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Abstract:
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Weighted least squares regression is tied to normality of the residuals. The two main motivations for weights are scale families and convolution families. In normal-theory regression, normality is retained under both scaling and convolution. Outside of normal-theory regression, the two motivations have differing implications. Under a scale model, the standardized residual distribution is static; under a convolution model, the distribution moves toward normality as more units are convolved. Empirically, we have observed both static and dynamic behavior. Some dynamic behavior shows slower movement toward normality than convolution would suggest. We develop a family of nonparametric Bayesian models that include the scale and convolution families as endpoints, and that include a continuum of distributions in between. Implications of the models for case-specific decisions are discussed.
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