056 0.060

0.106 0.096

0 0

0 0

III

100 200

0.064 0.046

0.000 0.000

0.000 0.000

0.012 0.026

0.126 0.106

0.002 0.000

0.006 0.006

0.044 0.062

287 312

0 0

IV

100 200

0.076 0.056

0.000 0.000

0.000 0.000

0.000 0.000

0.152 0.108

0.000 0.000

0.000 0.000

0.000 0.000

500 500

198 25

Setting

0.10 ep

ga

Failure times ep

ga

Fu

Fe

Tn : Proposed test; ep ga Tn,u and Tn,u : U-process tests with Epanechnikov and Gaussian kernel functions; Tn,e : Empirical process test with the simple indicator weighting function; ep ga F u : Failure times of tests Tn,u and Tn,u ; F e : Failure time of test Tn,e .

Table 6.3 Empirical powers of different testing methods for Setting I under different censoring rates and sample sizes. % Cens.

20%

40%

α:

0.05

n

Tn

T n, u

T n, u

T n, e

Tn

T n, u

T n, u

T n, e

Fu

Fe

0.5

100 200

0.640 0.920

0.152 0.400

0.154 0.394

0.638 0.922

0.760 0.962

0.226 0.490

0.228 0.490

0.738 0.960

0 0

0 0

1.0

100 200

0.984 1.000

0.574 0.956

0.592 0.954

0.980 1.000

0.996 1.000

0.664 0.974

0.690 0.980

0.992 1.000

0 0

0 0

1.5

100 200

1.000 1.000

0.870 1.000

0.892 1.000

1.000 1.000

1.000 1.000

0.916 1.000

0.922 1.000

1.000 1.000

0 0

0 0

0.5

100 200

0.540 0.832

0.096 0.208

0.096 0.222

0.578 0.842

0.674 0.910

0.168 0.312

0.164 0.304

0.664 0.910

0 0

0 0

1.0

100 200

0.934 1.000

0.430 0.852

0.452 0.856

0.922 1.000

0.958 1.000

0.548 0.890

0.554 0.898

0.968 1.000

0 0

0 0

1.5

100 200

1.000 1.000

0.700 0.994

0.736 0.994

0.998 1.000

1.000 1.000

0.780 0.998

0.806 1.000

1.000 1.000

0 0

0 0

d

0.10 ep

ga

Failure times ep

ga

Tn : Proposed test; ep ga Tn,u and Tn,u : U-process tests with Epanechnikov and Gaussian kernel functions; Tn,e : Empirical process test with the simple indicator weighting function; ga ep F u : Failure times of tests Tn,u and Tn,u ; e F : Failure time of test Tn,e .

6

setting IV with 13-dimensional predictor, the U-process tests and the test with the SI weighting function often have empirical powers zero and the failure times are quite large. Clearly, these tests often lost effect in setting IV. But the proposed test still has satisfying empirical powers, which avoids the ‘‘curse of dimensionality’’. From the analysis of Tables 6.2–6.6, we can conclude that the proposed test has empirical sizes close to the test levels, its empirical powers are higher than those of the existing estimators except for the situation that the data are very concentrated; even if the predictors are moderate or high dimensional, it still performs well.

7

7. Real data analyses

1 2 3 4 5

8 9

In this section, we apply the proposed test and the existing tests in Stute et al. (2000) and Lopez and Patilea (2009) to analyze two real data sets. Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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Table 6.4 Empirical powers of different testing methods for Setting II under different censoring rates and sample sizes. % Cens.

20%

40%

α:

