Consider a window function and a density estimate Approximate this estimate by factoring the window function and expanding the factor in a Taylor series about the origin as follows: (a) Show that...

Consider a window function

and a density estimate

Approximate this estimate by factoring the window function and expanding the factor

in a Taylor series about the origin as follows:

(a) Show that in terms of the normalized variable u = x/h_n
the m-term approximation is given by

Where

(b) Suppose that the n samples happen to be extremely tightly clustered about u = u₀. Show that the two-term approximation peaks at the two points where

(c) Show that one peak occurs approximately at u = u₀, as desired, if u₀
‑1, but that it moves only to

(d) Confirm your answer to part (c) by plotting p_n2(u) versus u for u₀
= 0.01, 1, and 10.