  The distribution of the maximum of a zero truncated Poisson number of i.i.d.
exponentially distributed random variables is known as a Poisson-Exponential
distribution. It arises for example as a model for monotone hazard rates, and
also as a limiting distribution for example, of a Generalized
Poisson-Exponential distribution and a scaled Poisson-Geometric distribution.
To assess these distributional approximations, we first derive a Stein equation
for the distribution of the maximum of a random number of random variables, and
detail it for Poisson-Exponential distributions. We then provide upper bounds
on the approximation errors in total variation distance when approximating
Poisson-Exponential distributions and a Generalized Poisson-Exponential
distribution by different Poisson-Exponential distributions. Moreover,
employing standardised Stein equations we obtain upper bounds on the bounded
Wasserstein distance when using a Poisson-Exponential distribution to
approximate a scaled Poisson-Geometric distribution. With this result we bound
the bounded Wasserstein distance between the scaled distribution of the maximum
of a random number of waiting times of the occurrence of pattern in Bernoulli
sequences and a Poisson-Exponential distribution.
