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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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migrated from mathoverflow.net Dec 11 '13 at 20:35

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It seems that what you are asking for is actually a PhD thesis of someone! Check out this. –  Stat Dec 11 '13 at 22:37
    
Thank you! This thesis is very helpful for me! –  simoco Dec 12 '13 at 0:01

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