本文選題：傘形約束 + 優(yōu)化理論�。� 參考：《北京交通大學》2017年碩士論文

【摘要】：統(tǒng)計推斷是統(tǒng)計學中的一類重要課題.在統(tǒng)計推斷問題中,通常被估計的量或待檢驗的假設會有一些先驗條件,這些條件作為約束條件放在統(tǒng)計推斷模型中,形成有約束條件的統(tǒng)計推斷問題.帶約束的統(tǒng)計推斷應用背景非常廣泛,在醫(yī)藥,經(jīng)濟,環(huán)境科學,基因研究等領域起著很重要的作用,所以至今仍然是統(tǒng)計界研究的一個熱門.劑量—反應模型是一類經(jīng)典的帶約束的統(tǒng)計推斷問題,研究的是如何設置藥物的劑量使藥物的使用即安全又有效,副作用最小.處理好這類問題,對藥品的制造和使用有很大作用.在藥物的劑量—反應研究中,藥物的反應—劑量比率并不總是隨著劑量的增加而一直增加,當劑量達到一定程度后,藥物的毒副作用也會增加,其反應—劑量比率反而會下降.這樣,反應—劑量比率形成一種先增后減的傘形順序,其作為約束條件我們稱之為傘形約束.從數(shù)學實質(zhì)上看,帶傘形約束的劑量—反應模型就是一類帶約束的最優(yōu)化問題.不同于統(tǒng)計中的保序回歸方法,本文通過優(yōu)化理論對帶傘形約束的劑量—反應模型進行分析,用優(yōu)化的思想解決統(tǒng)計學中的經(jīng)典模型,為求解此類統(tǒng)計問題提供了一種新思路.首先,本文研究了帶傘形約束的劑量—反應模型的性質(zhì),證明了其解的唯一存在性,給出了其一階最優(yōu)性條件—KKT條件,二階最優(yōu)性條件,并研究了該模型的對偶性理論.其次,本文給出了該模型的三種優(yōu)化算法—序列二次規(guī)劃法,Rosen梯度投影法和乘子法,也給出了確定該模型傘形頂點的方法.最后,本文對一組真實數(shù)據(jù)進行了數(shù)值實驗,并對三種算法進行了比較.
[Abstract]:Statistical inference is a kind of important subject in statistics. In statistical inference, there are usually some prior conditions for the estimated quantity or the hypothesis to be tested. These conditions are placed in the statistical inference model as constraints, and form a statistical inference problem with constraints. Medicine, economics, environmental science, gene research and other fields play an important role. So far, it is still a hot topic in the field of statistics. The dose response model is a class of classic statistical inference problems with constraints. It is studied how to set the dosage of drugs to make the use of drugs safe and effective and the side effects are minimal. The problem has a great effect on the manufacture and use of drugs. In the dose response study of the drug, the reaction dose ratio of the drug does not always increase as the dose increases. When the dose reaches a certain extent, the drug side effects will increase, and the reaction dose ratio will decrease. In this way, the reaction dose ratio is also reduced. In fact, the dose response model with umbrella shaped constraints is a class of constrained optimization problems. Different from the sequential regression method in statistics, the dose response model with umbrella constraints is introduced in this paper. This paper provides a new idea for solving such statistical problems. First, this paper studies the properties of the dose response model with umbrella shaped constraints, proves the only existence of the solution, and gives the first order optimality conditions, the KKT condition, the two order optimality condition, and the study. The duality theory of the model. Secondly, three optimization algorithms of the model - sequence two times programming, Rosen gradient projection method and multiplier method are given in this paper. The method of determining the apex of the model umbrella is also given. Finally, this paper carries out numerical experiments on a group of real data and compares the three algorithms.
【學位授予單位】：北京交通大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O212.1

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相關期刊論文 前1條

1 史寧中;;保序回歸與最大似然估計[J];應用概率統(tǒng)計;1993年02期

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本文編號：1950518