# Multiply broken power-law densities as survival functions: An alternative to Pareto and lognormal fits

*Roman Tomaschitz*

*Physica A: Statistical Mechanics and its Applications*, 2020, vol. 541, issue C

**Abstract:**
Survival functions defined by multiply broken power-law densities are introduced to model firm-size data sets of four historical censuses reported in Montebruno et al., Physica A 523 (2019) 858. The survival functions (complementary cumulative distributions) are obtained by least-squares regression. The probability distributions are inferred from the analytic survival functions. The mean firm growth, the growth volatility and entropy evolution are calculated over a 30-year period covered by the censuses. A representation of the survival functions as single power-law density with varying exponent depending on firm size is derived. Index functions (i.e. rescaled logarithmic derivatives of survival functions) are used to demonstrate that neither Pareto power laws nor lognormal densities can accurately reproduce the tails of the firm-size distributions. Like the survival functions, the empirical rank–size relations of the four censuses also admit representation by broken power-law densities. A generating mechanism for empirically obtained broken power-law densities, based on the Fokker–Planck equation, is proposed as well.

**Keywords:** Size distribution of firms; Multi-parameter distribution; Rank–size relation; Multiply broken power law; Varying power-law index; Nonlinear least-squares regression (search for similar items in EconPapers)

**Date:** 2020

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:phsmap:v:541:y:2020:i:c:s0378437119317935

**DOI:** 10.1016/j.physa.2019.123188

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