# Infant Mortality and Income per Capita of World Countries for 1998–2016: Analysis of Data and Modeling by Increasing Returns

## Abstract

Annual cross-country data from the World Bank database during the years 1998 and 2016 demonstrate wide variation in infant mortality rates (IMR) and gross domestic product per capita (GDPpc). The row data, a descriptive analysis of the data and a *K*-means clustering of the countries into groups depending on the GDPpc and IMR measurements draw attention to the underlying structure of the data. We find that IMR follows a convex descent trend, where there is a range of high infant mortality rates at low GDPpc levels and there is a range of high GDPpc levels with low infant mortality rates. Thus, we assume that GDPpc is subject to increasing returns. This is equivalent to assuming that IMR comes from an unknown underlying convex relationship on the IMR observations at the GDPpc observations. Hence we consider the application of a method that makes least the sum of the squares of the errors to each dataset subject to the constraints from the assumption of non-decreasing returns. Our numerical results show that the estimated IMR values reach a point at which they obtain the lowest value and then increase again while GDPpc is at the highest levels. Thus they suggest that our assumption not only adequately describes reality, but also the ensuing model is able to capture imperceptible features of the underlying process.

## Keywords

Convexity Divided difference Gross domestic product Increasing returns Infant mortality rates*K*-means clustering Least squares Quadratic programming

## References

- 1.T. Callen, Gross domestic product: an economy’s all. IMF. http://www.imf.org/external/pubs/ft/fandd/basics/gdp.htm. Accessed 10 Nov 2016
- 2.Mortality Rate, Infant, The World Bank. http://data.worldbank.org/indicator/sp.dy.imrt.in. Accessed 20 Oct 2017
- 3.I.C. Demetriou, P. Tzitziris, Infant mortality and economic growth: modeling by increasing returns and least squares. In:
*Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2017, WCE 2017*, 5–7 July 2017, London, U.K., ed. by S.I. Ao, L. Gelman, D.W.L. Hukins, A. Hunter, A.M. Korsunsky, vol. II, pp. 543–548Google Scholar - 4.C. Hildreth, Point estimates of ordinates of concave functions. J. Am. Stat. Assoc.
**49**, 598–619 (1954)MathSciNetCrossRefGoogle Scholar - 5.C. de Boor,
*A Practical Guide to Splines*, Revised Edition (Springer, New York, 2001)Google Scholar - 6.I.C. Demetriou, M.J.D. Powell, The minimum sum of squares change to univariate data that gives convexity. IMA J. Numer. Anal.
**11**, 433–448 (1991)MathSciNetCrossRefGoogle Scholar - 7.I.C. Demetriou, Algorithm 742: L2CXFT, a Fortran 77 subroutine for least squares data fitting with non-negative second divided differences. ACM Trans. Math. Softw.
**21**(1), 98–110 (1995)CrossRefGoogle Scholar - 8.L. Bertinelli,
*Urbanization and Growth*(Duncan Black, London, 2000)Google Scholar - 9.I. Chakraborty, Living standards and economic growth: a fresh look at the relationship through the nonparametric approach. Centre for Development Studies. https://opendocs.ids.ac.uk/opendocs/handle/123456789/2942. Accessed 10 Nov 2016
- 10.G. James, D. Witten, T. Hastie, R. Tibshirani,
*An Introduction to Statistical Learning with Applications in R*(Springer, Heidelberg, 2013)CrossRefGoogle Scholar