Weighted Median of the Data in Solving Least Absolute Deviations Problems

Abstract

We consider the weighted median problem for a given set of data and analyze its main properties. As an illustration, an efficient method for searching for a weighted Least Absolute Deviations (LAD)-line is given, which is used as the basis for solving various linear and nonlinear LAD-problems occurring in applications. Our method is illustrated by an example of hourly natural gas consumption forecast.

Keywords:

• LAD
• Least absolute deviations
• Median
• Weighted LAD-line
• Weighted median

Mathematics Subject Classification:

• 65D10
• 65C20
• 62J05
• 90C27
• 90C56
• 90B85
• 34K29

Acknowledgment

We would like to thank an anonymous referee for useful comments and remarks, which helped us improve the article significantly.

This work was supported by the Ministry of Science, Education and Sports, Republic of Croatia, through research grant 235-2352818-1034.

Notes

Available at http://www.archive.org/details/mcaniquecles02laplrich

For subsets A, B ⊆ ℝ and real numbers α, β, and γ, we denote αA + βB: = {αa + βb: a ∈ A, b ∈ B} and A + γ: = {a + γ: a ∈ A}.