Can anybody tell me about the remainder term in a taylor series expansion and derivatives of the function f(x-ht) and how this relates to the order of taylor expansion?
Hi, I am doing some work on density estimation using kernels and am looking at bias and variance in MISE to find optimal bandwidth. I have a book which deals with this but only gives an approximation to a taylor expansion. The book gives
int K(t){f(x-ht)-f(x)} dt
I need a taylor expansion of the form f(x-ht)= .....
I think it starts as f(x) - ht f'(x) + 1/2 h^2t^2f''(x)+...
I also have further questions about this as this needs to be rearranged based on assumptions about K.
Thanks
Taylor series expansion of any function f around x at is
where is the derivative of f evaluated at x.
See Wikipedia's Taylor series where their a is our x, and their x is our .
yea thanks but if the function has derivatives only of finite order, then it can be approximated by Taylor expansion i.e. by a polynomial. So if a function has only 3 derivatives it can be approximated by a cubic polynomial. Im not quite sure how far to go with this expansion and the remainder term?
Thanks in advance
It depends on the function, how small ht is, and how good of an approximation you want. I think that the question you are asking is too general. How good of an approximation do you need? There isn't a "one-size fits all answer". For example, if you are looking at
and you are going to let , then there is no need to approximate beyond the linear term. Since the higher order terms are going to have an ht term (after dividing the ht in the denominator into each term) which is going to zero as ht does.
As for the size of the remainder term see Wikipedia's Taylor's theorem
If
So is the remainder term. Then there exists a number such that
assuming that f is n times differentiable in and times differentiable in