有限混合模型中的若干問題研究
發(fā)布時(shí)間：2018-08-18 17:27
【摘要】：在分析包含兩個(gè)或更多子總體的數(shù)據(jù)時(shí),有限混合模型具有很大的靈活性和便利性.自然地,有限混合模型被廣泛應(yīng)用于許多領(lǐng)域中,比如天文學(xué),醫(yī)學(xué),遺傳學(xué),工程學(xué),社會(huì)科學(xué)等等.本文的研究背景多是來自醫(yī)學(xué)和遺傳學(xué),都和混合模型密切相關(guān),主要內(nèi)容為以下三個(gè)部分.第一部分中,我們研究?jī)蓸颖镜耐|(zhì)性檢驗(yàn)問題,這兩組樣本中有一組樣本可能來自混合分布.在這個(gè)問題下,我們假設(shè)核函數(shù)為一般的位置-尺度分布,核函數(shù)尺度參數(shù)可以不同.混合分布的出現(xiàn)和尺度參數(shù)的不同導(dǎo)致似然函數(shù)無界以及Fisher信息可能關(guān)于混合比例無窮大,這些特點(diǎn)使得檢驗(yàn)兩樣本的同質(zhì)性非常具有挑戰(zhàn)性.為此,我們構(gòu)造懲罰的似然函數(shù)并基于該懲罰的似然函數(shù)構(gòu)造EM檢驗(yàn)統(tǒng)計(jì)量,以同時(shí)檢驗(yàn)均值信息和方差信息.另外,我們?cè)敿?xì)研究了 EM檢驗(yàn)統(tǒng)計(jì)量在原假設(shè)和局部備擇假設(shè)下的極限分布,并討論了樣本量的推算.模擬結(jié)果和實(shí)例分析表明,所提出的EM檢驗(yàn)比現(xiàn)有方法更高效且適用性強(qiáng).本部分內(nèi)容在一般的位置-尺度混合分布下推廣并豐富了 EM檢驗(yàn)在兩樣本問題中的應(yīng)用.數(shù)量性狀位點(diǎn)(Quantitative trait locus,QTL)區(qū)間檢測(cè)中往往涉及到混合模型.第二部分中,在核函數(shù)為位置-尺度分布下,我們研究似然比檢驗(yàn)在QTL區(qū)間檢測(cè)中的應(yīng)用,這部分內(nèi)容分別在兩個(gè)遺傳學(xué)情形下研究:減數(shù)分裂中,同源染色體的非姐妹染色單體之間不存在雙重交叉和存在雙重交叉,這兩種情形對(duì)應(yīng)的內(nèi)容分別在第三章和第四章.在第三章中,我們推導(dǎo)出兩種情形下極大似然估計(jì)和似然比統(tǒng)計(jì)量的大樣本性質(zhì),情形(1)為核函數(shù)的位置參數(shù)和尺度參數(shù)可能都不同;情形(2)為核函數(shù)的位置參數(shù)可能不同但尺度參數(shù)相同且未知.這兩種情形下,我們證明似然比的極限分布分別是卡方過程χ22(θ)和χ12(θ)的上確界.根據(jù)這個(gè)結(jié)果,我們并不能很容易地計(jì)算似然比檢驗(yàn)的臨界值.因此,我們進(jìn)一步給出兩種情形下似然比極限分布的顯式形式,顯式形式的出現(xiàn)使得我們能夠很容易的找到臨界值,從而極大的減弱了 QTL區(qū)間檢測(cè)中找尋臨界值的難度.另外,我們也對(duì)局部備擇假設(shè)下似然比檢驗(yàn)統(tǒng)計(jì)量的極限分布做了研究.通過數(shù)值模擬,我們研究了似然比檢驗(yàn)統(tǒng)計(jì)量的有限樣本性質(zhì)并和現(xiàn)有方法做了比較.模擬結(jié)果側(cè)面驗(yàn)證了我們?cè)谒迫槐葯z驗(yàn)下推導(dǎo)出的優(yōu)良性質(zhì).最后,我們用似然比檢驗(yàn)分析了一個(gè)實(shí)際問題,分析結(jié)果表明,似然比檢驗(yàn)可適用性強(qiáng).第四章中,我們假設(shè)同源染色體的非姐妹染色單體之間存在雙重交叉,其他假設(shè)均和第三章相同.由于雙重交叉的存在,統(tǒng)計(jì)模型和第三章中的不再相同.我們繼續(xù)考慮第三章中的兩種情形:情形(1)和情形(2).在情形(2)下,構(gòu)造似然比的檢驗(yàn)過程和第三章中的類似,且大樣本性質(zhì)也類似.值得注意的是,在情形(1)下,我們并不能直接基于似然函數(shù)構(gòu)造似然比檢驗(yàn).因?yàn)樵诒菊碌慕y(tǒng)計(jì)模型下,似然函數(shù)是無界的,這導(dǎo)致我們無法得到相合的極大似然估計(jì).為此,我們對(duì)尺度參數(shù)添加懲罰函數(shù)從而構(gòu)造懲罰的似然函數(shù),進(jìn)而得到相合的懲罰極大似然估計(jì).最終基于懲罰的似然函數(shù),我們構(gòu)造似然比檢驗(yàn)并建立相應(yīng)的大樣本性質(zhì).類似于第三章,我們進(jìn)一步研究了兩情形下極限分布的顯式形式和局部備擇假設(shè)下似然比檢驗(yàn)統(tǒng)計(jì)量的極限分布.最后,分別通過數(shù)值模擬和實(shí)例分析,我們研究了似然比檢驗(yàn)統(tǒng)計(jì)量的有限樣本性質(zhì)并和現(xiàn)有方法做了比較.第三部分中,針對(duì)帶有結(jié)構(gòu)參數(shù)的有限位置-尺度混合模型,我們研究了極大似然估計(jì)的強(qiáng)相合性問題.在對(duì)參數(shù)空間不作任何限制的情形下,我們給出強(qiáng)相合性結(jié)果和詳細(xì)的證明.另外,我們給出了一些例子:核函數(shù)分別為正態(tài)分布,邏輯分布,極值分布和t分布的有限混合模型,并證明這些模型滿足假設(shè)條件.
[Abstract]:Naturally, finite mixing models are widely used in many fields, such as astronomy, medicine, genetics, engineering, social sciences, etc. The research background of this paper is mostly from medicine and genetics, and both of them are mixed models. In the first part, we study the homogeneity test of two samples, one of which may come from a mixed distribution. In this case, we assume that the kernel function is a general position-scale distribution, and the scale parameters of the kernel function can be different. The difference between the likelihood function and the scale parameter leads to the unbounded likelihood function and the infinite mixed proportion of Fisher information. These characteristics make it very challenging to test the homogeneity of two samples. In addition, we study the limit distribution of EM test statistics under the original hypothesis and the local alternative hypothesis in detail, and discuss the estimation of sample size. Simulation results and case studies show that the proposed EM test is more efficient and applicable than the existing methods. Quantitative trait locus (QTL) interval detection often