  Pulsar timing arrays (PTAs) have made tremendous progress and are now showing
strong evidence for the gravitational-wave background (GWB). Further probing
the origin and characteristics of the GWB will require more generalized
analysis techniques. Bayesian methods are most often used but can be
computationally expensive. On the other hand, frequentist methods, like the PTA
Optimal Statistic (OS), are more computationally efficient and can produce
results that are complementary to Bayesian methods, allowing for stronger
statistical cases to be built from a confluence of different approaches. In
this work we expand the capabilities of the OS through a technique we call the
Per-Frequency Optimal Statistic (PFOS). The PFOS removes the underlying
power-law assumption inherent in previous implementations of the OS, and allows
one to estimate the GWB spectrum in a frequency-by-frequency manner. We have
also adapted a recent generalization from the OS pipeline into the PFOS, making
it capable of accurately characterizing the spectrum in the intermediate and
strong GW signal regimes using only a small fraction of the necessary
computational resources when compared with fully-correlated Bayesian methods,
while also empowering many new types of analyses not possible before. We find
that even in the strong GW signal regime, where the GWB dominates over noise in
all frequencies, the injected value of the signal lies within the
50th-percentile of the PFOS uncertainty distribution in 41-45% of simulations,
remaining 3$\sigma$-consistent with unbiased estimation.
