The Open UniversitySkip to content

On point estimation of the abnormality of a Mahalanobis index

Elfadaly, Fadlalla G.; Garthwaite, Paul H. and Crawford, John R. (2016). On point estimation of the abnormality of a Mahalanobis index. Computational Statistics & Data Analysis, 99 pp. 115–130.

Full text available as:
PDF (Version of Record) - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (711kB) | Preview
DOI (Digital Object Identifier) Link:
Google Scholar: Look up in Google Scholar


Mahalanobis distance may be used as a measure of the disparity between an individual’s profile of scores and the average profile of a population of controls. The degree to which the individual’s profile is unusual can then be equated to the proportion of the population who would have a larger Mahalanobis distance than the individual. Several estimators of this proportion are examined. These include plug-in maximum likelihood estimators, medians, the posterior mean from a Bayesian probability matching prior, an estimator derived from a Taylor expansion, and two forms of polynomial approximation, one based on Bernstein polynomial and one on a quadrature method. Simulations show that some estimators, including the commonly-used plug-in maximum likelihood estimators, can have substantial bias for small or moderate sample sizes. The polynomial approximations yield estimators that have low bias, with the quadrature method marginally to be preferred over Bernstein polynomials. However, the polynomial estimators sometimes yield infeasible estimates that are outside the 0–1 range. While none of the estimators are perfectly unbiased, the median estimators match their definition; in simulations their estimates of the proportion have a median error close to zero. The standard median estimator can give unrealistically small estimates (including 0) and an adjustment is proposed that ensures estimates are always credible. This latter estimator has much to recommend it when unbiasedness is not of paramount importance, while the quadrature method is recommended when bias is the dominant issue.

Item Type: Journal Item
Copyright Holders: 2016 The Authors
ISSN: 0167-9473
Project Funding Details:
Funded Project NameProject IDFunding Body
Not SetMR/J013838/1MRC (Medical Research Council)
Extra Information: Open Access funded by Medical Research Council
Keywords: Bernstein polynomials; Mahalanobis distance; Median estimator; Plug-in maximum likelihood; Quadrature approximation; Unbiased estimation
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 49481
Depositing User: Fadlalla Elfadaly
Date Deposited: 26 May 2017 08:48
Last Modified: 16 Jun 2020 00:58
Share this page:


Altmetrics from Altmetric

Citations from Dimensions

Download history for this item

These details should be considered as only a guide to the number of downloads performed manually. Algorithmic methods have been applied in an attempt to remove automated downloads from the displayed statistics but no guarantee can be made as to the accuracy of the figures.

Actions (login may be required)

Policies | Disclaimer

© The Open University   contact the OU