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Concept# Average absolute deviation

Summary

The average absolute deviation (AAD) of a data set is the average of the absolute deviations from a central point. It is a summary statistic of statistical dispersion or variability. In the general form, the central point can be a mean, median, mode, or the result of any other measure of central tendency or any reference value related to the given data set.
AAD includes the mean absolute deviation and the median absolute deviation (both abbreviated as MAD).
Measures of dispersion
Several measures of statistical dispersion are defined in terms of the absolute deviation.
The term "average absolute deviation" does not uniquely identify a measure of statistical dispersion, as there are several measures that can be used to measure absolute deviations, and there are several measures of central tendency that can be used as well. Thus, to uniquely identify the absolute deviation it is necessary to specify both the measure of deviation and the measure of central tendency. The stat

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Generalized Linear Models have become a commonly used tool of data analysis. Such models are used to fit regressions for univariate responses with normal, gamma, binomial or Poisson distribution. Maximum likelihood is generally applied as fitting method. In the usual regression setting the least absolute-deviations estimator (L1-norm) is a popular alternative to least squares (L2-norm) because of its simplicity and its robustness properties. In the first part of this thesis we examine the question of how much of these robustness features carry over to the setting of generalized linear models. We study a robust procedure based on the minimum absolute deviation estimator of Morgenthaler (1992), the Lq quasi-likelihood when q = 1. In particular, we investigate the influence function of these estimates and we compare their sensitivity to that of the maximum likelihood estimate. Furthermore we particularly explore the Lq quasi-likelihood estimates in binary regression. These estimates are difficult to compute. We derive a simpler estimator, which has a similar form as the Lq quasi-likelihood estimate. The resulting estimating equation consists in a simple modification of the familiar maximum likelihood equation with the weights wq(μ). This presents an improvement compared to other robust estimates discussed in the literature that typically have weights, which depend on the couple (xi, yi) rather than on μi = h(xiT β) alone. Finally, we generalize this estimator to Poisson regression. The resulting estimating equation is a weighted maximum likelihood with weights that depend on μ only.

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Robust procedures increase the reliability of the results of a data analysis. We studied such a robust procedure for binary regression models based on the criterion of least absolute deviation. The resulting estimating equation consists in a simple modification of the familiar maximum likelihood equation. This estimator is easy to compute with existing computational procedures and gives a high degree of protection. (C) 2010 Elsevier B.V. All rights reserved.

The present study proposes a validation of a mathematical index Q able to identify optimal geographic places for economic activities, solely based on the location variable. This research work takes its roots in the 1970s with the statistical analysis of spatial patterns, or analysis of point processes, whose main goal is to understand if a resulting spatial distribution of points is due to chance or not. Indeed point objects are commonplace (towns in regions, plants in the landscape, galaxies in space, shops in towns) and the development of specific mathematical tools are useful to understand their own location processes. Spatial point deviations from purely random configurations may be analyzed either by quadrat or by distance methods. An interesting method of the second category – the cumulative function M – was developed recently for evaluating the relative geographic concentration and co-location of industries in a nonhomogeneous spatial framework. On this basis, and having quantified retail store interactions, The French physicist Pablo Jensen elaborated the Q-index to automatically detect promising locations. To test the relevance of this quality index, Jensen used location data from 2003 and 2005 for bakeries in the city of Lyon and discovered that between these two years, shops having closed were located on significantly lower quality sites. Here, using bankruptcy data provided by the Registrar of companies of the State of Valais in Switzerland and by the City Council of Glasgow in Scotland, we implemented a method based on univariate logistic regressions to systematically test for the relevance of the Q-index on the many commercial categories available. We show that the Q-index is reliable, although significance tests did not reach stringent levels. Access to trustable bankruptcy data remains a difficult task.

2011