本文選題：非參數(shù)回歸模型 + 核估計��； 參考：《西安電子科技大學(xué)》2015年博士論文

【摘要】：非參數(shù)方法因其良好的靈活性在經(jīng)濟(jì)、醫(yī)藥、生物等領(lǐng)域有著廣泛的應(yīng)用.近年來,隨著生物學(xué)的迅猛發(fā)展以及生物數(shù)據(jù)的大量采集,生物統(tǒng)計的重要性日益顯著,其中對艾滋病的研究已經(jīng)引起了科學(xué)家的極大關(guān)注.艾滋病(AIDS)威脅著全人類的生命安全,目前對艾滋病病毒(HIV)的動態(tài)研究已成為AIDS研究領(lǐng)域的熱門問題.描述HIV動態(tài)變化的HIV模型是一組非線性常微分方程,其中包括HIV病毒死亡率,CD4+T細(xì)胞的新生率等多個重要未知參數(shù),準(zhǔn)確估計這些參數(shù)可以給AIDS致病機(jī)理以及藥效評估提供重要依據(jù).然而相對于參數(shù)估計方法,一般的非參數(shù)估計方法,如核估計和局部線性估計的收斂速度較慢且對窗寬的選擇敏感；另一方面,由于實際測量數(shù)據(jù)的波動性,HIV模型的參數(shù)估計常常是不穩(wěn)健的.本文針對非參數(shù)估計方法和HIV模型統(tǒng)計診斷的相關(guān)問題進(jìn)行研究,主要研究成果如下：1.將復(fù)合方法的思想應(yīng)用于非參數(shù)估計法中,提出復(fù)合非參數(shù)估計法：首先選擇不同的窗寬作相應(yīng)的非參數(shù)估計,然后通過一個參數(shù)回歸方法將備選的估計量重新組合.當(dāng)回歸函數(shù)具有2k階連續(xù)導(dǎo)數(shù)時,所得到的新估計量的最優(yōu)均方收斂速度達(dá)到O(n-4k/(4k+1)),即當(dāng)回歸函數(shù)足夠光滑時,新估計量的收斂速度接近參數(shù)估計的收斂速度.最優(yōu)窗寬的階數(shù)為O(n-1/(4k+1)),并且即使采用的窗寬不是最優(yōu)的,但在條件hj=O(n-α),且1/(5k)α1/5成立時,新估計量仍具有比普通非參數(shù)估計更小的均方誤差,這說明新估計量對窗寬的選擇穩(wěn)健.由此從收斂速度和窗寬選擇兩方面改進(jìn)了通常的非參數(shù)估計.模擬研究證實了復(fù)合方法的有效性.2.對HIV模型的參數(shù)估計進(jìn)行了實證研究.針對觀測數(shù)據(jù)含有測量誤差的情形,利用兩步估計法和廣義光滑方法對HIV模型進(jìn)行了參數(shù)估計,隨機(jī)模擬實驗證實了兩種方法良好的估計效果.通過比較兩種方法的表現(xiàn)得出：兩步估計法的計算更加簡單,而廣義光滑方法的精度更高.3.針對HIV模型的兩步估計方法,研究了基于均值漂移模型的統(tǒng)計診斷問題.給出了漂移參數(shù)的計算公式,構(gòu)造了Score檢驗統(tǒng)計量,并給出其大樣本分布.隨機(jī)模擬實驗和實例研究結(jié)果表明：(1)所構(gòu)造的檢驗統(tǒng)計量的極限分布是經(jīng)驗分布的一個合理近似；(2)所構(gòu)造的統(tǒng)計量可有效地檢測中度以上的偏移；(3)相對于內(nèi)點,邊界點對HIV模型的參數(shù)估計具有更大的影響.基于以上結(jié)論,在對HIV模型進(jìn)行參數(shù)估計時,我們應(yīng)對邊界點加以格外的關(guān)注.4.針對HIV模型的廣義光滑估計方法,分別討論了基于數(shù)據(jù)刪除模型的統(tǒng)計診斷和基于似然距離的局部影響分析方法,并且分別給出了數(shù)據(jù)刪除模型和擾動模型下的似然距離近似計算公式.通過模擬數(shù)據(jù)和臨床數(shù)據(jù)分析發(fā)現(xiàn)：兩種方法均能有效地檢測出模型中的異常點.
[Abstract]:Non-parametric method has been widely used in economy, medicine, biology and so on because of its good flexibility. In recent years, with the rapid development of biology and a large number of biological data collection, the importance of biological statistics has become increasingly significant, among which the study of AIDS has attracted great attention of scientists. HIV / AIDS (HIV / AIDS) is a threat to the safety of human life. The dynamic study of HIV / AIDS (HIV) has become a hot issue in the field of AIDS research. The HIV model for describing the dynamic changes of HIV is a set of nonlinear ordinary differential equations, which include several important unknown parameters, such as the HIV virus mortality rate of CD4 T cells, and so on. Accurate estimation of these parameters can provide an important basis for the pathogenesis of AIDS and the evaluation of its efficacy. However, compared with parameter estimation methods, general nonparametric estimation methods, such as kernel estimation and local linear estimation, converge slowly and are sensitive to the choice of window width; on the other hand, Parameter estimation of HIV models is often not robust because of the volatility of measured data. In this paper, the nonparametric estimation method and the statistical diagnosis of HIV model are studied. The main research results are as follows: 1. The idea of compound method is applied to the nonparametric estimation method, and a compound nonparametric estimation method is proposed: firstly, different window widths are selected for the corresponding nonparametric estimation, and then the alternative estimators are recombined by a parametric regression method. When the regression function has a continuous derivative of order 2k, the optimal mean square convergence rate of the new estimator reaches O(n-4k/(4k 1, that is, when the regression function is smooth enough, the convergence rate of the new estimator is close to that of the parameter estimation. The order of the optimal window width is O(n-1/(4k 1 / 1, and even if the window width is not optimal, the new estimator still has a smaller mean square error than the ordinary nonparametric estimator when the condition hjn- 偽, and 1 / 5) 偽 / 1 / 5, is established, which shows that the new estimator is robust to the window width selection. Thus, the general nonparametric estimation is improved in terms of convergence rate and window width selection. The simulation results show that the composite method is effective. 2. The parameter estimation of HIV model is studied empirically. Two step estimation method and generalized smoothing method are used to estimate the parameters of HIV model for the case where the observed data contain measurement errors. The results of stochastic simulation show that the two methods have good estimation effect. By comparing the performance of the two methods, it is found that the two-step estimation method is simpler, while the generalized smoothing method has higher accuracy. For the two-step estimation of HIV model, the statistical diagnosis problem based on mean shift model is studied. The calculation formula of drift parameter is given, the Score test statistic is constructed, and its large sample distribution is given. The results of random simulation experiments and case studies show that the limit distribution of the test statistics constructed by: 1) is a reasonable approximation of the empirical distribution. The boundary point has more influence on the parameter estimation of HIV model. Based on the above conclusions, we should pay special attention to the boundary point when we estimate the parameters of HIV model. For the generalized smooth estimation method of HIV model, the statistical diagnosis based on data deletion model and the local impact analysis method based on likelihood distance are discussed respectively. The approximate formulas of the likelihood distance under the data deletion model and the perturbation model are given respectively. Through the analysis of simulation data and clinical data, it is found that both methods can effectively detect abnormal points in the model.
【學(xué)位授予單位】：西安電子科技大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O212.1

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