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Heavy-tailedness and threshold sex determination. (English) Zbl 1154.62084
Summary: This paper studies the properties of the sex ratio in two-period models of threshold (e.g., polygenic or temperature-dependent) sex determination under heavy-tailedness in the framework of possibly skewed stable distributions and their convolutions. We show that if the initial distribution of the sex determining trait in such settings is moderately heavy-tailed and has a finite first moment, then an excess of males (females) in the first period leads to the same pattern in the second period. Thus, the excess of one sex over the other one accumulates over two generations and the sex ratio in the total alive population in the second period cannot stabilize at the balanced sex ratio value of 1/2. These properties are reversed for extremely heavy-tailed initial distributions of sex determining traits with infinite first moments. In such settings, the sex ratio of the offspring oscillates around the balanced sex ratio value and an excess of males (females) in the first period leads to an excess of females (males) in the second period. In addition, the sex ratio in the total living population in the second period can stabilize at 1/2 for some extremely heavy-tailed initial distributions of the sex determining trait. The results in the paper are shown to also hold for bounded sex determining phenotypes.

##### MSC:
 62P10 Applications of statistics to biology and medical sciences 62G32 Statistics of extreme values; tail inference
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##### References:
 [1] An, M. Y.: Logconcavity versus logconvexity: A complete characterization. Journal of economic theory 80, 350-369 (1998) · Zbl 0911.90071 [2] Bacci, G.: Sex determination. (1965) [3] Beirlant, J.; Goegebeur, Y.; Segers, J.; Teugels, J.: Statistics of extremes. Theory and applications. (2004) · Zbl 1070.62036 [4] Bull, J. J.: Evolution of environmental sex determination from genotypic sex determination. Heredity 47, 173-184 (1981) [5] Bull, J. J.; Charnov, E. L.: Enigmatic reptilian sex ratios. Evolution 43, 1561-1566 (1989) [6] Bulmer, M. G.; Bull, J. J.: Models of polygenic sex determination and sex ratio control. Evolution 36, 13-26 (1982) [7] Cherfas, J.; Gribbin, J.: The redundant male. (1985) [8] Edlund, L.: Son preference, sex ratios, and marriage patterns. Journal of political economy 107, 1275-1302 (1999) [9] Embrechts, P.; Klüppelberg, C.; Mikosch, T.: Modelling extremal events for insurance and finance. (1997) [10] Gabaix, X., Ibragimov, R., 2007. Rank-1/2: a simple way to improve the OLS estimation of tail exponents. Harvard Institute of Economic Research Discussion Paper No. 2106. Available at http://www.economics.harvard.edu/faculty/ibragimov/files/LogLogRevised2.pdf [11] Grant, V. J.: Sex determination and the maternal dominance hypothesis. Human reproduction 11, 2371-2375 (1996) [12] Ibragimov, R., 2005. New majorization theory in economics and martingale convergence results in econometrics. Ph.D. dissertation, Yale University. Available at http://www.economics.harvard.edu/faculty/ibragimov/files/Dissertation.pdf [13] Ibragimov, R.: Efficiency of linear estimators under heavy-tailedness: convolutions of ${\alpha}$-symmetric distributions. Econometric theory 23, 501-517 (2007) · Zbl 1237.62042 [14] Ibragimov, R.: Thou shalt not diversity: why two of every sort?. Journal of applied probability 44, 58-70 (2007) · Zbl 1136.60010 [15] Ibragimov, R.; Walden, J.: The limits of diversification when losses May be large. Journal of banking and finance 31, 2551-2569 (2007) [16] James, W. H.: Parental dominance/social status, hormone levels, and sex ratio of offspring. Reproductive and interpersonal aspects of dominance and status 2, 63-74 (1994) [17] James, W. H.: What stabilizes the sex ratio?. Annals of human genetics 59, 243-249 (1995) [18] James, W. H.: Evidence that mammalian sex ratios at birth are partially controlled by parental hormone levels at the time of conception. Journal of theoretical biology 180, 271-286 (1996) [19] James, W. H.: A potential mechanism for sex ratio variation in mammals. Journal of theoretical biology 189, 253-255 (1997) [20] Janzen, F. J.; Paukstis, G. L.: Environmental sex determination in reptiles: ecology, evolution, and experimental design. The quarterly review of biology 66, 149-179 (1991) [21] Karlin, S.: Mathematical models, problems, and controversies of evolutionary theory. American mathematical society. Bulletin. new series 10, 221-274 (1984) · Zbl 0552.92013 [22] Karlin, S.: Stochastic comparisons between means and medians for i.i.d. Random variables. The art of statistical science. A tribute to G. S. Watson (1992) [23] Karlin, S.; Lessard, S.: Theoretical studies on sex ratio evolution. (1986) [24] Marshall, A. W.; Olkin, I.: Inequalities: theory of majorization and its applications. (1979) · Zbl 0437.26007 [25] Oster, E.: Hepatitis B and the case of the missing women. Journal of political economy 113, 1163-1216 (2005) [26] Zolotarev, V. M.: One-dimensional stable distributions. (1986) · Zbl 0589.60015