The LIL for the Bickel-Rosenblatt test statistic

Let {Xn,n⩾1} be independent identically distributed random variables with common density function, f. We are interested in testing the hypothesis H0:f=f0 vs. H1:f≠f0, where f0 is a given density. In this note, we discuss the law of the iterated logarithm (LIL) of the Bickel-Rosenblatt test statistic under fixed alternatives. These results are of particular importance if the null hypothesis cannot be rejected. Also the proof of the results shows that the test statistics are of different order under the null hypothesis and the alternative hypothesis.