【摘要】：在很多實(shí)際的統(tǒng)計(jì)問題中,由于觀測技術(shù)或者試驗(yàn)經(jīng)費(fèi)等客觀條件的限制無法得到觀測對象的精確值,只能觀測到它所處的范圍,統(tǒng)計(jì)學(xué)稱這類數(shù)據(jù)為刪失數(shù)據(jù)。譬如在醫(yī)學(xué)中,研究某種疾病發(fā)病時(shí)間時(shí)會(huì)遇到刪失數(shù)據(jù);在工程學(xué)中,研究產(chǎn)品壽命時(shí)會(huì)遇到刪失數(shù)據(jù);在金融經(jīng)濟(jì)學(xué)中,研究貿(mào)易關(guān)系持續(xù)時(shí)長、債券的發(fā)行與違約時(shí)間、企業(yè)生存周期時(shí)會(huì)遇到刪失數(shù)據(jù)。由于刪失數(shù)據(jù)在醫(yī)學(xué)、工程學(xué)、金融經(jīng)濟(jì)學(xué)、保險(xiǎn)精算學(xué)、人口統(tǒng)計(jì)學(xué)等領(lǐng)域中大量存在,因而對刪失數(shù)據(jù)進(jìn)行統(tǒng)計(jì)分析尤為重要。當(dāng)前對刪失數(shù)據(jù)的研究大多集中在較為簡單的右刪失數(shù)據(jù)上,而對更為普遍存在的區(qū)間刪失數(shù)據(jù)的研究結(jié)果不多。實(shí)證分析也多集中在醫(yī)學(xué)和工程學(xué)領(lǐng)域,對于金融經(jīng)濟(jì)和保險(xiǎn)精算領(lǐng)域涉及較少。本文利用生存分析中對壽命數(shù)據(jù)的研究思路,討論區(qū)間刪失數(shù)據(jù)下幾種生存分布的統(tǒng)計(jì)推斷問題。采用基于刪失似然函數(shù)的極大似然估計(jì)方法、EM算法以及對EM算法的M步進(jìn)行改進(jìn)的ECM算法,對區(qū)間刪失數(shù)據(jù)下指數(shù)分布、Weibull分布、對數(shù)Logistic分布進(jìn)行參數(shù)估計(jì)。利用Matlab和R軟件編程實(shí)現(xiàn)區(qū)間刪失數(shù)據(jù)下三種分布的參數(shù)估計(jì),對不同區(qū)間刪失樣本進(jìn)行數(shù)值模擬,比較三種方法的估計(jì)效果,模擬結(jié)果表明所給參數(shù)估計(jì)方法可行且估計(jì)效果相近。改進(jìn)的ECM算法具有良好的收斂性質(zhì),比EM算法有更廣泛的應(yīng)用。當(dāng)樣本容量較大、刪失比例較小的情況下估計(jì)效果最好。另一方面,在參數(shù)估計(jì)和基于缺失信息原則的期望信息矩陣下,利用方差-協(xié)方差矩陣的一致漸近性質(zhì),構(gòu)造參數(shù)的置信區(qū)間,并利用使期望信息矩陣包含最大信息量的原則,設(shè)計(jì)最優(yōu)刪失計(jì)劃,并進(jìn)行數(shù)值模擬。最后,采用本文參數(shù)估計(jì)、置信區(qū)間構(gòu)造方法,對金融經(jīng)濟(jì)領(lǐng)域2004-2014年中國對美國的農(nóng)產(chǎn)品出口貿(mào)易關(guān)系持續(xù)時(shí)長的區(qū)間刪失數(shù)據(jù)進(jìn)行了實(shí)證分析,估計(jì)得到的Weibull分布生存函數(shù)與理論生存函數(shù)一致,表明上述統(tǒng)計(jì)推斷方法在金融經(jīng)濟(jì)領(lǐng)域具有較好的理論意義和實(shí)際應(yīng)用價(jià)值。
[Abstract]:In many practical statistical problems, due to the limitation of objective conditions such as observation technology or test expenses, the exact value of the observation object can not be obtained, and the scope of the observation object can only be observed. This kind of data is called censored data in statistics. For example, in medicine, censored data will be encountered when studying the onset time of a disease, and in engineering, the deleted data will be encountered when studying the life span of a product. In financial economics, the study of trade relations lasts a long time, the issuance and default time of bonds, the life cycle of enterprises will encounter censored data. Because of the large number of deleted data in medicine, engineering, financial economics, insurance actuary, demographics and other fields, it is particularly important to carry out statistical analysis of deleted data. At present, most of the research on deleted data focuses on the relatively simple right deleted data, but the research results on the more common interval censored data are few. Empirical analysis also focuses on medicine and engineering, but less on financial economy and insurance actuarial field. In this paper, several statistical inference problems of survival distribution under interval censored data are discussed by using the research ideas of life data in survival analysis. The maximum likelihood estimation method based on censored likelihood function, the EM algorithm and the improved ECM algorithm for M-step of EM algorithm are used to estimate the parameters of exponential distribution, Weibull distribution and logarithmic Logistic distribution under interval censored data. Using Matlab and R software programming to realize the parameter estimation of three kinds of distribution under interval censored data, the numerical simulation of different interval censored samples is carried out, and the estimation effects of the three methods are compared. The simulation results show that the given parameter estimation method is feasible and the estimation effect is similar. The improved ECM algorithm has a good convergence property and is more widely used than the EM algorithm. When the sample size is large and the censored ratio is small, the estimation effect is the best. On the other hand, under the parameter estimation and the expected information matrix based on the principle of missing information, the confidence interval of the parameter is constructed by using the uniform asymptotic property of the variance-covariance matrix, and the principle that the expected information matrix contains the maximum amount of information is used. The optimal censored plan is designed and the numerical simulation is carried out. Finally, using the method of parameter estimation and confidence interval construction, this paper makes an empirical analysis of the interval censored data of China's agricultural export trade relationship to the United States from 2004 to 2014 in the financial and economic fields. The estimated survival function of Weibull distribution is consistent with the theoretical survival function, which indicates that the above statistical inference method has good theoretical significance and practical application value in the field of finance and economy.
【學(xué)位授予單位】：北京工商大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：C829.2

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