## Cramér‐von Mises regression

### Abstract

Consider a linear regression model with unknown regression parameters β0 and independent errors of unknown distribution. Block the observations into q groups whose independent variables have a common value and measure the homogeneity of the blocks of residuals by a Cramér‐von Mises q‐sample statistic Tq(β). This statistic is designed so that its expected value as a function of the chosen regression parameter β has a minimum value of zero precisely at the true value β0. The minimizer β of Tq(β) over all β is shown to be a consistent estimate of β0. It is also shown that the bootstrap distribution of Tq0) can be used to do a lack of fit test of the regression model and to construct a confidence region for β0

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