Smoothed empirical likelihood inference for the difference of two quantiles with right censoring

Journal of Statistical Planning and Inference 146 (2014) 95–101

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Smoothed empirical likelihood inference for the difference of two quantiles with right censoring Hanfang Yang a, Crystal Yau b, Yichuan Zhao c,n a b c

School of Statistics, Renmin University of China, Beijing, China Transmission Financial and Accounting Controls, Georgia Power, Atlanta, GA 30308, United States Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, United States

a r t i c l e i n f o

abstract

Article history: Received 9 March 2012 Received in revised form 26 June 2013 Accepted 20 September 2013 Available online 14 October 2013

In this paper, using a smoothed empirical likelihood method, we investigate the difference of quantiles in the two independent samples and construct the confidence intervals. We prove that the limiting distribution of the empirical log-likelihood ratio is a chi-squared distribution like Shen and He (2007). In the simulation studies, in terms of coverage accuracy and average length of confidence intervals, we compare the empirical likelihood and the normal approximation methods with the optimal bandwidth selected by crossvalidation. The empirical likelihood method has a better performance most of the time. Finally, a real clinical trial data is used to illustrate how to generate empirical likelihood confidence bands using bootstrap method. & 2013 Elsevier B.V. All rights reserved.

Keywords: Censored data Difference of two quantiles Smoothed empirical likelihood Confidence interval

1. Introduction In probability theory, a useful quantity related to cumulative distribution function F is the quantile function. Assuming a continuous and strictly monotonic distribution function, F : R-ð0; 1Þ, there is an unique number xp for each p such that Fðxp Þ ¼ PðX rxp Þ ¼ p. xp is a number on the set of values of the random variable X, such that the fraction p of the total probability is assigned to the interval ð  1; xp Þ. xp is called a pth quantile of the distribution. In general, the pth quantile of F is defined by F  1 ðpÞ ¼ inffx A R : p rFðxÞg. The analysis of F  1 ðpÞ has played one of the major roles in statistics, economics, finance, risk management, etc. Empirical likelihood (EL) is a nonparametric method of statistical inference based on a data-driven likelihood ratio function. The empirical likelihood method is one of the most famous methodologies for nonparametric inference. The deployment of the EL method with respect to survival analysis can be traced back to Thomas and Grunkemeier (1975). The EL method was completely summarized in Owen (1988, 1990). In addition, EL confidence bands for quantile function and survival function have been derived in Li et al. (1996) and in Hollander et al. (1997), respectively. For complete data, Chen and Hall (1993) developed the smoothed EL inference for quantiles. Jing (1995), Qin and Zhao (1997) and Qin (1997) respectively proposed methodologies for the confidence intervals of the difference between two sample means and quantiles. For censored data, McKeague and Zhao (2005) proposed EL simultaneous confidence band for the difference of distribution functions in two-sample case. Zhao and Zhao (2011) developed EL confidence intervals for the contrast of two hazard functions motivated by Shen and He (2008). Moreover, Zhou and Jing (2003) proposed the smoothed EL method for the difference of quantiles for one sample complete data. Then Shen and He (2007)

n

Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Zhao).

0378-3758/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jspi.2013.09.010

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proposed a smoothed EL method on the same setting for one sample with right censoring. In this paper, we investigate the interval estimate of the quantile difference of two samples θ0 ¼ F 1 1 ðpÞ F 2 1 ðpÞ. Adopting the method proposed by Shen and He (2007), we develop the EL confidence interval for the difference of quantiles θ0 with two samples. The paper is organized as follows. In Section 2, the empirical likelihood procedure for the θ0 is developed. In Section 3, simulation studies are presented for comparing the performance between the smoothed empirical likelihood confidence intervals and the traditional normal approximation confidence intervals. It is then followed by an analysis on a real data application in Section 4. Section 5 is a summary of conclusions and discussion. The proofs are contained in the Appendix.

