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

Famous quotes containing the words relation to the, relation to, relation and/or function:

    “Unaware of the absurdity of it, we introduce our own petty household rules into the economy of the universe for which the life of generations, peoples, of entire planets, has no importance in relation to the general development.”
    —Alexander Herzen (1812–1870)

    “Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
    —Honoré De Balzac (1799–1850)

    “There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
    —Adrienne Rich (b. 1929)

    “Uses are always much broader than functions, and usually far less contentious. The word function carries overtones of purpose and propriety, of concern with why something was developed rather than with how it has actually been found useful. The function of automobiles is to transport people and objects, but they are used for a variety of other purposes—as homes, offices, bedrooms, henhouses, jetties, breakwaters, even offensive weapons.”
    —Frank Smith (b. 1928)