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16 - Regression Assumptions
from Part 4
Daniel S. Scheller, Texas Tech University
Book:

Elementary Statistics for Public Administration

Published online:

01 November 2024

Print publication:

12 September 2024, pp 325-344
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Summary

Using linear regression requires assumptions that must be met.The criteria for using regression is discussed including the need for the dependent variable to be interval and to have a linear relationship with the independent variable(s).Omitting relevant variables and problems are discussed, along with explaining the importance of the error term in a regression.Detecting multicollinearity in the R Commander is illustrated, along with implications of and solutions for multicollinearity.The effects of heteroscedasticity are discussed with an illustration of it.

Primitive Points in Rational Polygons
Part of
Imre Bárány, Greg Martin, Eric Naslund, Sinai Robins
Journal:

Canadian Mathematical Bulletin / Volume 63 / Issue 4 / December 2020

Published online by Cambridge University Press:

30 January 2020, pp. 850-870

Print publication:

December 2020
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Let ${\mathcal{A}}$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t{\mathcal{A}}$ is asymptotically $\frac{6}{\unicode[STIX]{x1D70B}^{2}}\text{Area}(t{\mathcal{A}})$ as $t\rightarrow \infty$. We show that the error term is both $\unicode[STIX]{x1D6FA}_{\pm }(t\sqrt{\log \log t})$ and $O(t(\log t)^{2/3}(\log \log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler’s $\unicode[STIX]{x1D719}(n)$.

Monotonicity of the error term in Gauss-Turán quadratures for analytic function
Gradimir V. Milovanović, Miodrag M. Spalević
Journal:

The ANZIAM Journal / Volume 48 / Issue 4 / April 2007

Published online by Cambridge University Press:

17 February 2009, pp. 567-581
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For Gauss–Turán quadrature formulae with an even weight function on the interval [−1, 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than I, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some ℓ2-error estimates are considered.