A tale of two tails: peakedness properties in inheritance models of evolutionary theory
Tóm tắt
In this paper, we study transmission of traits through generations in multifactorial inheritance models with sex- and time-dependent heritability. We further analyze the implications of these models under heavy-tailedness of traits’ distributions. Among other results, we show that in the case of a trait (for instance, a medical or behavioral disorder or a phenotype with significant heritability affecting human capital in an economy) with not very thick-tailed initial density, the trait distribution becomes increasingly more peaked, that is, increasingly more concentrated and unequally spread, with time. But these patterns are reversed for traits with sufficiently heavy-tailed initial distributions (e.g., a medical or behavioral disorder for which there is no strongly expressed risk group or a relatively equally distributed ability with significant genetic influence). Such traits’ distributions become less peaked over time and increasingly more spread in the population. The proof of the results in the paper is based on the general results on majorization properties of heavy-tailed distributions obtained recently in Ibragimov (Econom Theory 23: 501–517, 2007) and also presented in the author’s Ph.D. dissertation (Ibragimov, New majorization theory in economics and martingale convergence results in econometrics. Yale University, 2005) and several their extensions derived in this work.
Tài liệu tham khảo
Adler RJ, Feldman RE, Gallagher C (1998) Analysing stable time series. In: Adler RJ, Feldman RE, Taqqu MS (eds) A practical guide to heavy tails. Statistical techniques and applications. Birkhauser, Boston, MA, pp 133–158
Allison DB, Fernandez JR, Heo M, Beasley TM (2000) Testing the robustness of the new Haseman–Elston quantitative-trait loci-mapping procedure. Am J Hum Genet 67:249–252
Allison DB, Neale MC, Zannolli R, Schork NJ, Amos CI, Blangero J (1999) Testing the robustness of the likelihood-ratio test in a variance-component quantitative-trait loci-mapping procedure. Am J Hum Genet 65:531–544
An MY (1998) Logconcavity versus logconvexity: a complete characterization. J Econ Theory 80:350–369
Becker GS (1974) A theory of social interactions. J Polit Econ 82:1063–1091
Becker GS (1976) Altruism, egoism and genetic fitness: economics and sociobiology. J Econ Lit 14:817–826
Becker GS (1993) Human capital: a theoretical and empirical analysis, with special reference to education. Chicago University Press, Chicago, IL
Bergstrom TC (1995) On the evolution of altruistic ethical rules for siblings. Am Econ Rev 85:58–81
Bergstrom TC (2002) Evolution of social behavior: individual and group selection. J Econ Perspect 16:67–88
Bernheim BD, Ray D (1987) Economic growth with intergenerational altruism. Rev Econ Stud 54:227–241
Birnbaum ZW (1948) On random variables with comparable peakedness. Ann Math Stat 19:76–81
Bull JJ (1981) Evolution of environmental sex determination from genotypic sex determination. Heredity 47:173–184
Bull JJ, Charnov EL (1989) Enigmatic reptilian sex ratios. Evolution 43:1561–1566
Bulmer MG (1980) The mathematical theory of quantitative genetics. Oxford University Press, New York, NY
Bulmer MG, Bull JJ (1982) Models in polygenic sex determination and sex ratio evolution. Evolution 36:13–26
Cherfas J, Gribbin J (1985) The redundant male. Pantheon Books, New York, NY
Currie J (2000) Child health in developed countries. In: Culyer AJ, Newhouse JP (eds) Handbook of health economics, vol 1B. Elsevier, New York, NY, pp 1053–1139
Dharmadhikari SW, Joag-Dev K (1988) Unimodality, convexity and applications. Academic, Boston, MA
Ehrman L, Parsons PA (1977) The genetics of behavior. Sinauer, Sunderland, MA
Frank RG, McGuire TG (2000) Economics and mental health. In: Culyer AJ, Newhouse JP (eds) Handbook of health economics, vol 1B. Elsevier, New York, NY, pp 892–954
Fuller JL, Simmel EC (1983) (eds) Behavior genetics: principles and applications. Lawrence Erlbaum, Hillsdale, NJ
