May 16th, 2016, 12:05 PM  #1 
Member Joined: Jan 2010 Posts: 44 Thanks: 0  Local quadratic approximation
I wanted to implement some penalized regression parameter estimation algorithm by Fan&Li (http://sites.stat.psu.edu/~rli/research/penlike.pdf, section 3.3), but cannot catch the idea of some details. More specifically, I don't quite understand how local quadratic approximation is derived. Assume that $\displaystyle p(x)$ is concave penalty function s.t. $\displaystyle p(0)=0$ and $\displaystyle p$ is not differentiable at origin. It is stated that given initial value $\displaystyle x_0$ we can approximate $\displaystyle \left[p(x)\right]'=p'(x)\text{sgn}(x)\approx \lbrace{p'(x_0)/x_0\rbrace}x$, $\displaystyle (1)$ when $\displaystyle x\neq 0$ (actually it should be that both $\displaystyle x,x_0\neq 0$, shouldn't it?) and the approximation leads to $\displaystyle p(x)\approx p(x_0)+\tfrac{1}{2}\lbrace{p'(x_0)/x_0)\rbrace}(x^2x_0^2)$, for $\displaystyle x\approx x_0$. $\displaystyle (2)$ I think that $\displaystyle (1)$ follows from assumption that if $\displaystyle x\approx x_0$ then $\displaystyle p'(x)\text{sgn}(x)/x\approx p'(x_0)/x_0$. But how one gets to $\displaystyle (2)$? Why the second term in Taylor approximation is lost and how is second derivative approximated? Thanks in advance! 

Tags 
approximation, local, quadratic 
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