Accurate Approximations for Nonparametric Tests in Linear Models

Abstract: 

In general nonparametric tests have been based on test statistics chosen to have good properties in related parametric problems. Bootstrap or asymptotic approximations for the distribution of the test statistics, or robust versions of them, are used to obtain p-values. We propose a likelihood like test statistic based on a tilted bootstrap. Approximations for the distribution of this test statistic can be made by Monte Carlo approximations to the bootstrap or by a general multivariate saddlepoint approximation which is accurate in the tails of the distribution. We will concentrate on robust multiple regression for both fixed and random independent variables, describing first the tilted bootstrap approach. Then we will consider a general result on saddlepoint approximations for certain statistics based on empirical exponential families and use this to give an approximation to the bootstrap distribution. Finally we will examine the numerical properties of the tests.