Adaptive Regression

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While there have been a large number of estimation methods proposed and developed for linear regression, none has proved good for all purposes. This text focuses on the construction of an adaptive combination of two estimation methods so as to help users make an objective choice and combine the desirable properties of two estimators.

Adaptive Regression Review

In introductory statistics courses students are taught linear regression. Basic linear regression in an elementary course is simply least squares estimation of the regression line. Least squares has optimal properties when the conditional distribution of the variable of interest has a normal distribution, conditional on the predictor variables. However, when the distribution is non-normal, the least squares estimates can sometimes have very undesirable properties. It is well-known that least squares estimates are very sensitive to individual outliers and leverage points. Consequently any regression analysis requires diagnostic checking of the model using various diagnostic statistics. Although students are often taught about the diagnostics used to check the modeling assumptions, most introductory texts do not go into the modeling alternatives to least squares. There are now many alternatives in the literature that are robust to departures from the normal distribution and/or multicollinearity. These include least absolute deviations,ridge regression, M, L,S and GM regression. The authors review least squares and these other methods in Chapter 2. The next four chapters of the book discusses four new alternative methods that the author's refer to as adaptive methods. These methods involve combining two of the estimation procedures discussed in Chapter two. The various methods involve taking convex combinations of two of these estimates. Adaptive methods allow the weights in these convex combinations to be determined from the data. Chapters 3 - 6 deal with adaptive weights for various pairs of estimates. Chapter 7 deals with adaptive trimmed estimates. Chapter 8 covers adaptive ways for combining hypothesis testing methods. Chapter 9 deals with computational aspects and software devised for these methods. Chapter 10 provides some asymptotic properties of estimators discussed in earlier chapters particularly the studentized M-estimators of Chapter 2. The authors provide many concrete examples and compare the various methods.
This is a nice little monograph for anyone interested in regression analysis methods that are practical and advance the theory of regression. This should interest both the applied statistician and the statistical researcher.

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