Ng Cheuk Fai

Ng Cheuk Fai

PhD candidate in Economics

Absolute Continuities

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Absolutely continuity is particularly useful in statistics because any function that is absolute continuous must be differentiable almost everywhere and satisfies the fundamental theorem of calculus. As a result, any random variable that has an absolute continuous density function will be a continuous random variable.

There are various types of continuties and their relative restrictivenesses vary. Below summarizes this relationship.

\[\begin{aligned} \text{lipschitz cont.}\subseteq\text{absolute cont.}\subseteq \text{uniformly cont.} \subseteq \text{continuous} \end{aligned}\]
Reference

https://en.wikipedia.org/wiki/Absolute_continuity

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Below is a plot of \(D(r)\).

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