The productivity of top researchers: a semi-nonparametric approach

Abstract

Research productivity distributions exhibit heavy tails because it is common for a few researchers to accumulate the majority of the top publications and their corresponding citations. Measurements of this productivity are very sensitive to the field being analyzed and the distribution used. In particular, distributions such as the lognormal distribution seem to systematically underestimate the productivity of the top researchers. In this article, we propose the use of a (log)semi-nonparametric distribution (log-SNP) that nests the lognormal and captures the heavy tail of the productivity distribution through the introduction of new parameters linked to high-order moments. The application uses scientific production data on 140,971 researchers who have produced 253,634 publications in 18 fields of knowledge (O’Boyle and Aguinis in Pers Psychol 65(1):79–119, 2012) and publications in the field of finance of 330 academic institutions (Borokhovich et al. in J Finance 50(5):1691–1717, 1995), and shows that the log-SNP distribution outperforms the lognormal and provides more accurate measures for the high quantiles of the productivity distribution.

Appendices

Appendix 1

This appendix lists the first eight d _s parameters in terms of the central moments of the SNP distribution. For more information, see Del Brio and Perote (2012).

$$d_{1} = \mu_{1}$$

(18)

$$d_{2} = \frac{1}{2}\left( {\mu_{2} - 1} \right)$$

(19)

$$d_{3} = \frac{1}{6}\left( {\mu_{3} - 3\mu_{1} } \right)$$

(20)

$$d_{4} = \frac{1}{24}\left( {\mu_{4} - 6\mu_{2} + 3} \right)$$

(21)

$$d_{5} = \frac{1}{120}\left( {\mu_{5} - 10\mu_{3} + 15\mu_{1} } \right)$$

(22)

$$d_{6} = \frac{1}{720}\left( {\mu_{6} - 15\mu_{4} + 45\mu_{2} - 15} \right)$$

(23)

$$d_{7} = \frac{1}{5040}\left( {\mu_{7} - 21\mu_{5} + 105\mu_{3} - 105\mu_{1} } \right)$$

(24)

$$d_{8} = \frac{1}{40320}\left( {\mu_{8} - 28\mu_{6} + 210\mu_{4} - 420\mu_{2} + 105} \right)$$

(25)

Appendix 2

This appendix derives the cdf of the SNP distribution.

$$\begin{aligned} G_{x} \left( a \right) = & \int\limits_{ - \infty }^{a} {g\left( {x;\varvec{d}} \right)dx = \int\limits_{ - \infty }^{a} {\phi \left( x \right)dx} + \sum\limits_{s = 1}^{n} {d_{s} \int\limits_{ - \infty }^{a} {H_{s} \left( x \right)\phi \left( x \right)dx} } } \\ = & \int\limits_{ - \infty }^{a} {\phi \left( x \right)dx - \left. {\sum\limits_{s = 1}^{n} {d_{s} H_{s - 1} \left( x \right)\phi \left( x \right)} } \right|}_{ - \infty }^{a} \\ = & \int\limits_{ - \infty }^{a} {\phi \left( x \right)dx - \phi \left( a \right)\sum\limits_{s = 1}^{n} {d_{s} H_{s - 1} \left( a \right)} } \\ \end{aligned}$$

Given that $\mathop {\lim }\limits_{x \to \pm \infty } H_{s} \left( x \right)\phi \left( x \right) = 0 \quad \forall s \ge 1,$ it follows that

$$\begin{aligned} \int {H_{s} \left( x \right)\phi \left( x \right)dx} = & \int {\left( { - 1} \right)^{s} \frac{{d^{s} \phi \left( x \right)}}{{dx^{s} }}dx_{t} = \left( { - 1} \right)^{s} \frac{{d^{s - 1} \phi \left( x \right)}}{{dx^{s - 1} }}} \\ = &\, \left( { - 1} \right)^{s} \left( { - 1} \right)^{s - 1} H_{s - 1} \left( x \right)\phi \left( x \right) = - H_{s - 1} \left( x \right)\phi \left( x \right) \\ \end{aligned}$$

□