Optimal importance sampling for some quadratic forms of ARMA processes

The determination of the asymptotically efficient importance sampling distribution for evaluating the tail probability P(L_n>u) for large n by Monte Carlo simulations, is considered. It is assumed that L_n is the likelihood ratio statistic for the optimal detection of signal with spectral density sˆ from noise with spectral density cˆ, L_n=(2n)^-1X_n^t{T_n (cˆ)^-1IT_n(cˆ+sˆ)^-1 }X_n, cˆ and sˆ being both modeled as invertible Gaussian ARMA processes, and X_n being a vector of n consecutive samples from the noise process. By using large deviation techniques, a sufficient condition for the existence of an asymptotically efficient importance sampling ARMA process, whose coefficients are explicitly computed, is given. Moreover, it is proved that such an optimal process is unique