A maximum likelihood method is used to deal with the combined estimation of multi-measurements of a branching ratio, where each result can be presented as an upper limit. The joint likelihood function is constructed using observed spectra of all meas
urements and the combined estimate of the branching ratio is obtained by maximizing the joint likelihood function. The Bayesian credible interval, or upper limit of the combined branching ratio, is given in cases both with and without inclusion of systematic error.
We study $D^0overline{D}{}^0$ pairs produced in $e^+e^-$ collisions at $sqrt{s}=3.773$ GeV using a data sample of 2.92 fb$^{-1}$ collected with the BESIII detector. We measure the asymmetry $mathcal{A}^{CP}_{Kpi}$ of the branching fractions of $D to
K^-pi^+$ in $CP$-odd and $CP$-even eigenstates to be $(12.7pm1.3pm0.7)times10^{-2}$. $mathcal{A}^{CP}_{Kpi}$ can be used to extract the strong phase difference $delta_{Kpi}$ between the doubly Cabibbo-suppressed process $overline{D}{}^{0}to K^-pi^+$ and the Cabibbo-favored process $D^0to K^- pi^+$. Using world-average values of external parameters, we obtain $cosdelta_{Kpi} = 1.02pm0.11pm0.06pm0.01$. Here, the first and second uncertainties are statistical and systematic, respectively, while the third uncertainty arises from the external parameters. This is the most precise measurement of $delta_{Kpi}$ to date.