0.05

n

Tn

Tn,u

T n, u

T n, e

Tn

T n, u

Tn,u

T n, e

Fu

Fe

0.5

100 200

0.484 0.790

0.060 0.064

0.072 0.080

0.400 0.694

0.604 0.868

0.156 0.152

0.140 0.166

0.526 0.790

0 0

0 0

1.0

100 200

0.926 0.996

0.260 0.420

0.284 0.446

0.832 0.978

0.956 1.000

0.410 0.600

0.444 0.624

0.890 0.986

0 0

0 0

1.5

100 200

0.996 1.000

0.604 0.878

0.626 0.846

0.954 0.998

0.998 1.000

0.764 0.934

0.794 0.928

0.976 1.000

0 0

0 0

0.5

100 200

0.440 0.622

0.058 0.086

0.072 0.082

0.372 0.580

0.566 0.734

0.124 0.154

0.136 0.180

0.478 0.684

0 0

0 0

1.0

100 200

0.902 0.978

0.246 0.370

0.272 0.378

0.810 0.956

0.940 0.988

0.394 0.538

0.446 0.548

0.880 0.972

0 0

0 0

1.5

100 200

0.988 0.998

0.516 0.754

0.562 0.750

0.924 0.980

0.996 0.998

0.680 0.850

0.712 0.850

0.946 0.984

0 0

0 0

d

0.10 ep

ga

Failure times ep

ga

Tn : Proposed test; ep ga Tn,u and Tn,u : U-process tests with Epanechnikov and Gaussian kernel functions; Tn,e : Empirical process test with the simple indicator weighting function; ep ga F u : Failure times of tests Tn,u and Tn,u ; e F : Failure time of test Tn,e .

Table 6.5 Empirical powers of different testing methods for Setting III under different censoring rates and sample sizes. % Cens.

20%

40%

d

α:

0.05

0.10 ga T n, u

T n, e

Failure times

n

Tn

ep Tn,u

Tn

ep T n, u

ga Tn,u

T n, e

Fu

Fe

0.5

100 200

0.834 0.988

0.000 0.000

0.000 0.000

0.134 0.624

0.914 0.992

0.000 0.000

0.006 0.014

0.286 0.778

307 310

0 0

1.0

100 200

0.984 1.000

0.000 0.000

0.000 0.000

0.280 0.854

0.998 1.000

0.000 0.000

0.008 0.020

0.450 0.942

306 304

0 0

1.5

100 200

0.992 1.000

0.000 0.000

0.000 0.000

0.266 0.874

0.996 1.000

0.000 0.000

0.012 0.024

0.442 0.952

301 331

0 0

0.5

100 200

0.674 0.938

0.000 0.000

0.000 0.000

0.104 0.394

0.784 0.966

0.000 0.000

0.006 0.012

0.210 0.590

316 332

0 0

1.0

100 200

0.930 1.000

0.000 0.000

0.000 0.000

0.128 0.708

0.964 1.000

0.000 0.000

0.004 0.008

0.276 0.848

314 327

0 0

1.5

100 200

0.964 1.000

0.000 0.000

0.000 0.000

0.176 0.722

0.980 1.000

0.000 0.000

0.016 0.014

0.334 0.882

325 319

0 0

Tn : Proposed test; ep ga Tn,u and Tn,u : U-process tests with Epanechnikov and Gaussian kernel functions; Tn,e : Empirical process test with the simple indicator weighting function; ep ga F u : Failure times of tests Tn,u and Tn,u ; F e : Failure time of test Tn,e .

7.1. Analysis of PBC data The first data set comes from the Mayo Clinic trial on primary biliary cirrhosis (PBC) of the liver from 1974 to 1984. See Fleming and Harrington (2011). We only take 276 patients into account who met eligibility criteria for the randomized placebo controlled trial of the drug D-penicillamine. The data set is of competing risk types: death and liver transplantation. As suggested by one reviewer, we consider the combined endpoint rather than interpreting the competing endpoint as a censoring variable. So the survival time variable Y is the number of days between registration and the death or liver transplantation, which is censored due to the end of the study. And the censoring rate is 53.26%. The existing works, for example, Tibshirani (1997) and Zhang and Lu (2007), chosen the survival time variable in a similar way. Seventeen covariates are taken into consideration: X1 : Treatment Code, X2 : Age in years, X3 : Sex, X4 : Presence of ascites, X5 : Presence of hepatomegaly, X6 : Presence of spiders, X7 : Presence of edema, X8 : Serum bilirubin, X9 : Serum cholesterol, X10 : Albumin, X11 : Urine copper, X12 : Alkaline phosphatase, X13 : SGOT, X14 : Triglycerides, X15 : Platelet count, X16 : Prothrombin time, X17 : Histologic stage of disease. For X7 , we regroup the data: if edema is present, then X7 = 1; otherwise X7 = 0. For X17 , we recombine grades 1 and 2 as the low grade, and grades 3 and 4 as the high grade. Also let X17 = 1 for the high grade and X17 = 0, otherwise. Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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Table 6.6 Empirical powers of different testing methods for Setting IV under different censoring rates and sample sizes. % Cens.

