Test Based on Empirical Distribution Function

Abstract

The chapter-wise summary of the thesis is as follows: Chapter 1 includes introduction about the Tests based on Empirical Distribution Func- tions. The main focus of this chapter is on Non-parametric tests. This chapters includes basic concepts, de nitions, and brief discription about the Goodness-of- t problem. Goodness-of- t tests are used to check the compatibility of a set of observed sample values with a normal distribution or any other distribution. These tests are designed for a null hypothesis which is the statement about the form of probability function or cu- mulative distribution function of the parent population from which the sample is drawn. Here 2 Goodness-of- t test and its applications are described in details. In Chapter 2 the second goodness-of- t test the Kolmogorov-Smirnov test is discussed in details. The Kolmogorov-Smirnov statistics are used as general goodness-of- t tests which are known to be more sensitive to location than to scale alternatives. This test is based on vertical deviation between observed and expected cumulative distribution func- tions. In this chapter the Kolmogorov-Smirnov one-sample statistic, the Kolmogorov- Smirnov two-sample statistic and their applications are discussed. Then, in Chapter 3 the two-sample, distribution-free statistics of Smirnov (1939) are used to de ne a new statistic. While the Smirnov statistics are used as a general goodness-of- t test, a distribution-free scale test based on this new statistic is devel- oped. It is shown that this new test has higher power than the two-sided Smirnov statistic in detecting di erences in scale for some symmetric distributions with equal means/medians.