involves mixed models. In the second part, we study the application of likelihood ratio test in QTL interval detection under the location-scale distribution of kernel function, which is in two genetic cases respectively. In the third chapter, we derive the large sample properties of maximum likelihood estimators and likelihood ratio statistics in two cases, case (1) is a nuclear functor. In both cases, we prove that the limit distribution of likelihood ratio is the upper bound of chi-square process_22 (theta) and_12 (theta), respectively. According to this result, we can not easily calculate the likelihood ratio. Therefore, we further give the explicit form of likelihood ratio limit distribution in two cases. The appearance of the explicit form makes it easy to find the critical value, thus greatly reducing the difficulty of finding the critical value in QTL interval detection. Likelihood ratio test statistics are studied by numerical simulation and compared with the existing methods. The simulation results verify the good properties derived from the likelihood ratio test. Finally, a practical problem is analyzed by likelihood ratio test. In Chapter 4, we assume that there is a double crossover between the non-sister chromatids of homologous chromosomes, and the other assumptions are the same as in Chapter 3. Because of the double crossover, the statistical model is no longer the same as in Chapter 3. We continue to consider two cases in Chapter 3: case (1) and case (2). (2) The process of constructing likelihood ratio is similar to that in Chapter 3, and the properties of large samples are similar. It is noteworthy that in the case (1), we can not construct likelihood ratio test directly based on the likelihood function. Because the likelihood function is unbounded in the statistical model of this chapter, we can not get the consistent maximum likelihood estimation. Finally, based on the penalty likelihood function, we construct the likelihood ratio test and establish the corresponding large sample properties. Similar to Chapter 3, we further study the explicit limit distribution in two cases. Limit distributions of likelihood ratio test statistics under formal and local alternative assumptions are studied. Finally, we study the finite sample properties of likelihood ratio test statistics and compare them with existing methods by numerical simulation and case analysis. In the third part, we study the finite position-scale mixed model with structural parameters. The strong consistency problem of maximum likelihood estimators is discussed. In the case of no restriction on parameter space, we give strong consistency results and detailed proof. In addition, we give some examples: the finite mixed models of kernel functions are normal distribution, logical distribution, extremum distribution and t distribution, and prove that these models satisfy the hypothesis. Pieces.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O212.1
本文編號(hào)：2190166
[Abstract]:Naturally, finite mixing models are widely used in many fields, such as astronomy, medicine, genetics, engineering, social sciences, etc. The research background of this paper is mostly from medicine and genetics, and both of them are mixed models. In the first part, we study the homogeneity test of two samples, one of which may come from a mixed distribution. In this case, we assume that the kernel function is a general position-scale distribution, and the scale parameters of the kernel function can be different. The difference between the likelihood function and the scale