20%

40%

α:

0.05

n

Tn

T n, u

Tn,u

Tn,e

Tn

Tn,u

T n, u

T n, e

Fu

Fe

0.5

100 200

0.546 0.882

0.000 0.000

0.000 0.000

0.000 0.000

0.710 0.940

0.000 0.000

0.000 0.000

0.002 0.000

500 500

220 28

1.0

100 200

0.872 0.994

0.000 0.000

0.000 0.000

0.002 0.000

0.938 0.996

0.000 0.000

0.000 0.000

0.010 0.008

500 500

191 26

1.5

100 200

0.908 1.000

0.000 0.000

0.000 0.000

0.002 0.002

0.970 1.000

0.000 0.000

0.000 0.000

0.008 0.016

499 500

218 27

0.5

100 200

0.458 0.792

0.000 0.000

0.000 0.000

0.000 0.000

0.612 0.870

0.000 0.000

0.000 0.000

0.004 0.000

500 500

219 26

1.0

100 200

0.708 0.954

0.000 0.000

0.000 0.000

0.000 0.002

0.826 0.992

0.000 0.000

0.000 0.000

0.006 0.004

500 500

201 22

1.5

100 200

0.788 0.984

0.000 0.000

0.000 0.000

0.000 0.002

0.888 0.992

0.000 0.000

0.000 0.000

0.000 0.004

500 500

210 23

d

0.10 ep

ga

Failure times ep

ga

Tn : Proposed test; ep ga Tn,u and Tn,u : U-process tests with Epanechnikov and Gaussian kernel functions; Tn,e : Empirical process test with the simple indicator weighting function; ep ga F u : Failure times of tests Tn,u and Tn,u ; e F : Failure time of test Tn,e .

Fig. 7.1. PBC data: Scatterplots of the response variable Y versus X βˆ n and the estimated studentized residuals eˆ n versus X βˆ n .

1 2 3 4 5 6 7 8 9 10

We aim for testing the null hypothesis H0 : E [Y |X ] = X τ β , where X = (X1 , X2 , . . . , X17 )τ , and β is an unknown regression parameter. We compute the P-value of the proposed test: Pn = 0.6450. We also compute the P-values of the U-process test and the empirical process test with the SI weighting function: Pu = 1 (degenerated) and Ps = 0.3320. For the U-process test, we choose Epanechnikov kernel function and the bandwidth 2.34std(X )n−1/3 . Therefore, the linear model hypothesis cannot be rejected. We plot the scatter plots of the response variable Y versus X βˆ n and the estimated studentized residuals eˆ n versus X βˆ n , which are shown in Fig. 7.1. From Fig. 7.1, we can find that the linear relationship between the response and predictors is reasonable. The proposed method provides higher P-value than the empirical process test with the SI weighting function does. The U-process test is invalid for this testing problem. Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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Fig. 7.2. Melanoma data: Scatterplots of the response variable Y versus X βˆ n and the estimated studentized residuals eˆ n versus X βˆ n .

7.2. Analysis of malignant melanoma data The second data set is a malignant melanoma data set. Patients experienced radical operations performed from 1962 to 1977 at Department of Plastic Surgery, University Hospital of Odense, Denmark. A detailed description of this data set can be found in Andersen et al. (2012). There are 205 measurements. Five attributes were recorded: X1 : Sex, X2 : Age, X3 : Year of Operation, X4 : Tumor Thickness, X5 : Ulceration. The survival time variable Y was uncensored if the patient was dead and censored if the patient was alive at the end of the study. The censoring rate is 65.37%. The methods to choose the kernel function and the bandwidth are the same as those in Section 7.1. We compute the P-values of the proposed test, the U-process test and the empirical process test with the SI weighting function: Pn = 0.0420, Pu = 0.2957 and Ps = 0.1460. The proposed method rejects the null hypothetical linear model. But two existing methods cannot reject the null hypothesis. We plot scatter plots of the response variable Y versus X βˆ n and the estimated studentized residuals eˆ n versus X βˆ n as shown in Fig. 7.2. From Fig. 7.2, we can observe that the linear model is indeed unreasonable. In this example, we can conclude that the proposed method is more reliable and sensitive than the other two existing tests. 8. Conclusion In this paper, we investigated the model checking problem of right-censored parametric models. By considering the projection-based empirical process, we proposed an integrated testing method. The asymptotic properties of the test statistic under the null hypothesis, local and global alternative hypotheses were studied. A wild bootstrap method was applied to determine the critical value. The numerical studies showed that the proposed tests outperform the existing testing methods in terms of empirical sizes and empirical powers. The numerical studies also validated that the proposed test has the high capacity to deal with the moderate-high dimensional data. The methods in Stute et al. (2000) and Lopez and Patilea (2009) tend to degenerate for moderate-high dimensional data. The proposed tests perform well for low or moderate-high dimensional data. In recent years, the research on semiparametric models has attracted a lot of attention. The checking of the adequacy of semiparametric models is a necessary procedure before further statistical analysis. However, the existing testing methods focus on the low dimensional data. Hence it is interesting to extend the proposed testing method to check semiparametric models with a moderate-high dimensional predictor. Furthermore, in reality, the data are often with complex structure. For example, the data are measured with error, longitudinal or missing. It would be desirable to develop testing methods with the dimensionality reduction effect to check the candidate models with complex data structures. Acknowledgments The authors would like to thank the Editor, the Associate Editor and the reviewers for their careful review and insightful comments that have led to significant improvement of this article. Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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2 3 4 5 6 7 8 9 10 11 12 13

