The statistical methods discussed in the previous chapter are restricted by many assumptions. As an example, the inference is valid only if the assumed distribution is also the underlying truth distribution. Non-parametric methods are versatile and not restricted by many assumptions.
In this chapter we first consider estimation problems and then the testing problems. We will begin with the importance of the empirical distribution function (edf) and state the fundamental theorem of statistics in Section 8.2. The edf is further explored for estimation of statistical functionals. The jackknife and bootstrap methods are considered in the next Section 8.3. Smoothing techniques for estimation problems are covered in Section 8.4. Finally, we conclude the chapter with some of the very important non-parametric tests in Section 8.5.
Let be a random sample from an unknown distribution function on the real line. The empirical distribution function, abbreviated as edf, is then defined by
where is an indicator function. The edf is very intuitive and plays a highly useful tool in ...