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 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 $\displaystyle p(|x|)\approx p(|x_0|)+\tfrac{1}{2}\lbrace{p'(|x_0|)/|x_0|)\rbrace}(x^2-x_0^2)$, 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! Tags approximation, local, quadratic Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post life24 Calculus 7 May 16th, 2016 03:34 PM mastermind Calculus 1 February 27th, 2015 03:14 PM Paul4763 Calculus 7 April 29th, 2011 09:54 AM remeday86 Calculus 3 April 28th, 2009 10:34 AM sds234 Calculus 1 November 5th, 2008 12:55 PM

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