New Tests Based On Bernstein Distribution Estimator

In this paper,new tests for goodness-of-fit based on the Bernstein distribution estimator are introduced.Noting that the empirical distribution function may not be appropriate for estimating continuous distributions in small sample cases,we develop Kolmogorov–Smirnov and Cram?er–von Mises test statistics based on the Bernstein distribution estimator.Furthermore,the asymptotic distributions under the null hypothesis and a sequence of alternatives are studied.Simulations of sizes and powers comparison are also conducted to show the performance of the proposed tests.A real data set is analyzed for illustration.This paper is organized as follows.In Section 1,the related background of the goodnessof-fit test and the Bernstein distribution estimator are introduced.We also introduce the main work of this paper.In Section 2,we construct the new test statistics based on the Bernstein distribution estimator.The asymptotic distributions under the null and a sequence of alternatives are also investigated in this section.In Section 3,extensive Monte Carlo simulation studies are carried out to illustrate the performance(empirical sizes and powers)of the proposed test statistics.A numerical example is provided in Section 4.In Section 5,the conclusion and discussion are given.All proofs are collected in the Appendix.