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The lifetime of quasars can be estimated by means of their proximity zonesizes, which are regions of enhanced flux bluewards of the Lyman-$\alpha$emission line observed in the rest-frame UV spectra of high-redshift quasars,because the intergalactic gas has a finite response time to the quasars'radiation. We estimate the effective lifetime of the high-redshift quasarpopulation from the composite transmitted flux profile within the proximityzone region of a sample of $15$ quasars at $5.8\leq z\leq 6.6$ with precisesystemic redshifts, and similar luminosities, i.e. $-27.6\leqM_{1450}\leq-26.4$, and thus a similar instantaneous ionizing power. We developa Bayesian method to infer the effective lifetime from the composite spectrum,including robust estimates of various sources of uncertainty on the spectrum.We estimate an effective lifetime of the quasar population as a whole of$\log_{10}(t_{Q}/{yr}) = 5.7^{+0.5 (+0.8)}_{-0.3 (-0.5)}$ given by the medianand $68$th ($95$th) percentile of the posterior probability distribution. Whileour result is consistent with previous quasar lifetime studies, it posessignificant challenges on the current model for the growth of supermassiveblack holes (SMBHs) located in the center of the quasars' host galaxies, whichrequires that quasar lifetimes are more than an order of magnitude longer.

Estimating the effective lifetime of the $z\sim6$ quasar population from the composite proximity zone profile