本文關鍵詞：線性回歸模型參數(shù)有偏估計的研究　出處：《東北農業(yè)大學》2014年碩士論文　論文類型：學位論文

  更多相關文章： 有偏估計 均方誤差 相對效率 I-divergence估計

【摘要】：線性模型作為統(tǒng)計模型的一類,由于其應用廣泛、形式簡單和易于處理,得到了廣泛的關注與研究。在線性模型中,由于參數(shù)估計的基礎性地位,作為線性模型首先要面臨的問題,決定了估計回歸參數(shù)具有重要意義。參數(shù)估計最基本、最常見的方法是最小二乘估計。近些年來當數(shù)據(jù)存在復共線性時最小二乘估計均方誤差異常大導致估計值變壞,提出了有偏估計的概念。有偏估計在一定條件下減小了均方誤差,改進了最小二乘估計的不足。參數(shù)有偏估計對于線性模型理論的發(fā)展和完善具有非常重要的意義。 本文結合國內外參數(shù)有偏估計的相關研究理論、存在的問題,主要在相對效率意義下推導優(yōu)于最小二乘估計效率上界,針對已有的有偏估計只從局部改進最小二乘估計,提出了帶有約束的有偏估計類,主要工作如下： 首先,用有偏估計代替最小二乘估計后雖然減小了估計的均方誤差,但是估計的精度受到一些損失,相對效率能更好的度量有偏估計代替最小二乘估計損失的大小。因此本文重點在相對效率評價準則意義下分別推導出廣義嶺估計和Liu估計優(yōu)于最小二乘估計的效率上界。為推導有偏估計優(yōu)于最小二乘估計的條件提供了新的思路。 其次,已有的有偏估計雖然改進了最小二乘估計,但得出的均方誤差是未知參數(shù)的函數(shù),有偏估計得到的結果是探索性的,而不是確證性的。當變量存在約束情況下,已有的有偏估計并不適用,得出的估計值很不理想。針對上述這些問題本文從函數(shù)差異性角度出發(fā),采用新的度量函數(shù),在變量有約束時最終提出I-divergence估計。并在Kuhn-Tucker理論基礎上,設計迭代算法,得出迭代解,并證明了迭代過程的收斂性。如果所涉及的數(shù)據(jù)均為非負實值約束,并且數(shù)據(jù)呈正相關性,則I-divergence準則是僅有的一致性選擇。進一步用仿真數(shù)據(jù)驗證新估計的優(yōu)劣,計算嶺估計、Liu估計的均方誤差和新估計比較充分證明了在變量非負約束情況下新估計比已有的有偏估計更好的減小了均方誤差。I-divergence估計理論的提出又進一步豐富和發(fā)展了參數(shù)估計理論。 最后,結合股票定價模型實例給出了I-divergence估計在股票定價中的應用,分析結果說明了該估計的可行性和優(yōu)越性。該估計能夠幫助投資人有效描述和跟蹤市場變化,將逐漸為證券界認可和接受,其意義重大,必將為如何更有效地為企業(yè)價值評估提供一些有益的思考。在金融領域初次應用該估計有力說明了統(tǒng)計理論知識的實用性和創(chuàng)新性。 帶約束參數(shù)有偏估計對于線性模型理論的發(fā)展和完善具有非常重要的意義。做為線性模型理論有益補充,將廣泛應用于農業(yè)、管理,經濟、軍事,工程技術等領域,進一步豐富了統(tǒng)計理論。對社會作出巨大的貢獻。
[Abstract]:As a kind of statistical model, linear model has been widely studied because of its wide application, simple form and easy to deal with. In the linear model, because of the basic position of parameter estimation. As the first problem of linear model, it is important to estimate regression parameters, which is the most basic. The most common method is the least square estimation. In recent years, when the data is complex collinearity, the mean square error of the least squares estimation is very large, which leads to the deterioration of the estimation value. The concept of biased estimation is proposed, which reduces the mean square error under certain conditions. The parameter biased estimation is very important for the development and improvement of the linear model theory. In this paper, combined with the domestic and foreign research theory of biased estimation of parameters, the existing problems, mainly in the sense of relative efficiency than the least square estimation of the efficiency of the upper bound. In this paper, a class of constrained biased estimators is proposed, which only improves the least square estimators from the local level. The main work is as follows: First, though the mean square error of estimation is reduced by using biased estimation instead of least square estimation, the accuracy of estimation is reduced. The relative efficiency can better measure the loss of biased estimation instead of least square estimation. Therefore, the generalized ridge estimate and Liu estimate are derived respectively in the sense of relative efficiency evaluation criterion, which is superior to the least square estimate. A new idea is provided for the derivation of the condition that the biased estimation is superior to the least square estimation. Secondly, although the existing biased estimation improves the least square estimation, the mean square error obtained is a function of unknown parameters, and the results obtained by the biased estimation are exploratory. When the variables are constrained, the existing biased estimation is not applicable, and the estimated value is not ideal. In view of these problems, this paper starts from the point of view of function difference. Using a new metric function, the I-divergence estimation is proposed when the variables are constrained. Based on the Kuhn-Tucker theory, an iterative algorithm is designed and the iterative solution is obtained. The convergence of the iterative process is proved if the data involved are non-negative real value constraints and the data are positively correlated. The I-divergence criterion is the only consistent choice. Further, the new estimation is verified by simulation data, and the ridge estimation is calculated. The mean square error of Liu estimator and the new estimator fully prove that the new estimator can reduce the mean square error better than the existing biased estimator in the case of variable nonnegative constraint. The theory of parameter estimation is further enriched and developed. Finally, the application of I-divergence estimation in stock pricing is given with an example of stock pricing model. The analysis results show the feasibility and superiority of this estimate, which can help investors to effectively describe and track market changes, and will gradually be recognized and accepted by the securities industry, which is of great significance. It will provide some useful thoughts on how to evaluate the enterprise value more effectively. The first application of this estimate in the field of finance can explain the practicability and innovation of statistical theory knowledge. The biased estimation of constrained parameters is of great significance to the development and improvement of linear model theory. As a useful supplement of linear model theory, it will be widely used in agriculture, management, economy and military. Engineering technology and other fields have further enriched statistical theory and made great contributions to society.
【學位授予單位】：東北農業(yè)大學
【學位級別】：碩士
【學位授予年份】：2014
【分類號】：O212.1;F830.91

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