Fourier Transform by Distribution Function in Statistics

Abstract

The Fourier transform is one of the most important methods, which has the ability to transform complex integral equations into simple algebraic equations and is frequently used in both mathematics and statistics. Although the Fourier transform is valid for every function in mathematics under certain conditions, this situation can become more complicated in statistics because of the fact that statistics are very different from mathematics. While in statistics, different observation values, that is, different x values, are considered for each situation, in mathematics for each x a function is defined. Because in statistics, random variables are concerned rather than functions, and the density functions of the observed values of interest should also be known. In statistics, it is seen that the Fourier transform is used in non-parametric models in which asymptotic properties are examined. In the Fourier transform, which can be performed using both distribution and density functions, it is not possible to use the density function when there are unknown or non-integrable density functions or very slow convergence rate (considering asymptotic properties). In such cases, it would be more appropriate to perform the Fourier transform with the distribution function. In this study, suggestions are presented on under which conditions it would be more appropriate to perform the Fourier transform with the distribution function.