發(fā)布時(shí)間：2018-03-30 16:51

  本文選題：半?yún)?shù)方法　切入點(diǎn)：經(jīng)驗(yàn)似然方法　出處：《哈爾濱工業(yè)大學(xué)》2016年博士論文

【摘要】：半?yún)?shù)經(jīng)驗(yàn)似然方法廣泛的被用于統(tǒng)計(jì)學(xué)中,此方法將半?yún)?shù)方法與經(jīng)驗(yàn)似然方法有效的融合在一起,該方法具有如下優(yōu)點(diǎn):一方面,半?yún)?shù)方法能彌補(bǔ)參數(shù)方法對(duì)回歸函數(shù)需要具有較強(qiáng)基本假設(shè)這一缺點(diǎn),又能彌補(bǔ)非參數(shù)模型不能充分利用已知信息的缺點(diǎn);另一方面,由于經(jīng)驗(yàn)似然方法在應(yīng)對(duì)復(fù)雜問(wèn)題時(shí)不必受分布函數(shù)制約,因此經(jīng)驗(yàn)似然方法能夠描述一些不確定性的問(wèn)題或一些無(wú)法用具體函數(shù)描述的問(wèn)題。本文以多變點(diǎn)模型與整值時(shí)間序列模型為研究對(duì)象,對(duì)半?yún)?shù)經(jīng)驗(yàn)似然檢驗(yàn)問(wèn)題進(jìn)行探討。主要研究?jī)?nèi)容有以下幾個(gè)方面:1.針對(duì)相同參數(shù)權(quán)函數(shù)下含有兩個(gè)變點(diǎn)的多變點(diǎn)問(wèn)題,給出了半?yún)?shù)經(jīng)驗(yàn)似然函數(shù)。利用Lagrange乘子方法得到了變點(diǎn)估計(jì)值,以及變點(diǎn)估計(jì)的最大似然檢驗(yàn)統(tǒng)計(jì)量。并利用強(qiáng)大數(shù)定律獲得了經(jīng)驗(yàn)似然比檢驗(yàn)統(tǒng)計(jì)量的漸近分布以及變點(diǎn)估計(jì)值的p-值,并證明了最大似然函數(shù)與一個(gè)連續(xù)的凸函數(shù)漸近相等,以及變點(diǎn)估計(jì)漸近的服從三點(diǎn)分布。除此之外,通過(guò)數(shù)值模擬驗(yàn)證了半?yún)?shù)經(jīng)驗(yàn)似然方法比非參數(shù)經(jīng)驗(yàn)似然方法能更好地檢驗(yàn)變點(diǎn)估計(jì)值,最后用實(shí)際數(shù)據(jù)診斷了模型具有較好的適用性。2.分析了不同參數(shù)權(quán)函數(shù)下含有兩個(gè)變點(diǎn)的多變點(diǎn)問(wèn)題,該問(wèn)題的數(shù)學(xué)模型是利用經(jīng)驗(yàn)似然方法,結(jié)合Lagrange乘子,構(gòu)造了半?yún)?shù)經(jīng)驗(yàn)似然函數(shù)。并通過(guò)最大似然估計(jì)得到了變點(diǎn)的估計(jì)值以及p-值,應(yīng)用強(qiáng)大數(shù)定律得到了經(jīng)驗(yàn)似然比和半?yún)?shù)經(jīng)驗(yàn)似然統(tǒng)計(jì)量的漸近分布。在數(shù)值模擬上,大量的實(shí)驗(yàn)表明,當(dāng)變點(diǎn)的真實(shí)值在隨機(jī)變量的相對(duì)中間位置時(shí),半?yún)?shù)經(jīng)驗(yàn)似然檢驗(yàn)比非參數(shù)的方法相對(duì)優(yōu)越,而當(dāng)真實(shí)值在相對(duì)兩端時(shí)半?yún)?shù)經(jīng)驗(yàn)似然方法的優(yōu)越性不那么明顯。而實(shí)際數(shù)據(jù)依然驗(yàn)證了模型有很好的適用性。3.對(duì)于不同參數(shù)權(quán)函數(shù)下的含有有限個(gè)變點(diǎn)的多變點(diǎn)模型,利用半?yún)?shù)經(jīng)驗(yàn)似然方法構(gòu)造了經(jīng)驗(yàn)似然函數(shù)。通過(guò)最大似然函數(shù)和強(qiáng)大數(shù)定律得到了變點(diǎn)的估計(jì)值、變點(diǎn)估計(jì)的p-值以及半?yún)?shù)經(jīng)驗(yàn)似然統(tǒng)計(jì)量,建立了關(guān)于有限個(gè)變點(diǎn)的極大似然估計(jì)的漸近結(jié)果,并利用bootstrap方法對(duì)有限個(gè)變點(diǎn)的數(shù)目進(jìn)行模擬估計(jì),模擬結(jié)果顯示文中所提出的經(jīng)驗(yàn)似然方法對(duì)變點(diǎn)的估計(jì)是有效的。在參數(shù)估計(jì)的精度上,通過(guò)數(shù)值模擬,得出當(dāng)變點(diǎn)在相對(duì)中間的位置時(shí)半?yún)?shù)經(jīng)驗(yàn)似然有較好的效用,而當(dāng)變點(diǎn)在相對(duì)兩端時(shí)卻無(wú)法判斷。實(shí)例也驗(yàn)證了模型擬合有較好的能力。4.探討了對(duì)含有間歇性噪聲的整值時(shí)間序列INAR(k)模型,基于雙似然方法建立了半?yún)?shù)經(jīng)驗(yàn)似然函數(shù)。利用Lagrange乘子的方法得到了參數(shù)的估計(jì)值,并通過(guò)強(qiáng)大數(shù)定律和中心極限的性質(zhì)證明了參數(shù)的經(jīng)驗(yàn)似然比統(tǒng)計(jì)量漸進(jìn)的服從自由度為k+2的χ2分布,且證明了參數(shù)的置信區(qū)間是凸集。除此之外,數(shù)值實(shí)驗(yàn)說(shuō)明了半?yún)?shù)經(jīng)驗(yàn)似然的效用,以及非間歇性噪聲對(duì)模型的影響是顯著的,而間歇性噪聲對(duì)模型的影響是不顯著的。綜上所述,在多變點(diǎn)模型和整值時(shí)間序列模型中半?yún)?shù)經(jīng)驗(yàn)似然方法都具有較高的效率。論文依次闡述了半?yún)?shù)經(jīng)驗(yàn)似然方法在相同參數(shù)權(quán)、不同參數(shù)權(quán),有限個(gè)變點(diǎn)的不同參數(shù)權(quán)以及整值時(shí)間序列模型中的應(yīng)用。大量的數(shù)值模擬實(shí)驗(yàn)和實(shí)例證明了該方法的效用,體現(xiàn)了半?yún)?shù)經(jīng)驗(yàn)似然方法的優(yōu)越性。同時(shí)說(shuō)明了本文所研究的方法適用于多變點(diǎn)和整值時(shí)間序列模型。
[Abstract]:A semiparametric empirical likelihood method is used widely in statistics, this method of semi parametric method and empirical likelihood method effectively together, this method has the following advantages: on the one hand, the semi parametric method can compensate for parameter method has a strong assumption that the disadvantages of regression function, and make up the non parametric model cannot make full use of the known information shortcomings; on the other hand, due to the empirical likelihood method without restricting the distribution function in dealing with complex problems, so the empirical likelihood method can describe some uncertain problems or some with a specific function to describe the problem. The thesis uses variable point model and integer valued time series model as the research object. Semiparametric empirical likelihood test problems are discussed. The main contents are as follows: 1. for the same power function with two parameters change point changing point to ask Questions, gives a semiparametric empirical likelihood function. By using the Lagrange method to get the change point estimation, maximum likelihood