On the reliability of the chi-square test
The author studies the reliability of the \(\chi^ 2\) test when the end points of class intervals change but the number R of such intervals is kept fixed. The variation of this test involves \[ v(x)=1/(nh)\sum^{R- 1}_{i=0}(N(x+ih)-ih)^ 2=\sum^{R-1}_{i=0}y^ 2_ n(x+ih) \] with \(h=1/R\) and N(t) is the number of variables \(X_ i\) with \(t\leq X_ i<t+h\); v(x) is nothing but the \(\chi^ 2\) test using \(\{x,x+h,...,x+(R- 1)h\}\) as a partition of [0,1). The author first computes the correlation between v(x) and v(y) then he obtains the following expression for the variance of v(x): \[ var v(x)=E\int (v(x)-\bar v)^ 2 d\mu +var \bar v, \] where \(\bar v=\int v(x)d\mu\) for x in [0,h]. The first term can be interpreted as the part of the variance of v(x) due to the choice of the position of the partition on the interval [0,1]. Finally the author proposes the use of a Cramer-von Mises type statistic to eliminate the effects of this fluctuation.
scientific article; zbMATH DE number 3994771