2. Main results Throughout this paper, we are using the standard two samples with right censored data. We adopt the same notations as Shen and He (2007) and Yang and Zhao (2012) did for simplicity. First of all, let T ji Z0, i ¼ 1; …; nj be i.i.d. failure times with continuous distribution Fj, j¼1,2. Let Cji, i ¼ 1; …; nj be i.i.d. censoring times with continuous distribution function Gj, j¼1,2, and independent of Tji. We observe each sample in the form of ðX ji ; δji Þ, where X ji ¼ minðT ji ; C ji Þ and δji ¼ IðT ji r C ji Þ. Like Shen and He (2007), we denote X ðjiÞ as the order statistics of the jth sample. The concomitant of δji is denoted as δðjiÞ and the number of risk sets is rji. Let η0 ¼ F 2 1 ðpÞ. Since F is strictly monotone increase function and θ0 ¼ θðpÞ ¼ F 1 1 ðpÞ  F 2 1 ðpÞ, we have F 1 1 ðpÞ ¼ η0 þθ0 ; F 2 1 ðpÞ ¼ η0 : The likelihood ratio is defined as Shen and He (2007) did similarly, Rðθ0 ; η; pÞ ¼

supφji A Φ fLðF 1 ; F 2 Þ : F 1 ðη þ θ0 Þ ¼ p; F 2 ðηÞ ¼ pg supφji A Φ LðF 1 ; F 2 Þ;

where φj1 ; φj2 ; …; φjnj are the hazard values at X ðj1Þ ; X ðj2Þ ; …; X ðjnj Þ and φj ¼ ðφj1 ; φj2 ; …; φjnj Þ A Φ, the space of hazard values. It is ~ 0 ; η; pÞ is proposed in this case. Let K(t) be a smooth computationally difficult to maximize Rðθ0 ; η; pÞ and a smoothed Rðθ distribution function, and hj be a bandwidth. Denote K hj ðtÞ ¼ Kðt=hj Þ. Like Shen and He (2007) and Yang and Zhao (2012), we define ( ) Φ2 ¼

n1

n2

i¼1

i¼1

φ1 ; φ2 A Φ : ∑ K h1 ðη þ θ0  X ð1iÞ Þlnð1  φ1i Þ ¼ lnð1 pÞ; ∑ K h2 ðη  X ð2iÞ Þlnð1  φ2i Þ ¼ lnð1 pÞ :

~ 0 ; η; pÞ is given like Shen and He (2007) and Yang and Zhao (2012). Define And hence the smoothed EL ratio Rðθ ^ ~ Rðθ0 ; pÞ ¼ supη Rðθ0 ; η; pÞ. Denote   n1   δð1iÞ  lnð1 pÞ ¼ 0; ð2:1Þ q1n1 ðη; λ1 Þ ¼ ∑ K h1 η þ θ0 X ð1iÞ ln 1  r 1i þ λ1 K h1 ðη þ θ0  X ð1iÞ Þ i¼1  n2   q2n2 ðη; λ2 Þ ¼ ∑ K h2 η  X ð2iÞ ln 1  i¼1

 δð2iÞ  lnð1  pÞ ¼ 0; r 2i þ λ2 K h2 ðη X ð2iÞ Þ

 δð1iÞ r 1i þ λ1 K h1 ðη þθ0  X ð1iÞ Þ i¼1   n2   δð2iÞ ¼ 0; þ ∑ λ2 K ′h2 η  X ð2iÞ ln 1  r 2i þ λ2 K h2 ðη X ð2iÞ Þ i¼1

ð2:2Þ

 n1   q3n1 n2 ðη; λ1 ; λ2 Þ ¼ ∑ λ1 K ′h1 η þθ0  X ð1iÞ ln 1 

    n1   λ1 K h1 ðη þ θ0  X ð1iÞ Þ λ1 K h1 ðη þ θ0  X ð1iÞ Þ ln R^ ðθ0 ; pÞ ¼ ∑ r 1i  δð1iÞ ln 1 þ  r 1i ln 1 þ r 1i δð1iÞ r 1i i¼1     n2   λ2 K h2 ðη X ð2iÞ Þ λ2 K h2 ðη  X ð1iÞ Þ  r 2i ln 1 þ ; þ ∑ r 2i  δð2iÞ ln 1 þ r2i  δð2iÞ r 2i i¼1

