Some statistics in common use take a form of a ratio of two statistics such as sample correlation coefficient, Pearson's coefficient of variation and so on. In this paper, obtaining an asymptotic representation of the ratio statistic until the third order term, we will discuss asymptotic mean squared errors of the ratio statistics. We will also discuss bias correction of the sample correlation coefficient and the sample coefficient of variation. Mean squared errors of the corrected estimators are also obtained.

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In this paper we propose new smoothed sign and Wilcoxon's signed rank tests, which are based on a kernel estimator of the underlying distribution function of data. We discuss approximations of p-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon's tests in the sense of the Pitman's asymptotic relative efficiency, and the differences of the ordinary and the new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Though the smoothed tests are not distribution-free, we can obtain Edgeworth expansions with residual term o(n −1 ), which do not depend on the underlying distribution.