Uniformity test based on the empirical Bernstein distribution

In this paper, we firstly review the origin of Bernstein polynomial and the various application of it. Then we review the importance of goodness-of-fit test, especially the uniformity test, and we examine lots of different test statistics proposed by far. After that we suggest two new statistics for testing the uniformity. These two statistics are based on Komogorov-Smirnov test type and Cramér-Von Mises test type, respectively. Also we embed Bernstein polynomial into those test type and take advantage of great approximation performance of this polynomial. Finally, we run a Monte-Carlo simulation to compare the performance of our statistics to those without embedding the Bernstein polynomials. We compare their performance in term of powers and inefficiencies. We found that by choosing suitable value for parameter, our statistics can perform better than the original form in most of the cases. The suggestion of choosing optimal value will be given