ð2:3Þ

ð2:4Þ

where η; λ1 and λ2 satisfy Eqs. (2.1)–(2.3). Now we have the following theorems. Theorem 2.1 (Yang and Zhao, 2012). Assume regularity conditions (C.1)–(C.5) in the Appendix hold. Then, the maximum of LðF 1 ; F 2 Þ with the constraint condition Φ2 for large n ¼ n1 þ n2 is achieved at a unique φ1 ; φ2 , a.s. Theorem 2.2. Assume regularity conditions (C.1–C.5) in the Appendix hold. For fixed p, it exists a solution ηE , i.e., ^ 0 ; pÞ. We have ~ 0 ; η ; pÞ ¼ sup Rðθ ~ 0 ; η; pÞ ¼ Rðθ Rðθ E η D 2 ~ 0 ; η ; pÞ⟶ 2 ln Rðθ χ1: E

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97

Thus, the asymptotic 100ð1 αÞ% smoothed EL confidence interval for θ0 at the fixed p is ^ pÞ r χ 2 ðαÞg; Rα ¼ fθ : 2 ln Rðθ; 1

ð2:5Þ

where χ 21 ðαÞ is the upper αquantile of χ 21 . 3. Simulation study In this section, we compare the performance of empirical likelihood (EL) and the normal approximation (NA) methods by Monte Carlo simulation in terms of coverage accuracy and average length of confidence intervals. The normal approximation (NA) method is commonly employed to construct the confidence interval. Let   δðjiÞ : ð3:1Þ F jnj ðxÞ ¼ 1  ∏ 1  r ji X ðjiÞ r x 1 1 Define θ^ ¼ F 1n ðxÞ  F 2n ðxÞ. From Shen and He (2007), Andersen et al. (1993) and Corollary 1 of Veraverbeke (2001), we 1 2 D

know ðn1 þn2 Þ1=2 ðθ^  θ0 Þ⟶Nð0; s2 Þ, where s2 and its consistent estimator s^ 2 are defined as equations (3.2) and (3.4) in Shen and He (2007) with a slight adjustment. Therefore, the asymptotic 100ð1  αÞ% confidence interval for θ0 based on the normal approximation (NA) method is s^ θ^ 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqð1 αÞ; n1 þn2 where qð1 αÞ is the ð1  α=2Þ quantile of the standard normal distribution Nð0; 1Þ. We use the Epanechnikov kernel (   3 1  u2 if jujr 1 wðuÞ ¼ 4 0 otherwise  1=3

. and the smoothing bandwidth is chosen to be hj ¼ cj n nj In the simulation study, we adopt the same setting in Shen and He (2007) and Yang and Zhao (2012). The distribution of the failure time for sample j, j ¼1,2 is denoted as Fj, which is distributed as exponential. While the distribution of censoring time for sample j is denoted as Gj. For each method, we generate 1000 random samples of size n1 ðn1 ¼ 30; 50; 100Þ, where n1 ¼ n2 . Throughout the study, we let the distribution of censoring time be exponential distribution with mean θj , which will

Table 1 Comparison of coverage probabilities for the 95% EL and NA confidence intervals for the difference of two quantiles θ0 ¼ F 1 1 ðpÞ  F 2 1 ðpÞ with 10% and 30% censoring rates. pth

