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Least absolute deviations. Theory, applications, and algorithms. (English) Zbl 0536.62049
Progress in Probability and Statistics, Vol. 6. Boston-Basel-Stuttgart: Birkhäuser. XIV, 349 p. DM 74.00 (1983).
As opposed to the various methods based on ”the principle of arithmetic mean” for estimating the simple linear regression model $$y_ i=\alpha +\beta x_ i+u_ i$$, $$i=1,...,n$$, R. J. Boscovich proposed a ”pure objective” method in 1757. According to Boscovich’s principle the estimate of $$\beta$$ should minimize $$\sum^{n}_{i=1}|(y_ i-\bar y)-\beta(x_ i-\bar x)|$$. Denoting the estimates by a and b, respectively, the estimate of $$\alpha$$ was then found by $$a=\bar y-b\bar x$$. Here $$\bar x$$ and $$\bar y$$ denote the arithmetic mean of observations on x and y, respectively. Boscovich’s proposal is of course equivalent to what we today call the least absolute deviations (LAD) estimator constrained to pass through the mean (\=x,ȳ) of the data. For the more general linear regression model $$y_ i=\sum^{m}_{j=1} \beta_ jx_{ji}+u_ i$$, $$i=1,...,n$$, the LAD estimator of $$\beta$$ is defined as the minimizer of $$\sum^{n}_{i=1}| y_ i- \sum^{m}_{j=1}\beta_ jx_{ji}|.$$
It was early recognized that the LAD estimator was preferable over other estimators when some observations with outlying residuals appeared in the data, i.e. the residual distribution was characterized by fat tails. For example, in the well-known Théorie Analytique des Probabilités from 1818, Laplace derived the asymptotic variances of the LAD and least squares estimators of the slope parameter when the regression lines pass through the origin and when the residuals are assumed to be i.i.d. with distribution function F. Comparing the estimators on the basis of their asymptotic variances, Laplace found that the LAD estimator is preferable when $$\{2F'(0)\}^{-2}<V[U]$$. This relation has more recently been proved for the general linear regression model by G. Bassett and R. Koenker, J. Am. Stat. Assoc. 73, 618-622 (1978; Zbl 0391.62046).
One of the problems associated with the LAD estimator was the difficulty in computation. Another disadvantage was the scarcely known stochastic properties. However, the development of convex analysis and linear programming techniques have provided a fruitful tool for examining the mechanics of the LAD estimator and for the construction of efficient algorithms for its computation. Also, recent statistical research has provided asymptotic stochastic properties of the LAD estimator. This makes LAD to a useful alternative estimator, robust against outliers.
This book is a very well structured, systematic and up-to-date presentation of known properties of LAD. The material is separated into three parts, one for theory, one for applications and one for algorithms. In the first part, the uniqueness and optimality properties are discussed. Further, asymptotic theory for estimators of parameters in regression and autoregression models is given. The theory is nicely illustrated by means of Monte Carlo experiments. The second part outlines some ideas of how LAD can be used to analyze models for two-way tables and for the construction of spline functions. The main part of the book is devoted to the final part dealing with algorithmic considerations. I would also like to point out that the authors contribute with some interesting results of their own. The book is certainly of great value for all interested in LAD or in robust procedures in general and the authors deserve respectful admiration for their work.
Reviewer: H.Nyquist

##### MSC:
 62J05 Linear regression; mixed models 62-02 Research exposition (monographs, survey articles) pertaining to statistics 65C99 Probabilistic methods, stochastic differential equations 62-04 Software, source code, etc. for problems pertaining to statistics 65K05 Numerical mathematical programming methods 90C05 Linear programming 62F35 Robustness and adaptive procedures (parametric inference) 90C25 Convex programming