parameter leads to the unbounded likelihood function and the infinite mixed proportion of Fisher information. These characteristics make it very challenging to test the homogeneity of two samples. In addition, we study the limit distribution of EM test statistics under the original hypothesis and the local alternative hypothesis in detail, and discuss the estimation of sample size. Simulation results and case studies show that the proposed EM test is more efficient and applicable than the existing methods. Quantitative trait locus (QTL) interval detection often involves mixed models. In the second part, we study the application of likelihood ratio test in QTL interval detection under the location-scale distribution of kernel function, which is in two genetic cases respectively. In the third chapter, we derive the large sample properties of maximum likelihood estimators and likelihood ratio statistics in two cases, case (1) is a nuclear functor. In both cases, we prove that the limit distribution of likelihood ratio is the upper bound of chi-square process_22 (theta) and_12 (theta), respectively. According to this result, we can not easily calculate the likelihood ratio. Therefore, we further give the explicit form of likelihood ratio limit distribution in two cases. The appearance of the explicit form makes it easy to find the critical value, thus greatly reducing the difficulty of finding the critical value in QTL interval detection. Likelihood ratio test statistics are studied by numerical simulation and compared with the existing methods. The simulation results verify the good properties derived from the likelihood ratio test. Finally, a practical problem is analyzed by likelihood ratio test. In Chapter 4, we assume that there is a double crossover between the non-sister chromatids of homologous chromosomes, and the other assumptions are the same as in Chapter 3. Because of the double crossover, the statistical model is no longer the same as in Chapter 3. We continue to consider two cases in Chapter 3: case (1) and case (2). (2) The process of constructing likelihood ratio is similar to that in Chapter 3, and the properties of large samples are similar. It is noteworthy that in the case (1), we can not construct likelihood ratio test directly based on the likelihood function. Because the likelihood function is unbounded in the statistical model of this chapter, we can not get the consistent maximum likelihood estimation. Finally, based on the penalty likelihood function, we construct the likelihood ratio test and establish the corresponding large sample properties. Similar to Chapter 3, we further study the explicit limit distribution in two cases. Limit distributions of likelihood ratio test statistics under formal and local alternative assumptions are studied. Finally, we study the finite sample properties of likelihood ratio test statistics and compare them with existing methods by numerical simulation and case analysis. In the third part, we study the finite position-scale mixed model with structural parameters. The strong consistency problem of maximum likelihood estimators is discussed. In the case of no restriction on parameter space, we give strong consistency results and detailed proof. In addition, we give some examples: the finite mixed models of kernel functions are normal distribution, logical distribution, extremum distribution and t distribution, and prove that these models satisfy the hypothesis. Pieces.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O212.1
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