| For measuring the accuracy of a continuous diagnostic test in diagnostic medicine, the receiver operating characteristic (ROC) curve is often used. And in medical statistics the people concern how to estimate the ROC curve. The most commonly used non-parametric estimator for ROC curve is the empirical ROC curve. But one major weakness of the empirical ROC curve estimator is its jagged form, since the true ROC curve is a smooth function. Lloyd proposed using a kernel smoothing method and certified that the kernel estimators of the ROC curve are better than empirical. However, Lloyd's estimator has some drawbacks. First, it uses two kernel estimators for the distribution functions in the diseased and nondiseased populations and then takes the inverse of the kernel estimator for the distribution function of nondiseased population. Therefore, the resulting ROC curve estimator is not invariant under a monotone transformation of the data. Second, this method focuses on optimal smoothing of estimator for a single distribution and does not address optimal estimation of the ROC curve. In chapter 1, we introduce the ROC curve and some method of estimating the ROC curve. In chapter 2, we propose a new kernel method, local polynomial estimators, to estimate the ROC curve. Using the method can decrease the mean squared error(MSE) and overcome the shortcoming of Lloyd's estimator because of just using one kernel. In chapter 3, we model the test results as belonging to a location-scale family. There is a functional relation between the covariates and the datum which transform from the mean through an unknown function. We estimate unknown function and parameters use quasi-likelihood type methods and the local polynomial fitting, then we can estimate the ROC curve. |