Censoring

Method

0.4

10%

EL

NA

0.4

30%

EL

NA

0.5

10%

EL

NA

0.5

30%

EL

NA

n

Data 1

Data 2

Data 3

30 50 100 30 50 100

0.950 0.956 0.934 0.959 0.952 0.938

0.938 0.949 0.940 0.958 0.956 0.945

0.942 0.939 0.941 0.933 0.929 0.953

30 50 100 30 50 100

0.953 0.954 0.944 0.959 0.949 0.951

0.942 0.957 0.955 0.954 0.952 0.957

0.920 0.939 0.955 0.947 0.949 0.964

30 50 100 30 50 100

0.943 0.945 0.955 0.909 0.924 0.936

0.948 0.954 0.954 0.924 0.927 0.925

0.942 0.947 0.965 0.887 0.913 0.940

30 50 100 30 50 100

0.952 0.951 0.965 0.933 0.940 0.960

0.954 0.944 0.953 0.926 0.932 0.951

0.941 0.952 0.945 0.916 0.934 0.936

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Table 2 Comparison of average lengths for the 95% EL and NA confidence intervals for the difference of two quantiles θ0 ¼ F 1 1 ðpÞ  F 2 1 ðpÞ with 10% and 30% censoring rates. pth

Censoring

Method

0.4

10%

EL

NA

0.4

30%

EL

NA

0.5

10%

EL

NA

0.5

30%

EL

NA

n

Data 1

Data 2

Data 3

30 50 100 30 50 100

2.6417 2.1816 1.6445 3.4246 2.5238 1.7486

2.1261 1.7190 1.2333 2.4519 1.8518 1.2752

1.5360 1.3552 1.0800 1.9603 1.5651 1.1259

30 50 100 30 50 100

2.5039 2.0829 1.5534 3.0693 2.2996 1.6257

1.9934 1.6022 1.1511 2.2564 1.6984 1.2010

1.4624 1.3001 1.0246 1.8909 1.4554 1.0582

30 50 100 30 50 100

2.9454 2.5782 2.0780 3.7062 2.9990 2.2024

2.3607 2.0380 1.5865 2.7896 2.1386 1.5576

1.5312 1.4246 1.1802 2.4584 1.9045 1.4102

30 50 100 30 50 100

2.7910 2.4594 1.9458 3.5023 2.7620 2.0089

2.3249 1.9131 1.4653 2.5102 2.0065 1.4514

1.5387 1.3613 1.1285 2.2349 1.7648 1.2695

be adjusted to stabilize the censoring rates. Moreover, we choose the nominal level 1  α ¼ 0:95, and let p ¼0.4 and p ¼0.5 for the pth quantile. The following are the detailed descriptions of the three simulation settings: (1) F1 and F2 are exponential distributed with mean 1 and 3; (2) F1 and F2 are exponential distributed with mean 2 and 4; (3) F1 and F2 are exponential distributed with mean 3 and 5. The censoring time is exponential distributed and those mean parameters are selected automatically to simulate data with 10% or 30% censoring rate for three simulations. A bandwidth selection is one of the critical issues in using smoothing technique. In this paper, we employ a crossvalidation method to determine the optimal bandwidth as Faraway and Jhun (1990), Chen et al. (2009) and Xue (2009) did in the simulation part. Because of the censoring, we employ the kernel quantile estimator under right censoring, the equation (3.2) in Padgett (1986), rather than the standard kernel quantile estimator. As Chen et al. (2009) illustrated the cross-validation procedure, we minimize the mean square error of kernel smoothing estimation and select the optimal cj of  1=3 . hj ¼ cj n nj The simulation results for empirical likelihood have a good performance (Tables 1 and 2). Regardless of the sample size and censoring rate, its coverage probability is relatively close to the nominal level, and its average length becomes narrower when the sample size increases. The simulation study shows that the empirical likelihood average lengths of 95% confidence intervals are shorter than the corresponding normal approximation average lengths in the most cases. Our simulation result indicates that the empirical likelihood method has a better performance than normal approximation method most of the time. 4. Real application In this section we illustrate our approach using a clinical trial data. The data is obtained from a Mayo Clinic trial about a treatment for primary biliary cirrhosis (PBC) of the liver for a 10-year interval, from 1974 to 1984. In that 10-year trial, total of PBC patients were chosen to participate in the randomized placebo controlled trial for the drug D-penicillamine. For detailed description, see Fleming and Harrington (1991), McKeague and Zhao (2002, 2005, 2006). In this paper, only the 312 randomized participating patients are examined, 158 patients received the D-penicillamine treatment, and the rest 154 patients received a placebo. Theorem 2.2 has a limitation to construct the simultaneous confidence band. We propose a simple procedure to obtain confidence band for fθðpÞ; τ1 rp rτ2 g. Claeskens et al. (2003), Hall and Owen (1993) and Yang and Zhao (2012), etc. used the bootstrap method to establish empirical likelihood confidence band. As they mentioned, one can generate the bootstrap

