注册 登录  
 加关注
   显示下一条  |  关闭
温馨提示!由于新浪微博认证机制调整,您的新浪微博帐号绑定已过期,请重新绑定!立即重新绑定新浪微博》  |  关闭

平安甜橙博客

家俭则兴,人勤则健,能勤能俭,永不贫贱

 
 
 

日志

 
 
 
 

Redescending M-estimator  

2017-10-26 20:02:10|  分类: 概率论与数理统计 |  标签: |举报 |字号 订阅

  下载LOFTER 我的照片书  |

In statistics, Redescending M-estimators are Ψ-type M-estimators which have ψ functions that are non-decreasing near the origin, but decreasing toward 0 far from the origin. Their ψ functions can be chosen to redescend smoothly to zero, so that they usually satisfy ψ(x) = 0 for all x with |x| > r, where r is referred to as the minimum rejection point.

Due to these properties of the ψ function, these kinds of estimators are very efficient, have a high breakdown point and, unlike other outlier rejection techniques, they do not suffer from a masking effect. They are efficient because they completely reject gross outliers, and do not completely ignore moderately large outliers (like median).

Advantages

Redescending M-estimators have high breakdown points (close to 0.5), and their Ψ function can be chosen to redescend smoothly to 0. This means that moderately large outliers are not ignored completely, and greatly improves the efficiency of the redescending M-estimator.

The redescending M-estimators are slightly more efficient than the Huber estimator for several symmetric, wider tailed distributions, but about 20% more efficient than the Huber estimator for the Cauchy distribution. This is because they completely reject gross outliers, while the Huber estimator effectively treats these the same as moderate outliers.

As other M-estimators, but unlike other outlier rejection techniques, they do not suffer from masking effects.

Disadvantages

The M-estimating equation for a redescending estimator may not have a unique solution.

 

Choosing redescending Ψ functions

When choosing a redescending Ψ function, care must be taken such that it does not descend too steeply, which may have a very bad influence on the denominator in the expression for the asymptotic variance

Redescending M-estimator - 贾智博 - 平安甜橙博客

 where F is the mixture model distribution.This effect is particularly harmful when a large negative value of ψ'(x) combines with a large positive value of ψ2(x), and there is a cluster of outliers near x.

Examples

1. Hampel's three-part M estimators have Ψ functions which are odd functions and defined for any x by:

Redescending M-estimator - 贾智博 - 平安甜橙博客
 

This function is plotted in the following figure for a=1.645, b=3 and r=6.5.

Hampel.png

2. Tukey's biweight or bisquare M estimators have Ψ functions for any positive k, which defined by:

Redescending M-estimator - 贾智博 - 平安甜橙博客
 

This function is plotted in the following figure for k=5.

Tukey.png

3. Andrew's sine wave M estimator has the following Ψ function:

Redescending M-estimator - 贾智博 - 平安甜橙博客 

This function is plotted in the following figure.

Andrew.png

  评论这张
 
阅读(7)| 评论(0)
推荐 转载

历史上的今天

评论

<#--最新日志,群博日志--> <#--推荐日志--> <#--引用记录--> <#--博主推荐--> <#--随机阅读--> <#--首页推荐--> <#--历史上的今天--> <#--被推荐日志--> <#--上一篇,下一篇--> <#-- 热度 --> <#-- 网易新闻广告 --> <#--右边模块结构--> <#--评论模块结构--> <#--引用模块结构--> <#--博主发起的投票-->
 
 
 
 
 
 
 
 
 
 
 
 
 
 

页脚

网易公司版权所有 ©1997-2018