The zero-modified logarithmic distribution with prob \(= p\)
and p0 \(= p_0\) is a discrete mixture between a
degenerate distribution at zero and a (standard) logarithmic. The
probability mass function is \(p(0) = p_0\) and
$$%
p(x) = (1-p_0) f(x)$$
for \(x = 1, 2, \ldots\), \(0 < p < 1\) and \(0 \le
p_0 \le 1\), where \(f(x)\) is the probability mass
function of the logarithmic.
The cumulative distribution function is
$$P(x) = p_0 + (1 - p_0) F(x)$$
The special case p0 == 0 is the standard logarithmic.
The zero-modified logarithmic distribution is the limiting case of the
zero-modified negative binomial distribution with size
parameter equal to \(0\). Note that in this context, parameter
prob generally corresponds to the probability of failure
of the zero-truncated negative binomial.
If an element of x is not integer, the result of
dzmlogarithmic is zero, with a warning.
The quantile is defined as the smallest value \(x\) such that
\(F(x) \ge p\), where \(F\) is the distribution function.