1 Introduction

In the study of distributions of sources, the shape of the distribution is often assumed to be known. Thus, one searches for the mean value of some statistics of the distribution. For example, in checking the uniformity of the distribution of gamma-ray bursts, one looks for the mean value of $V/V_{\rm max}$, $<V/V_{\rm max}\gt$, where V is the volume enclosed at the observed distance of a source and $V_{\rm max}$ is the volume enclosed at the maximum distance where the source would be detectable (see, e.g., Higdon & Schmidt 1990). Also, in checking the uniformity of the distribution of quasars, one applies the test of $<V/V_{\rm max}\gt$, or its variant $<V^{\prime }/V_{\rm max}^{\prime }\gt$, (Schmidt 1968; Mathez 1976; Avni & Bahcall 1980; Hartwick & Schade 1990). The mean value reflects a collective, rather than partial, behavior of the distribution. An unsatisfactory feature of the method is the fact that a given mean value may correspond to different distributions. Aside from the mean value method, the $\chi ^2$ test and the Kolmogorov-Smirnov (K-S) test have been frequently applied to test distributions (see, e.g., Boyle et al. 1987, 1988; Hartwick & Schade 1990; Warren et al. 1994; Pei 1995). The two tests correspond to different aspects of distributions. The K-S test concerns the maximum value of the deviation between the observed distribution function of a sample and a given cumulative distribution function, while the $\chi ^2$ test reflects the behavior of intervals of a sample compared with a given cumulative distribution function. The K-S test is more sensitive than the $\chi ^2$test and is not affected by the artificial way of dividing intervals as the latter does.

In Sect. 2, we define and introduce a new statistical method for the test of known distributions. In Sect. 3, some properties of the new test are discussed and presented. An example of application of the new test is given in Sect. 4 and a summary of this paper is given in Sect. 5.