Bias of two estimators

Asked Dec 05 '15 at 21:01

Active Dec 06 '15 at 15:23

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I hope someone can help me. I have some trouble calculating the bias of two estimators.Unluckily it is really urgent because I hold a presentation next week. The topic is nonparametric local regression. In order to compare kernel estimators I have to calculate the bias of the following two estimators: $m_1(x,h)=n^{-1}f(x)^{-1}\sum_{i=0}^nK_{h}(X_i-x)Y_i$ and $m_2(x,h)=n^{-1}\sum_{i=0}^nf(X_i)^{-1}K_{h}(X_i-x)Y_i$. For the second estimator I had until now:

$$E[m_2(x,h)]=\int K_{h}(z-x)m(z)\frac{f(z)}{f(z)}dz=\int K(\frac{z-x}{h})m(z)dz=h\int K(u)m(x+uh)du$$ $$= hm(x)\int K(u)du+\frac{1}{2}h^3m´´(x)\int u^2K(u)du=hm(x)+\frac{1}{2}h^3m´´(x)\int u^2K(u)du+o(h^2)$$ Somehow I´m stuck here. I don´t know what to do. Please could someone help me!

edited Dec 06 '15 at 15:23

asked Dec 05 '15 at 21:01

user143352

1

Sorry it is my first time writing here. Is it better? – user143352 Dec 05 '15 at 21:40
I particularly dislike the factor h before m(x) – user143352 Dec 05 '15 at 22:11
It is much better, thank you. – Servaes Dec 05 '15 at 23:07
Please anyone an idea? – user143352 Dec 06 '15 at 03:52

Bias of two estimators

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