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15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

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Appendix A. Notations We give some notations which appear in the text or are needed in the proofs: H(t) = P(Z ≤ t);

4 5

H0 (t) = P(Z > t , δ = 0);

6

7

ω(Zi , δi , t) = (1 − G(Zi −))[

Z i ∧t

∫

(1 − H(s))−2 dH0 (s) + (1 − H(t))−1 I(Z1 ≤ t , δ1 = 0)];

0 8

9 10 11 12 13 14 15

16

17 18 19 20 21 22 23

24 25 26

27 28 29 30 31 32 33 34

35

36

37

for any i, j = 1, 2, . . . , n, Hu,θ (Zi , δi , Xi ; Zj , δj , Xj ; u, θ )

= δi /(1 − G(Zi −))ηi I(Xiτ θ ≤ u) + δj /(1 − G(Zj −))ηj I(Xjτ θ ≤ u) + δi /(1 − G(Zi −))2 (Zi − g(Xi , β )I(Xiτ θ ≤ u)ω(Zj , δj , Zi )) + δj /(1 − G(Zj −))2 (Zj − g(Xj , β )I(Xjτ θ ≤ u)ω(Zi , δi , Zj )) ˙ , β )τ I(X τ θ ≤ u))A−1 {δi /(1 − G(Zi −))g˙β (Xi , β )ηi + δi /(1 − G(Zi −))2 ηi g˙β (Xi , β ) + E(g(X × E(ω(Zi , δi , Zj )|Zi , δi ) + δj /(1 − G(Zj −))g˙β (Xj , β )ηj + δj /(1 − G(Zj −))2 ηj g˙β (Xj , β ) × E(ω(Zj , δj , Zi )|Zj , δj )}.

Appendix B. Proofs of the main theorems We begin this subsection by listing the conditions needed in the proofs of the theorems. (C1) The function g(x, β ) satisfies Lipschitz condition of order 2 with respect to β ; (C2) supx E(Y 2 |X = x) < ∞; ˙ , β )g(X ˙ , β )τ ] is a positive definite matrix; (C3) A = E [g(X (C4) Let τL = inf{t : L(t) = 1}. The distribution functions of Y and C , F (·) and G(·), satisfy 0 < τF < τG < ∞; Y and C are independent; Pr(δ = 0|X , Y ) = Pr(δ = 0|Y ); (C5) The density function of θ , f (·), is a continuous function with compact support. Remark 1. Conditions (C1)–(C3) guarantee the consistency of the estimation of the null hypothetical model. Condition (C4) is a common assumption for the random censorship model. Condition (C5) is needed to ensure the consistency of the proposed test. Remark 2. The independence condition in (C4) contains two parts: (i) The variables Y and C are independent; (ii) The variables Y and C are conditionally independent given the covariate X . The former works for the Kaplan–Meier estimator and δ the latter aims for the following properties: for any integrable function L(T , X ), the equation E [ 1−G(Z L(T , X )] = E [L(T , X )] −) holds. Both conditions are necessary for the construction of the objective function (2.2). Note that the condition Pr(δ = 0|X , Y ) = Pr(δ = 0|Y ) is the same as the Assumption (II) in Stute et al. (2000) and Lopez and Patilea (2009). Therefore Lopez and Patilea (2009), Stute et al. (2000) and our method assume the same independence condition: both independence and conditional independence. However, Pardo-Fernández et al. (2007) only assumed the independence of Y and C given the covariates. We first list two lemmas needed for the proofs of the main theorems. Lemma 1. Under Conditions (C1)–(C4), we have

ˆ − G(t) = n−1 G(t)

n ∑

ω(Zi , δi , t) + Op (n−3/4 log3/4 n)

i=1 38

with ω(Zi , δi , t) shown in Appendix A.

