Construct the kernel estimate of f (x), for each x ∈ ℜ, by using the U(−1, 1) kernel;

Question

Construct the kernel estimate of [latex]f(x),[/latex] for each [latex]{\mathcal{x}}\in\Re,[/latex] by using the U(−1, 1) kernel; i.e., by taking
[latex]K(x)={\frac{1}{2}},\quad{\mathrm{for-1}}\leq x\leq1,\mathrm{~and~}0,\ \mathrm{otherwise.}[/latex]

Accepted Answer

Here, it is convenient to use the indicator notation, namely, [latex]K(x)\,=\,I_{[-1,1]}(x)[/latex] (where, it is recalled, [latex]I_{A}(x)=1\;\mathrm{if}\;x\in A,\;\mathrm{and}\;0\;\mathrm{if}\;x\in A^{c}).[/latex] Then the estimate (31) becomes as follows:
[latex]\hat{f}_{n}(x)=\frac{1}{n h_{n}}\sum\limits_{i=1}^{n}K\left(\frac{x-X_{i}}{h_{n}} ight).\qquad\qquad\qquad\qquad\qquad(31)[/latex]

[latex]\hat{f}_{n}(x)=\frac{1}{n h_{n}}\sum\limits_{i=1}^{n}I_{[-1,1]}\left(\frac{x-X_{i}}{h_{n}} ight),\quad x\in R.\qquad\qquad\qquad(32)[/latex]
So, [latex]I_{[-1,1]}(\frac{x-X_{i}}{h_{n}})\,=\,1,[/latex] if and only if [latex]x-h_{n}\,\leq\,X_{i}\ \leq\ x+h_{n};[/latex] in other words, in forming [latex]{\hat{f}}_{n}(x),[/latex] we use only those observations [latex]X_{i}[/latex] which lie in the window [latex][x-h_{n},\,x+h_{n}].[/latex] The breadth of this window is, clearly, determined by [latex]h_{n},[/latex] and this is the reason that [latex]h_{n}[/latex] is referred to as the bandwidth.
Usually, the minimum of assumptions required of the kernel K and the bandwidth [latex]h_{n}[/latex] in order for us to be able to establish some desirable properties of the estimate [latex]{\hat{f}}_{n}(x)[/latex] given in (31), are the following:

[latex]\left.\begin{array}{l}~~~~~~~~~~~~~~~~K\mathrm{~is~ bounded;~i.e.,~sup}\left\{K(x);x\in \Re ight\}\lt \infty.\~~~~~~~ xK(x)~\mathrm{tends~to}~0~\mathrm{as}~x o \pm \infty;\mathrm{i.e.},|xK(x)|_{\overrightarrow{|x| o \infty}}0.\ K~\mathrm{is~symmetric~about}~0;~\mathrm{i.e.},~K(-x)=K(x),~x\in R.\end{array} ight\}\qquad(33)[/latex]

[latex]\left.\begin{array}{l} \mathrm{As}~n o\infty : ~ (i)~~~(0\lt )h_n o 0\ \qquad\qquad\quad ~~ (ii) ~~~~~~ nh_n o \infty \ \qquad\qquad\quad~(iii) ~~~~~~ nh_n^2 o \infty.\end{array} ight\}\qquad(34)[/latex]

REMARK 4 Observe that requirements (33) are met for the kernel used in (32). Furthermore, the convergences in (34) are satisfied if one takes, e.g., [latex]h_{n}\,=\,n^{-\alpha}\,\mathrm{with}\,0\,\lt \,\alpha\,\lt \,1/2.[/latex] Below, we record three (asymptotic) results regarding the estimate [latex]{\hat{f}}_{n}(x)[/latex] given in (31).

Question 15.8: Construct the kernel estimate of f (x), for each x ∈ ℜ, by u......

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