Adaptive Distribution-free Regression Methods and their Applications

We consider the model Y = βx + Z, where the random variable Z has a continuoustype distribution that can be badly skewed, contaminated, or censored. To test the hypothesis H 0 : β = β0, we use the distribution-free statistic K(β0) = Σc(Q i )a(R i ), where c(·) and a(·) are increasing score functions and Q i and R i are the respective ranks of x i and y i – β0 x i . The score functions c(·) and a(·) can be adapted or chosen after observing the data without destroying the distribution-free nature of the test. A Monte Carlo study is presented which illustrates the excellent performance of an adaptive test when a wide range of distributions is considered for the residuals. Interval and point estimates of sβ can be found by employing the “inverse” of the testing procedure. These results are used to find estimates of the percentile lines. Two examples are given which involve lifetimes of electric motor insulation and grade point averages of beginning university students, respectively.