39

Proof. See References Chen et al. (2003) and Lo and Singh (1986). □

40

Lemma 2. Suppose that Conditions (C1)–(C4) hold. Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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(i) Under the alternative hypothetical model (4.1) with dn = n−1/2 , we have n1/2 (βˆ n − β ) = A−1 {E(g˙β (X , β )D(X )) + n−1/2

n ∑

1

δi /[1 − G(Zi −)]g˙β (Xi , β )ηi

2

δi /(1 − G(Zi −))2 ηi g˙β (Xi , β )E [ω(Zi , δi , Z1 )|Zi , δi ]}

3

i=1

+ n−1/2

n ∑ i=1

+ op (1)

(B.1)

where A and ω(Zi , δi , t) are shown in Appendix A.

5

(ii) Under the alternative hypothetical model (4.1) with 0 ≤ dn < n−1/2 , then n1/2 (βˆ n − β ) = Op (n1/2 dn ).

6

(B.2)

Proof. (i) For βˆ n , based on the objection function (2.2), we obtain 1 −1/2 n1/2 (βˆ n − β ) =: A− n n

n ∑

4

7

8

ˆ i −))g˙β (Xi , β )(Zi − g(Xi , β )) δi /(1 − G(Z

9

i=1 1 −1/2 + A− n n

n ∑

ˆ i −))(g˙β (Xi , βˆ n ) − g˙β (Xi , β )) δi /(1 − G(Z

10

i=1

× (Zi − g(Xi , β )) + op (1) 1 −1 =: A− n Jn + An Mn + op (1) with An = n

11

(B.3)

ˆ i −)]g˙β (Xi , βˆ n )g˙β (Xi , βˆ n )τ . It is easy to prove that δ /[1 − G(Z

∑n −1

13

i=1 i

An = A + op (1).

(B.4)

For Jn , under the alternative hypothetical models (4.1) with dn = n−1/2 , we have Jn = n−1/2

n ∑

12

14

15

δi /[1 − G(Zi −)]g˙β (Xi , β )(n−1/2 D(Xi ) + ηi )

16

i=1

+ n−1/2

n ∑

ˆ i −) − G(Ti −))/(1 − G(Z ˆ i −))(1 − G(Ti −))g˙β (Xi , β ) δi (G(T

17

i=1

× (n−1/2 D(Xi ) + ηi ) = E(g˙β (X , β )D(X )) + n−1/2

18

n ∑

δi /[1 − G(Zi −)]g˙β (Xi , β )ηi

19

i=1

+ n−1

+n

n ∑

ˆ i −) − G(Ti −))/(1 − G(Ti −))2 g˙β (Xi , β )D(Xi ) δi (G(T

i=1 n −1/2

∑

20

ˆ i −) − G(Zi −))/(1 − G(Zi −))2 g˙β (Xi , β )ηi δi (G(Z

21

i=1

+ op (1).

(B.5)

Let

22

23

n

Jn1 = n−1

∑

ˆ i −) − G(Ti −))/(1 − G(Ti −))2 g˙β (Xi , β )D(Xi ), δi (G(T

24

i=1

and

25

Jn2 = n−1/2

n ∑

ˆ i −) − G(Zi −))/(1 − G(Zi −))2 g˙β (Xi , β )ηi . δi (G(Z

i=1

Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.

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ˆ − G(t)| = Op (n−1/2 log log(n)). See Földes and Rejtő (1981). Then the strong law of large numbers Note that sup0≤t ≤τ |G(t) implies that ˆ − G(t)| ∗ n−1 |Jn1 | ≤ sup |G(t)

3

0≤t ≤τ

Therefore, we have Jn1 = op (1).

6 7

i=1

= Op (n−1/2 log log(n)).

