Corrigendum for The Distribution of Family Sizes Under a Time-Homogeneous Birth and Death Process

Corrigendum for The Distribution of Family Sizes Under a Time-Homogeneous Birth and Death Process

PANAGIS MOSCHOPOULOS¹ AND MAX SHPAK²

¹Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, USA

²Department of Biological Sciences, University of Texas at El Paso, El Paso, Texas, USA

Address correspondence to Max Shpak, Department of Biological Sciences, University of Texas at El Paso, El Paso, Texas 79968, USA; Email: mshpak@utep.edu

Copyright © Taylor & Francis Group, LLC

Regretably, there are a number of significant errors in Moschopoulos and Shpak, (2010). Communications in Statistics—Theory and Methods 39:1761–1775. Most are typographic errors that have the potential to cause confusion, and in some instances, have carried through to several equations.

1. There should be negative exponents in Eq. (1.2):

   $$p_n(t)=e^{-\lambda t}{(1-e^{-\lambda t})}^{n-1}.$$ (1.2)
2. In the equation defining Q_n immediately above Eq. (2.5), there is a missing negative exponent in the denominator exp[−ωt]; it should be:

   $$Q_n={\int }_0^{\infty }\frac{\rho \omega exp[-(\omega +\rho )t]}{1-\theta exp[-\omega t]}{\left(\frac{1-exp[-\omega t]}{1-\theta exp[-\omega t]}\right)}^{n-1}dt.$$
3. The coefficient of 1 − θ in Eq. (2.5) (and those derived from it) cancels with the change of variable, and should not appear in the equations. Note that this did not lead to errors in the numerics because of normalization. The equations should be:

   $$Q_n=\rho {\int }_0^1{y}^{n-1}{(1-y)}^{\rho /\omega }{(1-\theta y)}^{-\rho /\omega -1}dy.$$ (2.5)

   $$q_n=\frac{\rho }{\omega }{\int }_0^1{y}^{n-1}{(1-y)}^{\rho /\omega }{(1-\theta y)}^{-\rho /\omega }dy.$$ (2.6)

   $${{}_{2}F}_1(a,b,c,\theta )=\sum _{k=0}^{\infty }\frac{{(a)}_k{(b)}_k}{{(c)}_k}\frac{{\theta }^k}{k!}$$ (2.7a)

   $$Q_n=\rho \mathrm{\Gamma }\left(1+\frac{\rho }{\omega }\right)A\sum _{k=0}^{\infty }{\left(1+\frac{\rho }{\omega }\right)}_{k}B\frac{{\theta }^k}{k!}.$$ (2.8)
   Furthermore, there is an error in the position of the bracket relative to powers of θ in (2.9), which should be

   $$\begin{array}{l}{Q}_n=\rho \mathrm{\Gamma }\left(1+\frac{\rho }{\omega }\right)n^{-1-\frac{\rho }{\omega }}\left[1-\frac{\frac{\rho }{\omega }\left(1+\frac{\rho }{\omega }\right)}{2n}+\frac{\frac{\rho }{\omega }\left(1+\frac{\rho }{\omega }\right)\left(3{\left(\frac{\rho }{\omega }\right)}^2+\frac{\rho }{\omega }\right)}{24n^2}\right]\\ \times \sum _{k=0}^{\infty }{\left(1+\frac{\rho }{\omega }\right)}_k\left[1-\frac{k}{n}(1+\frac{\rho }{\omega })+\frac{\phi (\rho ,\omega ,k)}{n^2}\right]\frac{{\theta }^k}{k!}\times \left(1+O(n^{-3})\right).\end{array}$$ (2.9)
4. The factor 1 − θ is extraneous to the asymptotic 2.11–14 as well. Note the typo in the paragraph immediately before (2.11), i.e., we have θ > 1 rather than θ < 1 in this instance. Moreover, there are a number of other errors with the indices and signs (most significantly, the equations should have θ⁻¹ in place of θ).

   $$Q_n=\rho {\int }_0^{1/\theta }{y}^{n-1}{(1-y)}^{\rho /\omega }{(1-\theta y)}^{-\rho /\omega -1}dy.$$ (2.11)

   $$Q_n=\rho {\theta }^{-n}{\int }_0^1{z}^{n-1}{\left(1-\frac{z}{\theta }\right)}^{\rho /\omega }{(1-z)}^{-\rho /\omega -1}dz.$$ (2.12)
   Note that powers of θ should be negative, since θ > 1

   $$Q_n=\rho {\theta }^{-n}\mathrm{\Gamma }\left(-\frac{\rho }{\omega }\right)A\sum _{k=0}^{\infty }{\left(-\frac{\rho }{\omega }\right)}_{k}B\frac{{\theta }^{-k}}{k!}.$$ (2.13)
   For the above distribution, the coefficients A and B are

   $$A=\frac{\mathrm{\Gamma }(n)}{\mathrm{\Gamma }\left(n-\frac{\rho }{\omega }\right)},B(\rho ,\omega ,k,n)=\frac{{(n)}_k}{{\left(n-\frac{\rho }{\omega }\right)}_k}=\frac{\mathrm{\Gamma }(n+k)\mathrm{\Gamma }(n-\frac{\rho }{\omega })}{\mathrm{\Gamma }(n)\mathrm{\Gamma }(n+k-\frac{\rho }{\omega })}.$$
   As in (2.9), there is also an error in the position of the bracket relative to the powers of θ in the expansion

   $$\begin{array}{l}{Q}_n=\rho {\theta }^{-n}\mathrm{\Gamma }\left(-\frac{\rho }{\omega }\right)n^{\frac{\rho }{\omega }}\\ \times \left\{1-\frac{\frac{\rho }{\omega }\left(\frac{\rho }{\omega }+1\right)}{2n}+\frac{\left(\frac{\rho }{\omega }-1\right)\left(\frac{\rho }{\omega }-2\right)\left[3{\left(1+\frac{\rho }{\omega }\right)}^2-\frac{\rho }{\omega }-1\right]}{24n^2}\right\}\\ \times \sum _{k=0}^{\infty }{\left(-\frac{\rho }{\omega }\right)}_k\left[1+\frac{k}{n}\left(\frac{\rho }{\omega }+1\right)+\frac{\phi (\rho ,\omega ,k)}{n^2}\right]\frac{{\theta }^{-k}}{k!}+O(n^{-3}).\end{array}$$ (2.14)
   The asymptotic approximation to this distribution, calculated for n → ∞, is:

   $$Q_n\approx \rho {\theta }^{-n}{(1-{\theta }^{-1})}^{\rho /\omega }\mathrm{\Gamma }\left(-\frac{\rho }{\omega }\right)n^{\rho /\omega }.$$ (2.15)