# 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 *x*_{1}, *x*_{2}, …, *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 *p*th percentile of the continuous distribution. Let *x*_{p} be the *p*th percentile of the distribution and let be the traditional estimator of *x*_{p}. To clearly define we need ...