test statistic and estimate the change point. And get the empirical likelihood ratio test statistic and asymptotic distribution of point estimates of p- value using the strong law of large numbers, and proved that the maximum likelihood function and a a continuous convex function is asymptotically equal, and the change point estimation follows three asymptotic distribution. In addition, verify the semiparametric empirical likelihood method than the non parametric empirical likelihood method can better test of change point estimation through the numerical simulation, finally the actual data diagnosis model has good applicability of the.2. analysis of different parameters the right function contains two point variable problem, the mathematical model of the problem is to use the empirical likelihood method with Lagrange multipliers, constructed a semiparametric empirical likelihood function and through. The maximum likelihood estimates to get the estimation of change point value and p- value, the application of strong law of large numbers of the asymptotic distribution of empirical likelihood ratio and semiparametric empirical likelihood statistic. In numerical simulation, a large number of experiments show that when the change point in the middle position relative to the true value of random variables, semiparametric empirical likelihood test method the non parameter relative superiority, and when the true value of superiority in semiparametric empirical likelihood method is opposite less obvious. But the actual data still validate the model.3. is suitable for the variable point model a finite change point for different parameters of the weighting function containing, constructed by the likelihood function a semiparametric empirical likelihood method. The maximum likelihood function and the strong law of large numbers are estimated change point value, p- value and change point estimation semiparametric empirical likelihood statistic is established on a Co. Asymptotic results for maximum likelihood estimation of the change point, and use the bootstrap method on a number of limited change points in the simulated estimation, simulation results show that the empirical likelihood estimation method proposed in this paper on the change point is effective. The precision of parameter estimation, by numerical simulation, when the change point has a good effect in the middle of the relative position of semiparametric empirical likelihood, and when the change point in the opposite but not judgment. Results show the model has good fitting ability of.4. on the whole value of time series of INAR containing intermittent noise (k) model, semiparametric empirical likelihood function is established based on double likelihood method. By using the Lagrange obtained the estimates of the parameters, and the properties of the strong law of large numbers and central limit that the empirical likelihood ratio statistic parameters of progressive degrees of freedom for the k+2 distribution 2, and prove The confidence interval of the parameters is convex. Besides, numerical experiments show that the semiparametric empirical likelihood utility, and non intermittent noise of the model is significant, while the effect of intermittent noise on the model is not significant. In summary, the variable point model and integer valued time series model in semiparametric empirical likelihood the method has higher efficiency. This paper expounds the semiparametric empirical likelihood method in the same parameters are right, right of different parameters, different parameters of limited right to the change point and the whole value of application in time series model. Numerical simulation of a large number of quasi experiments and examples prove that the utility of this method reflects the superiority of semi parametric the empirical likelihood method. At the same time that this method is suitable for variable point and integer valued time series model.

【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O212.1

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