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May 16th, 2016, 12:05 PM   #1
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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

$\displaystyle p(|x|)\approx p(|x_0|)+\tfrac{1}{2}\lbrace{p'(|x_0|)/|x_0|)\rbrace}(x^2-x_0^2)$,
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!
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