H. Yang et al. / Journal of Statistical Planning and Inference 146 (2014) 95–101

99

Confidence bands of the difference of two quantiles with PBC 2000

1500

EL confidence band NA confidence band Estimator of the difference of two quantiles Horizontal axes

1000

500

0

−500

−1000

−1500 0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Fig. 1. 95% EL and NA confidence bands for the difference of quantiles with Mayo Clinic trial data.

data fX nji g, then have r nji and δnðjiÞ , j ¼ 1; 2 and i ¼ 1; …; nj . We can rewrite Eqs. (2.1)–(2.4) from those re-sampling data as follows: ! !

n1 λn1 K h1 ðηn þ θ0 X nð1iÞ Þ λn1 K h1 ðηn þθ0  X nð1iÞ Þ n r ln R^n ðθ0 ; pÞ ¼ ∑ r n1i δnð1iÞ ln 1 þ ln 1 þ 1i r n1i  δnð1iÞ r n1i i¼1 ! ! n

n2 λn2 K h2 ðηn  X ð2iÞ Þ λn2 K h2 ðηn X nð1iÞ Þ n  r ; ð4:1Þ ln 1 þ þ ∑ r n2i δnð2iÞ ln 1 þ 2i rn2i  δnð2iÞ rn2i i¼1 where ðλn1 ; λn2 ; ηn Þ is the solution to the following equations: qn1n1 ðηn ; λn1 Þ ¼ 0;

ð4:2Þ

qn2n2 ðηn ; λn2 Þ ¼ 0;

ð4:3Þ

qn3n1 n2 ðηn ; λn1 ; λn2 Þ ¼ 0:

ð4:4Þ n

n

For each bootstrap data, we can estimate the smoothed EL estimator θ^ p ¼ arg maxθ log R^ ðθ; pÞ. Repeating those bootstrap n n n data and estimation procedure, we have a list of values θ^ and the maximum EL ratio R^ ðθ^ ; pÞ. The critical value cn satisfies p

p

the following equation: n n Pf 2 log R^ ðθ^ p ; pÞ rcn ; p A ½τ1 ; τ2 jðX ji ; δji Þ; i ¼ 1; …; nj ; j ¼ 1; 2g ¼ 1 α:

Thus, the bootstrap EL confidence bands C can be obtained as follows: ^ pÞ r cn ; τ1 r p r τ2 g: C ¼ fθðpÞ :  2 log Rðθ; For the Mayo Clinic trial data, 95% bootstrap EL confidence bands for the difference of two quantiles are illustrated in Fig. 1. From Fig. 1, which is plotted simultaneously with 20th quantile up to 60th, we notice that the normal approximation confidence band and EL confidence band almost always contain zero. Thus, both methods show that there is no difference for the two quantile functions of the treatment and placebo groups. The EL confidence bands are asymmetric in contrast to NA-based confidence band. Moreover, comparing the EL and NA 95% confidence bands for the difference of two quantiles, we observe that the EL bands have slightly shorter interval length than the normal approximation bands. Therefore, it shows the superiority of the EL-based methods. 5. Discussion The simulation studies show that the EL method is superior to the normal approximation method in small samples. EL's coverage probabilities do not change much regardless of sample size, or quantiles, which illustrated the reliability of the empirical likelihood method. The EL confidence interval is also shown to have a better interval estimate than the normal approximation one. Thus, the empirical likelihood method has a better reliability and performance. Even though the EL method has a better performance, a numerical burden exists on itself. As cross-validation method involved, the computational intensity increased significantly to select the optimal bandwidth. The theoretical optimal bandwidth is very challenging but appealing in the future study. Moreover, the proposed EL method for the difference of two quantiles contains four parameters. In order to obtain the confidence interval, we have to solve four nonlinear equations

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simultaneously. The jackknife empirical likelihood method could be an alternative method to overcome the difficulty in solving multi-dimensional nonlinear system for right censored data.

