Abstract
In this case study, we would like to illustrate the utility of characteristic functions, using an example of a sample statistic defined for samples from Cauchy distribution. The derivation of the corresponding asymptotic probability density function is based on [1], elaborating and expanding the individual steps of their presentation, and including a small extension; our reason for such a plagiarism is to make the technique, its mathematical tools and ingenious arguments available to the widest possible audience.
Highlights
The problem of finding the distribution of self-normalized sum of a Cauchygenerated sample has been considered since at least 1969, but solved only in 1973 by [1], our key reference
We demonstrate how characteristic functions (CHF) are used in Statistics to find a distribution of a specific sample statistic
We do this by using a single comprehensive example, namely finding the n → ∞ limit of the probability density function (PDF) of
Summary
The problem of finding the distribution of self-normalized sum of a Cauchygenerated sample has been considered since at least 1969 (see, for example, [2]), but solved (by proposing a concrete numerical algorithm) only in 1973 by [1], our key reference. We demonstrate how characteristic functions (CHF) are used in Statistics to find a distribution of a specific sample statistic (a function of individual observations). We do this by using a single comprehensive example, namely finding the n → ∞ limit of the probability density function (PDF) of. This goal has already been achieved (in a more general setting) by [1]; our main (rather pedagogical) reason for extending their presentation is to make it accessible to graduate and advance undergraduate students. To learn more about the mathematical details of transforming PDF into CHF and back the reader may like to consult [4]
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