The variance of the coupon collection time T_m(N) for m sets is uniquely minimized at the uniform probability vector, with a stronger radial monotonicity property proven exactly via Poissonization and Gamma reverse hazard monotonicity.
Yu, Minimum variance in the coupon collector’s problem,Journal of Applied Probability 54(2017), 655–656
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Terminal Defects, Growing Multiplicity, and Variance Extremality in the Double Dixie Cup Problem
The variance of the coupon collection time T_m(N) for m sets is uniquely minimized at the uniform probability vector, with a stronger radial monotonicity property proven exactly via Poissonization and Gamma reverse hazard monotonicity.