Statistical Diagnosis Of HIV Model Based On Nonparametric Method

Nonparametric method is widely used in the fields of economic, medicine, biology for its flexibility. With the fast development of biology and the gathering of massive biological data in recent years, the biometrics has became more and more important. Among the re-search topic of biology, AIDS(Acquired Immune Deficiency Syndrome) has received sci-entists much attention. AIDS has threatened the safety of human beings. At present, the dynamic studies of HIV(human immunodeficiency virus) are hot issues in AIDS research. The dynamics of HIV can be described by ordinary differential equations(ODEs), in which the unknown parameters include the death rate of virus, the generated rate of new CD4+T cell etc. Accurate estimation of these parameters can provide important information for un-derstanding the pathogenesis of AIDS and evaluation of treatment efficacy. However, by contrast with the parametric methods, the nonparametric methods such as kernel estimation and local linear estimation, are sensitive to the bandwidth and can not achieve a satisfactory convergence rate. On the other hand, due to the volatility of the actually measured data, the parametric estimation of above HIV model are often not robust. In this thesis we will focus on the problems which are related to the nonparametric estimation method and statistical diagnosis of HIV model. The major contributions of this thesis are outlined as follows:1. A composite nonparametric estimation is proposed by applying the idea of composite method to nonparametric estimation:several kernel estimators with different bandwidths are first prepared and then are combined by a parametric regression technique. When the regression function has the 2k-order continual derivative, the optimal mean squared conver-gence rate of the newly proposed estimator has the order of O(n-4k/(4k+1)). This means that when the regression function is smooth enough, the convergence rate of the new estimator will nearly achieve the parametric convergence rate. The optimal bandwidth is the order of O(n-1/(4k+1)), furthermore, we find that if the bandwidths hj are not optimally selected but sat-isfy the following mild condition:hj=O(n-1/(4k+1)) with 1/5k<α<1/5, the new estimator still has smaller mean square error than the original one does. This means that the new estimator is robust to the bandwidth. Thus, the common nonparametric estimator has been improved in two main ways of convergence rate and selection of bandwidth. Simulation studies show that the new approach is effective.2. Empirical research on the parametric estimation of HIV model is done. According to the observational data with measuring error, the parametric estimations of HIV model are proposed by using the two-step approach and generalized smoothing approach. Random simulation suggests that the two methods have good performances. By comparing the per-formances of the two methods, one can find that the two-step approach is easier, and the accuracy of generalized smoothing approach is higher.3. The mean shift outlier model is used to detect outliers in HIV model based on the two-step estimation. Approximate formula for shift parameter is derived, furthermore, a Score test statistic is constructed. The limit distribution of such Score test statistic is established. The simulation results and case study also show that:(1) The limit distribution is a reason-able approximation to the empirical distribution; (2) The middle bias can be detected by the proposed statistic; (3) The boundary points have more impact on the parameter estima-tion relative to interior points. Based on these conclusions, we suggest that the boundary measurements should be paid more attentions for the parameters estimation of HIV model.4. The statistical diagnosis based on the case-deletion model and the local influence analy-sis based on the likelihood distance are discussed, according to the generalized smoothing approach of HIV model respectively. The approximate formulas for the likelihood distance of the case-deletion model and perturbation model are derived. By applying the proposed method to the simulated and clinical trial data, we find that the two proposed procedure can detect the outliers effectively.