ロバストなべき分布の指数の推定

http://www.wolframalpha.com/input/?i=int%5Blog%28x%29*x%5E%28-alpha%29%2Cx%5D
$\int \log(x) x^{-\alpha} dx=-\frac{x^{1-\alpha}\{(\alpha-1)\log(x)+1\}}{(\alpha-1)^2}+C$
$f(x;\theta)=\alpha x_{min} x^{\alpha-1}$
$s(x;\theta)=\frac{d}{d\alpha}\log(f(x;\alpha))=\frac{d}{d\alpha}\log(\alpha \cdot xmi{n}^{\alpha} \cdot x^{\alpha-1})$
$=1/|\alpha|+\log(x_{min})-\log(x)$

0= $\frac{1}{n} \sum^{n}_{i=1}f(x_i;\theta)^{\beta}s(x_i;\theta)-\int_{x_{min}}^{\infty} f(x;\theta)^{1+\beta}s(x;\theta)dx$

$\int f(x;\theta)^{1+\beta}s(x;\theta)dx=c^{1+\beta}(J_1+J_2)$ +C
$J_1=(\frac{1}{\alpha}+\log(x_{min}))\cdot x^{-q+1} \cdot \frac{1}{1-q}$
$J_2=\frac{x^{1-q}((q-1)\log(x)+1)}{(q-1)^2}$
$c=\alpha\cdot x_{min}^\alpha$
$q=(\alpha+1)(1+\beta)$

Robust parameter estimation with a small bias against heavy contamination
Hironori Fujisawa, , Shinto Eguchi

http://www.sciencedirect.com/science/article/pii/S0047259X08000456
http://www.eecs.berkeley.edu/~yang/courses/cs294-6/papers/Steward_Robust%20parameter%20estimation%20in%20computer%20vision.pdf


 fpower13<-function(alpha){
    alpha*xmin^alpha*(xxx)^(-alpha-1)
 }

 s1x<-function(alpha){
    1/alpha+log(xmin)-log(xxx)

 }

 ssx<-function(alpha){
	c<-alpha*xmin^alpha
      q<-(alpha+1)*(1+beta)
      j1<--(1/alpha+log(xmin))*xmin^(-q+1)*(1/(1-q))
      j2<-xmin^(1-q)*((q-1)*log(xmin)+1)/(q-1)/(q-1)
      c^(1+beta)*(j1-j2)
 }

 likeli<-function(alpha){
	(mean(fpower13(alpha)^beta*s1x(alpha))-ssx(alpha))^2

 }
 beta<-1
 xxx<-runif(10000)^(-1/0.8)
 xmin<-1
 optimize(likeli,c(0,5))