Acknowledgments Hanfang Yang's research is supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (13XNF059). Yichuan Zhao' research is supported by the National Security Agency Grant. The authors would like to thank the two referees and AE for their useful suggestions and comments which lead to a much better revision. The authors acknowledge the assistance and comments by Meng Zhao and Jenny Zhao. Appendix A. Proofs of theorems The distribution function of Xji is denoted as Hj. We denote aF 1 ¼ inffx : F 1 ðxÞ 4 0g, bH1 ¼ supfx : H 1 ðxÞ o 1g, aF 2 ¼ inffx : F 2 ðxÞ 4 0g and bH2 ¼ supfx : H 2 ðxÞ o1g. Let w(t) be the derivative of K(t). Denote τj ðtÞ ¼ F ′j ðtÞ=ð1 F j ðtÞÞ. The following regularity conditions similar to those in Shen and He (2007) and Yang and Zhao (2012) are required to attain the asymptotic result, which are displayed for completeness:

(C1) We assume maxfaF 1 ; aF 2 g o η0 o minfbH1 ; bH2 g and maxfaF 1 ; aF 2 g o η0 þ θ0 ominfbH1 ; bH2 g. (C2) w(t) is a bounded function having support ½ 1; 1, such that 8 i¼0 > Z 1 <1 i¼1 ui wðuÞ du ¼ 0 > 1 :C i ¼ 2; 0

where C 0 40. (C3) Let 0 o τ1 ðη0 þθ0 Þτ2 ðη0 Þ o1. τ′1 ðxÞ exists and is continuous in the neighborhood of η0 þ θ0 , and τ′2 ðxÞ exists and is continuous in the neighborhood of η0 . (C4) As nj -1, j¼ 1,2, n1 =n2 -γ; 0 o γ o 1. 4 1 1 (C5) As nj -1, we have hj -0; nj hj -1; nj hj -0, ln hj =ðnj hj Þ-0 and ln hj =ln ln nj -1, j¼1, 2. Proof of Theorem 2.1. Following the same argument in Lemma 2.1 of Shen and He (2007) and Yang and Zhao (2012), we can prove it. □ Like Yang and Zhao (2012), we denote ɛn1 ¼ n1 s and ɛn2 ¼ n2 s ; 1=3 o s o 1=2. Lemma A.1 (Lemma 4.2 of Shen and He, 2007). If jη η0 j rɛ n ¼ minfɛn1 ; ɛ n2 g, the solution λ ¼ ðλ1 ðηÞ; λ2 ðηÞÞT of Eqs. (2.1) and (2.2) satisfies

λj ðηÞ ¼ O ɛ nj ; j ¼ 1; 2: nj Proof. The proof is along the lines of Lemma 4.2 of Shen and He (2007) and Yang and Zhao (2012). Define Z η dF 1 ðuÞ : s21 ðηÞ ¼ 0 F 1 ðuÞð1 H 1 ðu ÞÞ Using the inequality in Shen and He (2007), we have  λ1 λ1 ðs21 ðη0 þθ0 Þ þ oð1ÞÞ λ1 q1n1 ðη; λ1 Þ  q1n1 ðη; 0Þ 4 :  1g n1 1 þλ1 maxfr1i

ðA:1Þ

As Shen and He (2007) and Yang and Zhao (2012) did, we get q1n1 ðη; 0Þ ¼ Oðɛn1 Þ

ðA:2Þ

a:s:

Combining (A.1) and (A.2), we can easily obtain 1  1 þλ1 maxfr1i g λ1 λ1 q1n1 ðη; λ1 Þ  q1n1 ðη; 0Þ r n1 λ1 fs21 ðη0 þθ0 Þ þ oð1Þg ¼ Oðɛ n1 Þ a:s:

Similarly we have λ2 =n2 ¼ Oðɛn2 Þ a.s.

