Table 3. Empirical likelihood confidence intervals for the ROC curve at

Table 4. Empirical likelihood confidence intervals for the ROC curve at

4. Discussion

In this paper, we developed the smoothing empirical likelihood method for the ROC curve with missing datawhich is a natural extension of Claeskens et al. [7].The key technique used to impute the missing data is the random hot deck imputation procedure. Under imputation, the proposed smoothed EL statistic converges to a scaled chi-square distribution. In addition, we carry out the simulation studies to evaluate the finite sample performance of the proposed EL interval estimation for the ROC curve. For either smaller or larger q, the EL confidence intervals for the ROC curve have good coverage probabilities which are close to the nominal level. In summary, the proposed EL interval estimation is a reliable and useful tool for the ROC curve analysis with missing data. In the future, we will use other imputation methods to achieve better interval estimation and improve the performance.

5. Acknowledgements

The author acknowledges partial support under a FY09 Research Initiation Grant in Georgia State University. The author would like to thank Dr. Yichuan Zhao for his supervision.

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Appendix. Proof of Theorem 1

To prove Theorem 1, we need some additional lemmassimilar to those in Qin and Qian [10]. We only give an outline of the proofs since they follow similar arguments as Qin and Qian [10].

Lemma A.1. Under the regularity conditions of Theorem 1, as, we have

where

.

Proof of Lemma A.1. We follow the similar lines as Qin and Qian [10]. Let, and. It follows that

Like Qin and Qian [10], we have that

We have

where and is the cumulative distribution function of. By Lemma A.1 of Qin and Qian [10], we have

As Qin and Qian [10], we have that,

The rest of Lemma A.1 can be proved following same lines. It is omitted.

Lemma A.2. (Qin and Qian [10]). Assume that Under the regularity conditions (i)-(v),

uniformly for as, where c is a positive constant.

Proof of Lemma A.2. We follow the same arguments as Qin and Qian [10]. The proof is omitted.

Lemma A.3. (Qin and Qian [10]). Assume that Under the regularity conditions (i)-(v), in probability there exists a root of Equation (1) such that,

as, and attains its local maximum value at.

Proof of Lemma A.3. We follow the similar lines as Qin and Qian [10]. The proof is omitted.

Lemma A.4. Assume that the regularity conditions are satisfied. Then, as

where, and c_{0} are defined in Lemma A.1 and Theorem 1.

Proof of Lemma A.4. We follow the similar lines as Qin and Qian [10]. Let, ,

,. Using the Taylor expansion, Lemma A.2 and Lemma A.3, we have

where i = 1, 2, 3,. As Lemma 4.5 of Qin and Qian [10], we can show that

Thus

where

It follows that

From Lemma A.1, we have

thus Lemma A.4 is proved.

Proof of Theorem 1. It is similar to the proof of Theorem 1 in Qin and Qian [10]. The proof of Theorem 1 is omitted.