  We analyze three different post-processing methods applied to a single-shot
qubit readout: the average-signal (boxcar filter), peak-signal, and
maximum-likelihood methods. In contrast to previous work, we account for a
stochastic turn-on time $t_i$ associated with the leading edge of a pulse
signaling one of the qubit states. This model is relevant to spin-qubit
readouts based on spin-to-charge conversion and would be generically reached in
the limit of large signal-to-noise ratio $r$ for several other physical
systems, including fluorescence-based readouts of ion-trap qubits and
nitrogen-vacancy center spins. We derive analytical closed-form expressions for
the conditional probability distributions associated with the peak-signal and
boxcar filters. For the boxcar filter, we find an asymptotic scaling of the
single-shot error rate $\varepsilon \sim \ln r/\sqrt{r}$ when $t_i$ is
stochastic, in contrast to the result $\varepsilon \sim \ln r/ r$ for
deterministic $t_i$. Consequently, the peak-signal method outperforms the
boxcar filter significantly when $t_i$ is stochastic, but is only marginally
better for deterministic $t_i$ (a result that is consistent with the widespread
use of the boxcar filter for fluorescence-based readouts and the peak-signal
for spin-to-charge conversion). We generalize the theoretically optimal
maximum-likelihood method to stochastic $t_i$ and show numerically that a
stochastic turn-on time $t_i$ will always result in a larger single-shot error
rate. Based on this observation, we propose a general strategy to improve the
quality of single-shot readouts by forcing $t_i$ to be deterministic.
