  For a unified analysis on the phase estimation, we focus on the limiting
distribution. It is shown that the limiting distribution can be given by the
absolute square of the Fourier transform of $L^2$ function whose support
belongs to $[-1,1]$. Using this relation, we study the relation between the
variance of the limiting distribution and its tail probability. As our result,
we prove that the protocol minimizing the asymptotic variance does not minimize
the tail probability. Depending on the width of interval, we derive the
estimation protocol minimizing the tail probability out of a given interval.
Such an optimal protocol is given by a prolate spheroidal wave function which
often appears in wavelet or time-limited Fourier analysis. Also, the minimum
confidence interval is derived with the framework of interval estimation that
assures a given confidence coefficient.
