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Given a non-negative random variable $W$ and $\theta>0$, let the generalizedDickman transformation map the distribution of $W$ to that of $$ W^*=_dU^{1/\theta}(W+1), $$ where $U \sim {\cal U}[0,1]$, a uniformly distributedvariable on the unit interval, independent of $W$, and where $=_d$ denotesequality in distribution. It is well known that $W^*$ and $W$ are equal indistribution if and only if $W$ has the generalized Dickman distribution ${\calD}_\theta$. We demonstrate that the Wasserstein distance $d_1$ between $W$, anon-negative random variable with finite mean, and $D_\theta$ havingdistribution ${\cal D}_\theta$ obeys the inequality $$ d_1(W,D_\theta) \le(1+\theta)d_1(W,W^*). $$ The specialization of this bound to the case$\theta=1$ and coupling constructions yield $$ d_1(W_{n,1},D) \le \frac{8\log(n/2)+10}{n} \quad \mbox{for all $n \ge 1$, where} \quadW_{n,1}=\frac{1}{n}C_{n,1}-1, $$ and $C_{n,m}$ is the number of comparisonsmade by the Quickselect algorithm to find the $m^{th}$ smallest element of alist of $n$ distinct numbers. A similar bound holds for $m \ge 2$, and togetherrecover the results of [12] that show distributional convergence of $W_n$ tothe standard Dickman distribution in the asymptotic regime $m=o(n)$. Bydeveloping an exact expression for the expected running time $E[C_{n,m}]$,lower bounds are provided that show the rate is not improvable for all $m \not= 2$.

Non asymptotic distributional bounds for the Dickman Approximation of the running time of the Quickselect algorithm