Empirical density functions

Definitions

The outcome of $N$ measurements of $x$ yields the sequence $x_{i}\,,\, i=1,\ldots,\, N$ (Sample measurements). Without any additional assumptions the probability density function (PDF) is \begin{equation} \rho_{e}(x)=\frac{1}{N}\sum_{i=1}^{N}\delta(x-x_{i})\label{eq:rhoe} \end{equation} Here $\delta(x)$ is the Dirac delta function [1]. The density (\ref{eq:rhoe}) is also called a raw density function [2]. The PDF (\ref{eq:rhoe}) is obviously normalized.

The corresponding cumulative probability function is

\begin{equation} P_{e}(x)=\frac{1}{N}\sum_{i=1}^{N}H(x-x_{i}) \end{equation} Here $H(x)$ is the Heaviside step function [3]. For any function $f(x)$ this PDF gives the following average values \begin{equation} \left\langle \, f\,\right\rangle =\intop_{a}^{b}f(x)\rho_{e}(x)dx=\frac{1}{N}\sum_{i=N}^{N}f(x_{i}) \end{equation} In particular, the moments are \[ \left\langle x^{n}\right\rangle =\frac{1}{N}\sum_{i=N}^{N}x_{i}^{n} \]

Remarks

• Obviously the function (\ref{eq:rhoe}) is not smooth, but the sample measurements do not give information about smoothness.
• Anything beyond this formula is based on some assumptions, theories or other experiments.
• Eq. (\ref{eq:rhoe}) is a real non-parametric estimation of the probability density functions.

 

References

1. The Dirac delta function.
2. Kernel bandwidth optimization in spike rate estimation.
3. The Heaviside step function.