Galton F (1886) Regression towards mediocrity in hereditary stature. J Anthropol Inst GB Irel 15:246–263
Gilger JW (2000) Contributions and promise of human behavioral genetics. Hum Biol 72:229–255
Goldberger AS (1989) Economic and mechanical models of intergenerational transmission. Am Econ Rev 79:504–513
Haveman R, Wolfe B (2000) The economics of disability and disability policy. In: Culyer AJ, Newhouse JP (eds) Handbook of health economics, vol 1B. Elsevier, New York, NY, pp 995–1051
Hedges LV, Nowell A (1995) Sex differences in mental test scores, variability, and numbers of high-scoring individuals. Science 269:41–45
Ibragimov R (2005) New majorization theory in economics and martingale convergence results in econometrics. Ph. D. dissertation, Yale University
Ibragimov R (2007) Efficiency of linear estimators under heavy-tailedness: convolutions of α-symmetric distributions. Econom Theory 23:501–517
Janzen FJ, Paukstis GL (1991) Environmental sex determination in reptiles: ecology, evolution, and experimental design. Q Rev Biol 66:149–179
Jovanovic B, Rob RR (1987) Demand-driven innovation and spatial competition over time. Rev Econ Stud 54:63–72
Kahneman D, Knetsch JL, Thaler R (1986) Fairness as a constraint on profit seeking: entitlements in the market. Am Econ Rev 76:728–741
Karlin S (1984) Mathematical models, problems, and controversies of evolutionary theory. Bull, New Ser, Am Math Soc 10: 221–274
Karlin S (1992) Stochastic comparisons between means and medians for i.i.d. random variables. In: Mardia KV (ed) The art of statistical science. A tribute to GS Watson. Wiley, Chichester, UK
Karlin S, Lessard S (1986) Theoretical studies on sex ratio evolution. Princeton University Press, Princeton, NJ
Lichtenstein P, Holm NV, Verkasalo PK, Iliadou A, Kaprio J, Koskenvuo M, Pukkala E, Skytthe A, Hemminki K (2000) Environmental and heritable factors in the causation of cancer analyses of cohorts of twins from Sweden, Denmark, and Finland. New Engl J Med 343:78–85
Marshall AW, Olkin I (1979) Inequalities: theory of majorization and its applications. Academic, New York, NY
Micceri T (1989) The unicorn, the normal curve, and other improbable creatures. Psychol Bull 105:156–166
Mulligan CB (1999) Galton versus the human capital approach to inheritance. J Polit Econ 107: S184–S224
Peters HE (1992) Patterns of intergenerational mobility in income and earnings. Rev Econ Stat 74:456–466
Plomin R, deFries JC, McClearn GE (1990) Behavioral genetics: a primer. Freeman, New York, NY
Plomin R, Owen MJ, McGuffin P (1994) The genetic basis of complex behaviors. Science 264: 1733–1739
Proschan F (1965) Peakedness of distributions of convex combinations. Ann Math Stat 36:1703–1706
Risch N (2001) The genetic epidemiology of cancer: interpreting family and twin studies and their implications for molecular genetic approaches. Cancer Epidemiol Biomark Prev 10:733–741
Robson AJ (1995) The evolution of strategic behavior. Can J Econ Revue canadienne d’economique 28:17–41
Robson AJ (2002) Evolution and human nature. J Econ Perspect 16:89–106
Roff DA (1997) Evolutionary quantitative genetics. Chapman & Hall, New York, NY
Roughgarden J (1979) Theory of population genetics and evolutionary ecology. An introduction. Macmillan, New York, NY
Rowe DC (1994) The limits of family influence: genes, experience and behavior. Guilford, New York, NY
Samuelson PA (1993) Altruism as a problem involving group versus individual selection in economics and biology. Am Econ Rev 83:143–148
Shaked M, Shanthikumar JG (1994). Stochastic orders and their applications. Probability and Mathematical Statistics. Academic, Boston, MA
Simon HA (1990) Mechanism for social selection and successful altruism. Science 250:1665–1668
Simon HA (1993) Altruism and economics. Am Econ Rev 83:156–161
Zak PJ (2002) Genetics, family structure, and economic growth. J Evol Econ 12:343–365
Zimmerman DJ (1992) Regression toward mediocrity in economic stature. Am Econ Rev 82:409–429
Zolotarev VM (1986) One-dimensional stable distributions. American Mathematical Society, Providence