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I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector of unknown parameters. I need to esimate vector of unknown parameters $\theta$ of characteristic function.

I tried to find a PDF using the inverse Fourier transform to use the maximum likelihood method, but the characteristic function is too complicated for that. I also thought about building the empirical characteristic function using the time series of assets and to estimate parameters using the least square method, but I do not know how to build the empirical characteristic function, because time series is not just a sample of random variables, it is a random process that depends on time.

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  • $\begingroup$ Is this question about Mathematica the software, or a general mathematics question? $\endgroup$ – Mr.Wizard Dec 11 '13 at 19:51
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    $\begingroup$ It is a general question, but I'm triyng to solve it using Mathematica. $\endgroup$ – Dmitry Pavliv Dec 11 '13 at 20:16
  • $\begingroup$ Could you add a specific small example, possibly a simplified version of your actual problem ? $\endgroup$ – b.gates.you.know.what Dec 12 '13 at 10:02
  • $\begingroup$ Also asked on math.se: math.stackexchange.com/questions/602898/… $\endgroup$ – Ben Dec 12 '13 at 10:48

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