利用Bernstein多項式來研究二元迴歸

Binary regression with Bernstein polynomials

蕭愛齡(Ai-ling Hsiao)

指導教授 ： 趙一峰 ； 張憶壽

國立中央大學/理學院/數學系/碩士(2008年)

摘要

傳統上，針對二元資料之分析多採用 logistic 迴歸模型。但此模型在事件發生之條件機率上有單調函數之限制，因此我們利用Bernstein 多項式來表達事件發生之條件機率，因而於本文中提出一個藉由Bernstein 多項式所建構的貝氏迴歸模型。在貝氏方法中，我們將先驗分佈建立在Bernstein 多項式的次數和係數所組成的參數空間上，並對統計推論所需的後驗分佈用MCMC 的方法做
抽樣。最後，在相同的模型與方法下，比較在不同樣本數及先驗分佈下的模擬結果；其次，對於logistic 迴歸模型的限制，經由模擬顯示本文所提出的貝氏迴歸有較小的均方誤差。

關鍵字

馬可夫鏈蒙地卡羅法 ； logistic 迴歸模型 ； Bernstein 多項式

並列摘要

Data analysis of binary response variables are often conducted by logistic regression model. Logistic regression model assumes that the conditional probability function of success is a monotonic function. In order to eliminate this sometimes unnecessary monotone restriction, we propose to use Bernstein polynomials to model the conditional probability of success. As a Bayesian approach, we put a prior on the space of Bernstein polynomials having values in [0,1] through their coe cients. The sample from the posterior distribution for inference purpose is obtained by MCMC methods. We conduct simulation studies to examine the e ects of sample size and priors, to indicate that the numerical performance of this method is generally good and to show that our model performs better than the logistic regression model when
the regression function is not monotone.

並列關鍵字

MCMC ； Bernstein polynomial ； logistic regression model

參考文獻

[1] Chang, I. S., Chien, L. C., Hsiung, C. A., Wen, C. C., and Wu, Y. J. (2007). Shape restricted regression with random Bernstein polynomials. In Complex Dataset and Inverse Problems. IMS Lecture Notes Monograph Series , 54, 187-202

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[5] Resnick, S. I. (1999). A Probability Path. Birkhauser, Boston.

[2] Chang, I. S., Hsiung, C. A., Wu, Y. J., and Yang, C. C. (2005). Bayesian survival analysis using Bernstein polynomials. Scandinavian Journal of Statistics, 32,

447-466