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Estimating a Probability Density from Cumulative Distribution Data

Article Quant Q&A · Author: quant1
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Summary

The document explains how to obtain a probability density function from a cumulative distribution function. When the CDF is available in an analytical form, differentiate it. When it is represented by numerical data, approximate the derivative with finite differences.

For numerical estimates, it recommends a symmetric, second-order central difference: compare CDF values at points on either side of the target and divide their difference by twice the spacing. The stated truncation error is of second order in that spacing. The note gives no data, worked example, or guidance on choosing the spacing; in practice, spacing affects the balance between discretization error and sensitivity to noisy CDF estimates.

Key ideas

  • An analytical CDF can be differentiated to obtain the density of a continuous distribution.
  • For numerical CDF data, finite differences provide an approximation to the derivative.
  • A symmetric central difference has second-order truncation error in the step size.
  • The choice of step size matters, but the document does not explain how to select it.

Tags

Statistics
Full text
# Interpolation for PDF from Cumulative Distribution


# Interpolation for PDF from Cumulative Distribution












How to interpolate PDF(Probability Distribution Functions) from CDF (without root finding method) ?

Please tell the steps to do so.

Thanks.

## Answer by Tyler Olsen (score 1, accepted)

https://quant.stackexchange.com/a/21515

If you have an analytical form of the CDF, you can simply take the first derivative to obtain the PDF (for a continuous distribution). If you have numerical data points representing a CDF, you can construct a numerical approximation to the first derivative by using a finite difference method. If you're going the numerical route, you should use at least a symmetric second-order finite difference, since it's just as easy as the first-order methods.

$$ PDF(x) = \frac{CDF(x+\delta x) - CDF(x-\delta x)}{2 \,\delta x} + O(\delta x^2) $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.