A $Z_1^2$ framework for rotational-parameter estimation and uncertainty quantification in high-energy pulsars

We present a $Z_1^2$-based framework for estimating the spin frequency and frequency derivative of high-energy pulsars from Poisson-limited photon event lists. The key point is that the width of a coherent detection peak is not, by itself, the statistical uncertainty on the recovered rotational parameters. We develop and compare three computationally efficient estimators: segmented frequency regression, a coherent derivative scan, and a localized two-dimensional coherent fit. For sinusoidal signals, we derive the local form of the Z-squared response as a function of frequency and frequency derivative, and show that expressing the frequency at the midpoint of the observation removes the leading-order covariance between the two parameters. This gives simple uncertainty estimates in terms of the fitted peak amplitude and local widths, without requiring an exhaustive Monte Carlo simulation for each observation. We test these estimates with Monte Carlo simulations over a range of observing spans, signal strengths, grid resolutions, and good-time-interval structures, and show that the predicted uncertainties reproduce the run-to-run scatter of the recovered parameters in the tested regimes. We then apply the framework to AstroSat/LAXPC event lists for the Crab pulsar, Swift J0243.6+6124, and SAX J1808.4-3658. The results provide a practical and statistically motivated route to rotational-parameter estimation for targeted high-energy pulsar searches.