□

Lemma A.2 (Lemma 4.3 of Shen and He, 2007). Under the regularity conditions (C.1–C.5), it exists a solution ηE to ~ 0 ; η; pÞ attains its maximum when n is large, a.s. Eqs. (2.1)–(2.3) such that Rðθ

H. Yang et al. / Journal of Statistical Planning and Inference 146 (2014) 95–101

101

Proof. The proof is same as Lemma 4.3 of Shen and He (2007) with some modifications. Along the lines of Shen and He (2007) and Yang and Zhao (2012), we prove it. □ Proof of Theorem 2.2. The proof is a modification of Theorem 2.1 of Shen and He (2007) and Yang and Zhao (2012). Denote β1 ¼ λ1 =n1 , β2 ¼ λ2 =n2 and

0 1

̌τ 1 ðη0 þθ0 Þ ̌s 21 ðη0 þ θ0 Þ 0   ∂ðq1n1 ; q2n2 ; q3n1 n2 Þ

B 0 ̌s 22 ðη0 Þ C T^ n η0 ¼

ðη0 ;0;0Þ ¼ @  ̌τ 2 ðη0 Þ A;

∂ðη; β1 ; β2 Þ

0  n1 ̌τ 1 ðη0 þ θ0 Þ n2 ̌τ 2 ðη0 Þ where ̌τ j and ̌s j , j¼1,2 are defined in Yang and Zhao (2012). Using Taylor expansion, we can obtain 0 1 0 1 q1n1 ðη0 ; 0Þ ηE  η0 1 B C B β C @ A ¼  T^ n ðη0 Þ@ q2n2 ðη0 ; 0Þ A þOðɛ 2n Þ a:s: 1 q3n1 n2 ðη0 ; 0; 0Þ β2 

 pffiffiffiffiffi     2  pffiffiffiffiffi n1 ̌τ 2 ðη0 Þ̌τ 1 ðη0 þ θ0 Þpffiffiffiffiffi pffiffiffiffiffi n2 q2n2 η0 ; 0  ̌τ 22 η0 n1 q1n1 η0 ; 0 þ n2

D

⟶τ22 ðη0 Þðτ22 ðη0 Þs21 þ γτ21 ðη0 þθ0 Þs22 ðη0 ÞÞχ 21 ; by the asymptotic normality of qjnj ðη0 ; 0Þ, j ¼ 1; 2. n22 λ21 ðηE Þ ¼ 2 n1 f  n1 ̌τ 21 ðη0 þθ0 Þ̌s 2 ðη0 Þ2  n2 ̌s 21 ðη0 þ θ0 Þ̌τ 22 ðη0 Þg   pffiffiffiffiffi     2  pffiffiffiffiffi   n1 ̌τ 2 ðη0 Þ̌τ 1 ðη0 þ θ0 Þpffiffiffiffiffi ̌τ 22 η0 n1 q1n1 ηE ; 0 þ pffiffiffiffiffi n2 q2n2 ηE ; 0 þ O nɛ 4n n2   1 D τ2 η0 χ 21 : ⟶ 2 fγτ1 ðη0 þ θ0 Þs22 ðη0 Þ þ s21 ðη0 þ θ0 Þτ22 ðη0 Þg 2 ( )   λ21 2   ̌τ 21 ðηE þθ0 Þn1 ̌s 22 ðηE Þ ~ þoð1Þ  2 ln R θ0 ; ηE ¼ ̌s 1 ηE þ θ0 1 þ n1 n2 ̌τ 22 ðηE þθ0 Þ̌s 21 ðηE Þ D

s21 ðηE þ θ0 Þτ22 ðηE Þ 2 ðγτ1 ðηE þθ0 Þs22 ðηE Þ þs21 ðηE þ θ0 Þτ22 ðηE ÞÞ ¼ χ 21 : □

⟶

χ 21

ðτ22 ðηE Þs21 ðηE þθ0 Þ þ τ21 ðηE þ θ0 Þγs22 ðηE ÞÞ τ22 ðηE Þs21 ðηE þθ0 Þ

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