Estimation of parameters in a partially whitened representation of a stochastic process

For a process which may not obey a stochastic linear difference equation (SDE) excited by white noise (i.e., ARMA equation), we will develop an SDE of prespecified order ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,m</tex> ), excited by an "approximate white" noise. (The sense of approximation will be made precise in the text.) The corresponding representation is called the partially whitened representation (PWR) of order ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,m</tex> ) for the process. A recursive method of parameter estimation is presented, and the accuracy of the estimates will be determined by analytical and simulation methods. The algorithm will asymptotically converge to the vector of coefficients of the ARMA ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,m</tex> ) equation for the process, provided the process obeys such an equation. Otherwise, the algorithm will converge to the vector of coefficients of the PWR of order ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,m</tex> ) for the process.