4 5

n ∑ |δi /(1 − G(Ti −))2 g˙β (Xi , β )D(Xi )|

(B.6)

Further, it follows from Lemma 1 that Jn2 = n−3/2

8

n ∑

δi /(1 − G(Zi −))2 g˙β (Xi , β )ηi

i=1

= n−1/2

9

n ∑

ω(Zj , δj , Zi ) + op (1)

j=1

n ∑

δi /(1 − G(Zi −))2 g˙β (Xi , β )ηi E [ω(Zj , δj , Zi )|Zi ]

i=1

+ n−3/2

10

n ∑

δi /(1 − G(Zi −))2 g˙β (Xi , β )ηi

(ω(Zj , δj , Zi ) − E [ω(Zj , δj , Zi )|Zi ])

j=1

i=1

+ op (1) = Jn2,1 + Jn2,2 + op (1).

11 12 13

n ∑

(B.7)

Note that E [ω(Zj , δj , Zi ) − E(ω(Zj , δj , Zi )|Zi )] = 0. Then it can be checked that Jn2,2 = n−3/2

14

n n ∑ ∑

δi /(1 − G(Zi −))2 g˙β (Xi , β )ηi (ω(Zj , δj , Zi ) − E [ω(Zj , δj , Zi )|Zi ])

i=1 j=1,j̸ =i

+ n−3/2

15

n ∑

δi /(1 − G(Zi −))2 g˙β (Xi , β )ηi (ω(Zi , δi , Zi ) − E [ω(Zi , δi , Zi )|Zi ])

i=1

[1 ] [1 ] = Jn2 ,2 + Jn2,2 .

16

17 18 19

20 21

22

Note that for any i ̸ = j, i, j = 1, 2, . . . , n, δi /(1 − G(Zi −))2 g˙β (Xi , β )ηi ω(Zj , δj , Zi )− E [ω(Zj , δj , Zi )|Zi ] has means 0 and is [1] independent from each other. Then we can prove that the second order moment of Jn2,2 converges to zero. Hence we have [1 ]

[2]

Jn2,2 = op (1). Furthermore, it is easy to see that Jn2,2 = op (1). Thus, we get Jn2,2 = op (1).

(B.8)

By (B.7) and (B.8), it follows that Jn2 = n−1/2

n ∑

δi /(1 − G(Zi −))2 ηi E [ω(Z , δ, Zi )|Zi ] + op (1).

i=1 23

24

This, together with (B.5) and (B.6), can yield Jn = E(g˙β (X , β )D(X )) + n−1/2

n ∑

δi /[1 − G(Zi −)]g˙β (Xi , β )ηi

i=1

25

+ n−1/2

26

+ op (1).

n ∑

δi /(1 − G(Zi −))2 g˙β (Xi , β )ηi E [ω(Z , δ, Zi )|Zi ]

i=1

27 28 29 30

(B.9)

Using Conditions (C1)–(C2) and Lemma 1, we can show that Mn = op (1). Then by (B.3)–(B.4), (B.9), Condition (C3) and the central limit theorem, Part (i) of Lemma 2 is proved. (ii) Under the alternative hypothetical model (4.1) with 0 ≤ dn < n−1/2 , we rewrite (4.1) as follows: H1n : Y = g(X , β ) + n−1/2 (n1/2 dn )D(X ) + η =: g(X , β ) + n−1/2 Υn + η

31

with Υn = n1/2 dn D(X ). By the similar method to prove Part (i) of Lemma 2, we can prove Part (ii) of Lemma 2. □

32

Proof of Theorem 1. By letting D(X ) = 0, we can prove Theorem 1 from the proof of Theorem 2 listed in the following.

□

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13

Proof of Theorem 2. We only prove the result with dn = n−1/2 . The other results can be proved similarly. When δi = 1, we

1

have Zi = Yi , for i = 1, 2, . . . , n. Recalling (3.1), by some simple computations, we can get

2

Rn (u, θ ) = n−1/2

n ∑

δi /(1 − G(Zi −))(Yi − g(Xi , β )I(Xiτ θ ≤ u))

3

i=1

+ n−1/2

n ∑

ˆ i −) − G(Zi −))/(1 − G(Zi −))2 (Yi − g(Xi , β )I(Xiτ θ ≤ u)) δi (G(Z

4

i=1 n

−n−1/2

∑

˙ i , β )τ (βˆ n − β )I(Xiτ θ ≤ u) δi /(1 − G(Zi −))g(X

5

ˆ i −) − G(Zi −))/(1 − G(Zi −))2 g(X ˙ i , β )τ (βˆ n − β )I(Xiτ θ ≤ u) δi (G(Z

6

i=1 n

−n−1/2

∑ i=1

+ op (1)

7

2

∑

4

Rn[j] (u, θ ) −

∑

Rn[j] (u, θ ) + op (1).

