From Statistics to Mathematical Finance by Dietmar Ferger Wenceslao González Manteiga Thorsten Schmidt & Jane-Ling Wang

Author:Dietmar Ferger, Wenceslao González Manteiga, Thorsten Schmidt & Jane-Ling Wang
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

holds. In particular, for and , one has

The next result summarizes properties of the local quadratic smoother if the bandwidth is proportional to .

Theorem 2

Suppose g is bounded away from zero on I, K has a Hölder continuous second derivative and the bandwidth satisfies . Then the following hold with the first coordinate of .

(11.8)

(11.9)

and, for ,

(11.10)

Moreover, for any square-integrable functions satisfying

we have the expansion

(11.11)

Proof

Claim (11.8) is a consequence of Proposition 1 and (11.10) of Proposition 3. Statement (11.9) follows from the bounds and

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