Classical least squares regression consists of minimizing the sum of the squared residuals. Many authors have produced more robust versions of this estimator by replacing the square by something else, such as the absolute value. In this article a different approach is introduced in which the sum is replaced by the median of the squared residuals. The resulting estimator can resist the effect of nearly 50% of contamination in the data. In the special case of simple regression, it corresponds to finding the narrowest strip covering half of the observations. Generalizations are possible to multivariate location, orthogonal regression, and hypothesis testing in linear models.

Least Median of Squares Regression

If household income is pooled and then allocated to maximize welfare then income under the control of mothers and fathers should have the same impact on demand. With survey data on family health and nutrition in Brazil, the equality of parental income effects is rejected. Unearned income in the hands of a mother has a bigger effect on her family's health than income under the control of a father; for child survival probabilities the effect is almost twenty times bigger. The common preference (or neoclassical) model of the household is rejected. If unearned income is measured with error and income is pooled then the ratio of maternal to paternal income effects should be the same; equality of the ratios cannot be rejected. There is also evidence for gender preference: mothers prefer to devote resources to improving the nutritional status of their daughters, fathers to sons.

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Intra-household resource allocation: an inferential approach

Let $(X, Y)$ be a pair of random variables such that $X$ is $\mathbb{R}^d$-valued and $Y$ is $\mathbb{R}^{d'}$-valued. Given a random sample $(X_1, Y_1), \cdots, (X_n, Y_n)$ from the distribution of $(X, Y)$, the conditional distribution $P^Y(\bullet \mid X)$ of $Y$ given $X$ can be estimated nonparametrically by $\hat{P}_n^Y(A \mid X) = \sum^n_1 W_{ni}(X)I_A(Y_i)$, where the weight function $W_n$ is of the form $W_{ni}(X) = W_{ni}(X, X_1, \cdots, X_n), 1 \leqq i \leqq n$. The weight function $W_n$ is called a probability weight function if it is nonnegative and $\sum^n_1 W_{ni}(X) = 1$. Associated with $\hat{P}_n^Y(\bullet \mid X)$ in a natural way are nonparametric estimators of conditional expectations, variances, covariances, standard deviations, correlations and quantiles and nonparametric approximate Bayes rules in prediction and multiple classification problems. Consistency of a sequence $\{W_n\}$ of weight functions is defined and sufficient conditions for consistency are obtained. When applied to sequences of probability weight functions, these conditions are both necessary and sufficient. Consistent sequences of probability weight functions defined in terms of nearest neighbors are constructed. The results are applied to verify the consistency of the estimators of the various quantities discussed above and the consistency in Bayes risk of the approximate Bayes rules.

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Consistent Nonparametric Regression

There are at least two reasons why robust regression techniques are useful tools in robust time series analysis. First of all, one often wants to estimate autoregressive parameters in a robust way, and secondly, one sometimes has to fit a linear or nonlinear trend to a time series. In this paper we shall develop a class of methods for robust regression, and briefly comment on their use in time series. These new estimators are introduced because of their invulnerability to large fractions of contaminated data. We propose to call them “S-estimators” because they are based on estimators of scale.

Robust regression by means of s-estimators

It is a matter of common experience that ore values often do not follow the normal (or lognormal) distributions assumed for them, but, instead, follow some other heavier-tailed distribution. In this paper we discuss the robust estimation of the variogram when the distribution is normal-like in the central region but heavier than normal in the tails. It is shown that the use of a fourth-root transformation with or without the use of M-estimation yields stable robust estimates of the variogram.

Robust estimation of the variogram: I

An appealing approach to the problem of estimating the regression coefficients in a linear model is to find those values of the coefficients which make the residuals as small as possible. We give some measures of the dispersion of a set of numbers, and define our estimates as those values of the parameters which minimize the dispersion of the residuals. We consider dispersion measures which are certain linear combinations of the ordered residuals. We show that the estimates derived from them are asymptotically equivalent to estimates recently proposed by Jureckova. In the case of a single parameter, we show that our estimate is a "weighted median" of the pairwise slopes $(Y_j - Y_i)/(c^j - c^i)$.

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Estimating Regression Coefficients by Minimizing the Dispersion of the Residuals

The problem of finding location estimators which are "robust" against deviations from normality has received increasing attention in the last several years. See, for example, Tukey (1960), Huber (1968), and papers cited therein. In the theoretical work done on the estimation of a location parameter, the underlying distribution is usually assumed to be symmetric, and the estimand is taken to be the center of symmetry, a natural quantity to estimate in this situation. Since the finite sample size properties of many proposed estimators are difficult to study analytically, most research has focussed on their more easily ascertainable asymptotic properties, which, it is hoped, will provide useful approximations to the finite sample size case. Most of the estimators commonly studied are, under suitable regularity conditions, asymptotically normal about the center of symmetry, with asymptotic variance depending on the underlying distribution. We thus have a simple criterion, the asymptotic variance, for comparing the performance of different estimators for a given underlying distribution, and of a given estimator for different underlying distributions. Huber (1964) has formulated and solved some minimax problems, in which the estimators are judged by their asymptotic variance. In Section 2 we define and state the asymptotic variances which have been found for the three most commonly studied types of location estimators. In Section 3 we demonstrate some relationships among the three types of estimators, and in Section 4 we show that Huber's minimax result applies to all three types. Then, in Section 5 we consider an aspect of the more general estimation problem in which the distributions are not assumed symmetric. A model of asymmetric contamination of a symmetric distribution is formulated, in which the amount of asymmetry tends to zero as the sample size increases. The estimators here are thought of as estimating the center of the symmetric component of the distribution. The maximum likelihood type estimators are shown to be asymptotically normal under this model, but with a bias that tends to zero as the sample size increases. The estimators may be judged by their asymptotic mean squared error, a concept which is made meaningful by the model. We conclude in Section 6 with a minimax result analogous to Huber's, for which we allow both symmetric and asymmetric contamination of a given distribution and judge the estimators by their asymptotic mean squared error.

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Robust Estimates of Location: Symmetry and Asymmetric Contamination

This paper considers two procedures for estimating the center of a symmetric distribution, which use the observations themselves to choose the form of the estimator. Both procedures begin with a family of possible estimators. We use the observations to estimate the asymptotic variance of each member of the family of estimators. We then choose the estimator in the family with smallest estimated asymptotic variance and use the value given by that estimator as the location estimate. These procedures are shown to be asymptotically as good as knowing beforehand which estimator in the family is best for the given distribution, and using that estimator.

/pdf/some-flexible-estimates-of-location-27hn206knc.pdf

Louis A. Jaeckel

Papers

Estimating Regression Coefficients by Minimizing the Dispersion of the Residuals

Robust Estimates of Location: Symmetry and Asymmetric Contamination

Some Flexible Estimates of Location