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In parametric statistics, the goal is to estimate the parameter of a population assuming that we know the form of the density of the population. What happens if we do not know the form of the density of the population? In all practical problems, we do not know the form of the population density. Can someone give me an example of a problem where we know the form of the density of the population and a problem where we do not know the form of the density of the population?

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non-parametric statistics? en.wikipedia.org/wiki/Non-parametric_statistics –  GEdgar Oct 15 '11 at 21:28
    
Look up "kernel density estimation" and "orthogonal series density estimation". –  Emre Oct 16 '11 at 4:17

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It's generally harder to give examples of situations where we know the density in question, and just need to estimate a parameter. This is because probability densities are in some sense just a mathematical idealisation, and don't exist in the real world.

Perhaps one example is the German Tank Problem. Here one knows that the samples come from a discrete uniform distribution $\{1,2,\dots,N\}$ and one wishes to estimate $N$.

There are many 'in-between' problems. For example, we don't know the density of heights of UK males aged 18-49, though it can be well approximated by a normal distribution. We don't know the density of times between cars passing on a certain stretch of road between 10am and 11am, though it can be well approximated by an exponential distribution.

Problems where we do not know the form of the density at all are much more common. For example:

  • The density of earthquake magnitudes
  • The density of the size and frequency of stock market crashes
  • The density of length, cost and number of casualties in wars

Frequently (and unfortunately) problems where we don't even know an approximate density are the most important ones to solve!

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