A modified gamma/Gompertz/NBD model for estimating technology lifetime

Abstract

For efficient research and development (R&D) management, estimating the economic value of patents is becoming increasingly necessary. When estimating the economic value of patents, technology lifetime is one of the most important factors to be considered. The Pareto/non-negative binomial distribution (NBD) model is a stochastic model that can estimate the technology lifetime based on patent citation data. However, the Pareto/NBD model has some limitations. First, the model assumes that the technology of a patent is active until it is cited by another patent even though the cited patent is expired. Second, the probability distribution of the technology lifetime for a patent group always has a mode of zero, which implies that patent technologies are immediately replaced by other technologies. To address these issues, we propose a more generalized method that estimates the technology lifetime of a patent based on a modified gamma Gompertz with NBD (G/G/NBD) model. We apply the proposed methodology to estimate the lifetime of US patents in three communication-related technology areas. The case study and sensitivity analysis showed reliable estimates by the proposed methodology, where technology lifetimes were estimated within the patents’ term based on the proposed model while the existing model often resulted in their estimated lifetime being greater than the patent term.

Appendix: estimation of mean, mode, and median of the technology lifetime

In this Appendix, the mean, mode, and median of the technology lifetime distribution are estimated based on the G/G/NBD model. First, the mean value of a probability distribution is a natural way to estimate the lifetime. The mean lifetime of the patent group is obtained by computing the expectation using the pdf in Eq. (3), as in (Bemmaor & Glady, 2012):

$$\mathrm{mean}\left(\tau |\widehat{b},\widehat{s},\widehat{\beta }\right)=\left\{\begin{array}{l}\frac{1}{\widehat{b}\widehat{s}}{{}_{2}F}_{1}\left(\widehat{s},1;\;\widehat{s}+1;\frac{\widehat{\beta }-1}{\widehat{\beta }}\right), \quad {\text{for}} \;\widehat{b}, \widehat{s} > 0\, \,and \,\,\widehat{\beta }\ne 1 \\ \frac{1}{\widehat{b}}\left[\frac{\widehat{\beta }}{\widehat{\beta }-1}\right]\mathrm{ln}\left(\widehat{\beta }\right), \qquad \qquad {\text{for}} \; \widehat{b} > 0,\;\widehat{s}=1\, and \,\widehat{\beta }\ne 1\\ \frac{1}{\widehat{b}\widehat{s}}, \qquad \qquad \qquad \qquad \quad\;\; {\text{for}} \;\widehat{b}, \widehat{s} > 0\; and \;\widehat{\beta }=1\end{array}\right.$$

where ${{}_{2}F}_{1}\left(\widehat{s},1;\widehat{s}+1;\frac{\widehat{\beta }-1}{\widehat{\beta }}\right)$ denotes a Gaussian hypergeometric function and $\hat{b}$, $\hat{s}$, and $\hat{\beta}$ denote the estimated parameters. In addition, we can estimate the mode of the technology lifetimes at which the probability of a patent in the group being inactive is maximized. To derive the mode of the technology lifetime, we set equal to zero the first derivative of Eq. (3) with respect to $\tau$. The estimated mode of the technology lifetime is given by

$$\mathrm{mode}\left(\tau |\widehat{b},\widehat{s},\widehat{\beta }\right)=\left\{\begin{array}{ll}\frac{1}{\widehat{b}}\mathrm{ln}\left(\frac{\widehat{\beta }-1}{\widehat{s}}\right),& \quad \mathrm{for}\; 0<F\left({\tau }^{*}\right)\le 0.632\mathrm{\, and \,}\widehat{\beta }>\widehat{s}+1\\ 0, & \quad \mathrm{for }\;0<\widehat{\beta }\le \widehat{s}+1\end{array}\right..$$

Lastly, we derive the median lifetime of a patent group. Let $F\left(\tau |b,s,\beta \right)$ be the cumulative distribution function of the G/G model pdf in Eq. (3):

$$F\left(\tau |b,s,\beta \right)=1-{\left(\frac{\beta }{\beta -1+{e}^{b\tau }}\right)}^{s}.$$

(6)

The median lifetime of a patent group can be obtained by finding ${\tau }^{*}$ such that $F\left({\tau }^{*}|b,s,\beta \right)$ is equal to 0.5:

$$1-{\left(\frac{\beta }{\beta -1+{e}^{b\tau }}\right)}^{s}=0.5.$$

(7)

When $b$, $s>0$ and $\beta \ne 1$, there exists a unique solution because the CDF in (6) increases monotonically in $\tau$. Solving (7) for $\tau$, the median of the technology lifetime is given by:

$$\mathrm{median}\,\left(\tau |b,s,\beta \right)=\frac{1}{b}\mathrm{log}\left\{\beta \left({2}^\frac{1}{s}-1\right)+1\right\}, \quad \mathrm{for}\; b, s>0\mathrm{\, and \,}\beta \ne 1.$$

When $b$, $s>0$ and $\beta =1$, Equation (7) becomes

$$1-{e}^{-bs\tau }=0.5,$$

and the median is given by

$$\mathrm{median}\,\left(\tau |b,s,\beta \right)=\frac{1}{bs}\mathrm{log}2, \quad \mathrm{for}\ b,s>0\mathrm{\, and \,}\beta =1.$$

Finally, we obtain the median of the technology lifetimes, as shown in Equation (8).

$$\mathrm{median}\left(\tau |\widehat{b},\widehat{s},\widehat{\beta }\right)=\left\{\begin{array}{ll}\frac{1}{\widehat{b}}\mathrm{log}\left\{\widehat{\beta }\left({2}^{\frac{1}{\widehat{s}}}-1\right)+1\right\},& \mathrm{for}\; \widehat{b}, \widehat{s}>0\mathrm{\, and \,}\widehat{\beta }\ne 1\\ \frac{1}{\widehat{b}\widehat{s}}\mathrm{log}2, & \mathrm{for}\;\widehat{b}, \widehat{s}>0\mathrm{\, and \,}\widehat{\beta }=1.\end{array}\right.$$

(8)