Adaptive Tests of Significance Using Permutations of Residuals with R and SAS

CHAPTER 2

SMOOTHING METHODS AND NORMALIZING TRANSFORMATIONS

In this chapter we will describe many of the fundamental techniques that will be used to perform adaptive tests of significance. We will focus our attention on a weighting procedure that is used to transform the data so that the transformed data is approximately normally distributed. We will see in subsequent chapters how this transformed data is used in the adaptive tests.

2.1 TRADITIONAL ESTIMATORS OF THE MEDIAN AND THE INTERQUARTILE RANGE

The adaptive weighting procedure relies heavily on percentile estimators. In this section we describe the traditional estimator of a percentile and in the next section we describe an improved estimator that is based on the smoothed cumulative distribution function (c.d.f.).

Let x₁, x₂, …, x_n be a random sample from a continuous distribution having a c.d.f. of F(x) and let x₍₁₎, x₍₂₎, …, x_(n) be the set of ordered observations, in ascending order, that we will call the order statistics. If n is odd, the usual estimator of the median is x_((n+1)/2); and if n is even, the usual estimator is .

Now suppose we want to estimate the pth percentile of the continuous distribution. Let x_p be the pth percentile of the distribution and let be the traditional estimator of x_p. To clearly define we need ...