Kernel Density Estimation - Relation To The Characteristic Function Density Estimator

Relation To The Characteristic Function Density Estimator

Given the sample (x1, x2, …, xn), it is natural to estimate the characteristic function φ(t) = E as

\hat\varphi(t) = \frac{1}{n} \sum_{j=1}^n e^{itx_j}

Knowing the characteristic function it is possible to find the corresponding probability density function through the inverse Fourier transform formula. One difficulty with applying this inversion formula is that it leads to a diverging integral since the estimate is unreliable for large t’s. To circumvent this problem, the estimator is multiplied by a damping function ψh(t) = ψ(ht), which is equal to 1 at the origin, and then falls to 0 at infinity. The “bandwidth parameter” h controls how fast we try to dampen the function . In particular when h is small, then ψh(t) will be approximately one for a large range of t’s, which means that remains practically unaltered in the most important region oft’s.

The most common choice for function ψ is either the uniform function ψ(t) = 1{−1 ≤ t ≤ 1}, which effectively means truncating the interval of integration in the inversion formula to, or the gaussian function ψ(t) = e−π t2. Once the function ψ has been chosen, the inversion formula may be applied, and the density estimator will be

\begin{align} \hat{f}(x) &= \frac{1}{2\pi} \int_{-\infty}^{+\infty} \hat\varphi(t)\psi_h(t) e^{-itx}dt = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \frac{1}{n} \sum_{j=1}^n e^{it(x_j-x)} \psi(ht) dt \\ &= \frac{1}{nh} \sum_{j=1}^n \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(ht)\frac{x-x_j}{h}} \psi(ht) d(ht) = \frac{1}{nh} \sum_{j=1}^n K\Big(\frac{x-x_j}{h}\Big), \end{align}

where K is the inverse Fourier transform of the damping function ψ. Thus the kernel density estimator coincides with the characteristic function density estimator.

Read more about this topic:  Kernel Density Estimation

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