After providing some background for the need to consider estimates other than those resulting from normal theory, there is a brief review of some nonadaptive robust estimators. We introduce adaptive estimators using those of Tukey and McLaughlin, Jaeckel, Johns, Birnbaum and Mike, Takeuchi, Hajek, van Eeden, and Beran. Adaptive estimators based on preliminary testing and Stein-like procedures are then considered, and recommendations are made on how to select the amount of trimming. Various proposals for estimating regression coefficients are also considered. Adaptive distrubution-free tests look very promising for improving the power of nonparametric tests, and some of these techniques can be used effectively in data analysis. Asymmetric trimmed means, adapted to the particular sample, can easily be used with data and provide good descriptive statistics having an approximate error structure. Finally, it is conjectured that estimators based on “cliff-hangers” might be extremely effective if there ...

Adaptive Robust Procedures: A Partial Review and Some Suggestions for Future Applications and Theory

In many applications, the underlying scientific question concerns whether the variances of $k$ samples are equal. There are a substantial number of tests for this problem. Many of them rely on the assumption of normality and are not robust to its violation. In 1960 Professor Howard Levene proposed a new approach to this problem by applying the $F$-test to the absolute deviations of the observations from their group means. Levene's approach is powerful and robust to nonnormality and became a very popular tool for checking the homogeneity of variances. This paper reviews the original method proposed by Levene and subsequent robust modifications. A modification of Levene-type tests to increase their power to detect monotonic trends in variances is discussed. This procedure is useful when one is concerned with an alternative of increasing or decreasing variability, for example, increasing volatility of stocks prices or "open or closed gramophones" in regression residual analysis. A major section of the paper is devoted to discussion of various scientific problems where Levene-type tests have been used, for example, economic anthropology, accuracy of medical measurements, volatility of the price of oil, studies of the consistency of jury awards in legal cases and the effect of hurricanes on ecological systems.

The Impact of Levene's Test of Equality of Variances on Statistical Theory and Practice

Nonparametric Methods in Statistics.

Parametric statistics are based on the assumption of normality. Recent findings suggest that Type I error and power can be adversely affected when data are non-normal. This paper aims to assess the distributional shape of real data by examining the values of the third and fourth central moments as a measurement of skewness and kurtosis in small samples. The analysis concerned 693 distributions with a sample size ranging from 10 to 30. Measures of cognitive ability and of other psychological variables were included. The results showed that skewness ranged between −2.49 and 2.33. The values of kurtosis ranged between −1.92 and 7.41. Considering skewness and kurtosis together the results indicated that only 5.5% of distributions were close to expected values under normality. Although extreme contamination does not seem to be very frequent, the findings are consistent with previous research suggesting that normality is not the rule with real data.

http://diposit.ub.edu/dspace/bitstream/2445/153945/1/603461.pdf

Skewness and Kurtosis in Real Data Samples

SUMMARY The paucity of distributions that can be used as an aid to modelling the structure of multivariate data is a recognized limitation. We present a relatively simple family of distributions for use in describing data which are not elliptically symmetric. Properties of the family are discussed and it is shown that multivariate Pareto, Burr and Logistic distributions are special cases. In addition, simulation methods for applications in Monte Carlo robustness studies are outlined and, finally, the fit to a data set on uranium exploration is compared to that obtained using the usual bivariate normal distribution.

A Family of Distributions for Modelling Non-Elliptically Symmetric Multivariate Data

An adaptive distribution-free test is proposed for the two-sample location problem First, the data are used to assess the tailweight and skewness of the underlying distributions This leads to the selection and then application, with the same data, of one of several common rank tests for shift, such as the Mann-Whitney-Wilcoxon test The preliminary selection is made in a way that insures the testing procedure is distribution-free A Monte Carlo study shows that the adaptive test has excellent power over a wide class of distributions and is preferable to certain prominent nonadaptive tests

A Two-Sample Adaptive Distribution-Free Test

Two new methods of constructing robust linear and quadratic discriminant functions are introduced. The first is a generalization of Fisher's procedure for finding a linear discriminant function. It places less weight on those observations that are far from the overlapping regions of the two populations. The second new method substitutes M-estimates of the means and the covariance matrices into the usual expressions for the linear and quadratic discriminant functions. Monte Carlo results indicate lower misclassification probabilities for these schemes compared to Fisher's linear discriminant function in cases of heavy-tailed or contaminated distributions.

Generalized Linear and Quadratic Discriminant Functions Using Robust Estimates

Some adaptive distribution-free tests about location parameters of symmetric distributions are processed for the one-and two sample cases. The preliminary inferences in these adaptive schemes is to classify the underlying distributions as having light, medium, or heave rails with a statistic that is assentialy the range of the sample divided by the mean deviation from the sample median. If this inference suggests medium tails, a wilcoxon-type test is used; for heavy tails, a median-type test is used. A modified Wilcoxon test (alirnating middle sample itams) is processed for ligntatailed distributions and portions of its null distribution are trasulated. The favorable performance of these adaotive rets is demanstrated by a monta carla study.

Adaptive distribution-free tests

A distribution-free rank procedure is proposed for use in partial discrimination problems involving two populations. It is shown that this procedure can be applied with virtually any discriminant function. Moreover, the discriminant function may be selected after observing the samples on which it is to be based. Using Monte Carlo methods the rank procedure is compared with a normal theory and a tolerance region procedure. The rank procedure was the only one that adequately controlled the probabilities of misclassification while maintaining relatively small probabilities of not classifying an observation.

Distribution-Free Partial Discriminant Analysis

A model-free rank procedure is proposed for the two-population discrimination problem that enables the practitioner to better control the balance between the two probabilities of misclassification The method is applied to the discriminant functions resulting from normal assumptions and also to an adaptive one which is a weighted average of the linear and quadratic discriminant functions, where the weights are determined from the data A Monte Carlo study shows that the rank method can greatly improve the balance between the two misclassification probabilities while keeping their average comparatively small

Ronald H. Randles

Papers

A Two-Sample Adaptive Distribution-Free Test

Generalized Linear and Quadratic Discriminant Functions Using Robust Estimates

Adaptive distribution-free tests

Distribution-Free Partial Discriminant Analysis

Discriminant Analysis Based on Ranks