(B.10)

8

For Rn (u, θ ), note that Yi − g(Xi , β ) = n−1/2 D(Xi ) + ηi under the alternative hypothetical model (4.1) with dn = n−1/2 . Then

9

=:

j=1

j=3

[1]

we have

10

n

R[n1] (u, θ ) = n−1/2

∑

δi /(1 − G(Zi −))(n−1/2 D(Xi ) + ηi )I(Xiτ θ ≤ u)

11

i=1

= E [D(X )I(X τ θ ≤ u)] + n−3/2

∑

{δi /(1 − G(Zi −))ηi I(Xiτ θ ≤ u)

12

j
+ δj /(1 − G(Zj −))ηj I(Xjτ θ ≤ u)} + op (1).

(B.11) [2 ]

The second term on the right-hand side of (B.11) is a U-statistic process. For Rn (u, θ ), by Lemma 1, we can obtain R[n2] (u, θ ) = n−1/2

n ∑

13

14

ˆ i −) − G(Zi −))/(1 − G(Zi −))2 (Zi − g(Xi , β )I(Xiτ θ ≤ u)) δi (G(Z

15

i=1

= n−3/2

∑ {δi /(1 − G(Zi −))2 (Zi − g(Xi , β )I(Xiτ θ ≤ u)ω(Zj , δj , Zi ))

16

i
+ δj /(1 − G(Zj −))2 (Zj − g(Xj , β )I(Xjτ θ ≤ u)ω(Zi , δi , Zj ))} + op (1).

(B.12)

[3]

For Rn (u, θ ), by the strong law of large numbers and Lemma 1, it follows that

17

18

√

˙ , β )τ I(X τ θ ≤ u)] n(βˆ n − β ) + op (1) R[n3] (u, θ ) = E [g(X

19

˙ , β )τ I(X τ θ ≤ u)]A−1 E(g˙β (X , β )D(X )) + E [g(X ˙ , β )τ I(X τ θ ≤ u)]A−1 = E [g(X n ∑ × n−1/2 {δi /[1 − G(Zi −)]g˙β (Xi , β )ηi + δi /(1 − G(Zi −))2 ηi g˙β (Xi , β )

20

21

i=1

× E [ω(Zi , δi , Z1 )|Zi , δi ]} + op (1). ˙ , β )τ I(X τ θ ≤ u)]A−1 E(g˙β (X , β )D(X )) + E [g(X ˙ , β )τ I(X τ θ ≤ u)]A−1 = E [g(X ∑ × n−3/2 {δi /[1 − G(Zi −)]g˙β (Xi , β )ηi + δi /(1 − G(Zi −))2 ηi g˙β (Xi , β )

22 23 24

j
× E [ω(Zi , δi , Zj )|Zi , δi ] + δj /[1 − G(Zj −)]g˙β (Xj , β )ηj + δj /(1 − G(Zj −))2 × ηj g˙β (Xj , β )E [ω(Zj , δj , Zi )|Zj , δj ]} + op (1).

25

(B.13)

ˆ − G(t)| = Op (n−1/2 log log(n)) and Lemma 3, we can prove that By the fact sup0≤t ≤τ |G(t) R[n4] (u, θ ) = op (1).

26

27

(B.14)

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28

COMSTA: 6513

14

1

2

Z. Sun et al. / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

Thus, it follows from (B.10)–(B.14) that

˙ , β )τ I(X τ θ ≤ u)]A−1 E(g˙β (X , β )D(X )) Rn (u, θ ) = E [D(X )I(X τ θ ≤ u)] − E [g(X + n−3/2

3

∑

Hu,θ (Zi , δi , Xi ; Zj , δj , Xj ; u, θ ) + op (1),

(B.15)

j
7

where Hu,θ (Zi , δi , Xi ; Zj , δj , Xj ; u, θ ) is given in Appendix A. Because the indicator function I(X τ θ ≤ u) is monotone with respect to u, it is easy to prove that Gu = {Hu,θ (Zi , δi , Xi ; Zj , δj , Xj ; u, θ ) : u ∈ R} is a V–C class of functions, see Nolan and Pollard (1988). By Theorem 3.1 of Arcones and Yu (1994), we can show that Rn (u, θ ) converges to a Gaussian process with the covariance showed in (4.2). Further by the continuous mapping theorem, we can prove the result for Tn . □

