It is well-known that the negative binomial distribution has the unique mode [(r-1)/p]+1, if (r-1)/p is not an integer, and two modes, (r-1)/p and (r-1)/p+1, if (r-1)/p is an integer. In their paper "A note on the modes of the negative binomial distribution of order k, type I," Georghiou, Philippou and Psillakis (2017) derived a formula for the modes, say m(k; r, p), when p = 1/2, and k, r = 2, 3, .... They also found that m(k; 1, p) = k, and m(k; r, p) = kr for k, r = 2, 3, 4, 5, and p = 0.90, 0,95, 0,99.
The modes are unknown for all other cases.
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