  We introduce innovative inference procedures for analyzing time series data.
Our methodology enables density approximation and composite hypothesis testing
based on Whittle's estimator, a widely applied M-estimator in the frequency
domain. Its core feature involves the (general Legendre transform of the)
cumulant generating function of the Whittle likelihood score, as obtained using
an approximated distribution of the periodogram ordinates. We present a testing
algorithm that significantly expands the applicability of the state-of-the-art
saddlepoint test, while maintaining the numerical accuracy of the saddlepoint
approximation. Additionally, we demonstrate connections between our findings
and three other prevalent frequency domain approaches: the bootstrap, empirical
likelihood, and exponential tilting. Numerical examples using both simulated
and real data illustrate the advantages and accuracy of our methodology.