8

References

4 5 6

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 42 41

Andersen, P.K., Borgan, O., Gill, R.D., Keiding, N., 2012. Statistical Models Based on Counting Processes. Springer Science & Business Media. Arcones, M.A., Yu, B., 1994. Central limit theorems for empirical and U-processes of stationary mixing sequences. J. Theoret. Probab. 7, 47–71. Bierens, H.J., 1982. Consistent model specification tests. J. Econometrics 20, 105–134. Bierens, H.J., Ploberger, W., 1997. Asymptotic theory of integrated conditional moment tests. Econometrica 65, 1129–1151. Chen, M., Yuen, K., Zhu, L., 2003. Asymptotics of the goodness-of-fit test for a partial linear model with randomly censored data. Sci. China Ser. A 46, 145–158. Dette, H., 1999. A consistent test for the functional form of a regression based on a difference of variance estimators. Ann. Statist. 27, 1012–1040. Escanciano, J., 2006. A consistent diagnostic test for regression models using projections. Econometric Theory 22, 1030–1051. Eubank, R.L., Hart, J.D., 1992. Testing goodness-of-fit in regression via order selection criteria. Ann. Statist. 20, 1412–1425. Fleming, T.R., Harrington, D.P., 2011. Counting Processes and Survival Analysis, Vol. 169. John Wiley & Sons. Földes, A., Rejtő, L., 1981. A LIL type result for the product limit estimator. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 56, 75–86. Härdle, W., Mammen, E., 1993. Testing parametric versus nonparametric regression. Ann. Statist. 21, 1926–1947. Härdle, W., Mammen, E., Müller, M., 1998. Testing parametric versus semiparametric modeling in generalized linear models. J. Amer. Statist. Assoc. 93, 1461–1474. Horowitz, J.L., Härdle, W., 1994. Testing a parametric model against a semiparametric alternative. Econometric Theory 10, 821–848. Koul, H., Susarla, V., Van Ryzin, J., 1981. Regression analysis with randomly right-censored data. Ann. Statist. 9, 1276–1288. Li, Q., Racine, J.S., 2007. Nonparametric Econometrics: Theory and Practice. Princeton University Press. Lo, S.H., Singh, K., 1986. The product-limit estimator and the bootstrap: some asymptotic representations. Probab. Theory Related Fields 71, 455–465. Lopez, O., Patilea, V., 2009. Nonparametric lack-of-fit tests for parametric mean-regression models with censored data. J. Multivariate Anal. 100, 210–230. Ma, S., Zhang, J., Sun, Z., Liang, H., 2014. Integrated conditional moment test for partially linear single index models incorporating dimension-reduction. Electron. J. Stat. 8, 523–542. Nolan, D., Pollard, D., 1988. Functional limit theorems for U-processes. Ann. Probab. 1291–1298. Pardo-Fernández, J.C., Van Keilegom, I., González-Manteiga, W., 2007. Goodness-of-fit tests for parametric models in censored regression. Canad. J. Statist. 35, 249–264. Stute, W., González Manteiga, W., Presedo Quindimil, M., 1998. Bootstrap approximations in model checks for regression. J. Amer. Statist. Assoc. 93, 141–149. Stute, W., González Manteiga, W., Sánchez Sellero, C., 2000. Nonparametric model checks in censored regression. Comm. Statist. Theory Methods 29, 1611–1629. Stute, W., Zhu, L.-X., 2002. Model checks for generalized linear models. Scand. J. Stat. 29, 535–545. Sun, Z., Wang, Q., 2009. Checking the adequacy of a general linear model with responses missing at random. J. Statist. Plann. Inference 139, 3588–3604. Tibshirani, R., 1997. The lasso method for variable selection in the cox model. Stat. Med. 16, 385–395. Wu, C.-F.J., 1986. Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Statist. 1261–1295. Xia, Y., Li, W., Tong, H., Zhang, D., 2004. A goodness-of-fit test for single-index models. Statist. Sinica 14, 1–28. Zhang, H.H., Lu, W., 2007. Adaptive lasso for cox’s proportional hazards model. Biometrika 94, 691–703. Zheng, J.X., 1996. A consistent test of functional form via nonparametric estimation techniques. J. Econometrics 75, 263–289.

Please cite this article in press as: Sun Z., et al., Consistent test for parametric models with right-censored data using projections. Computational Statistics and Data Analysis (2017), https://doi.org/10.1016/j.csda.2017.09.005.