Simultaneous analysis of several complete samples. The V/V/sub max/ and V/sub e//V/sub a/ variables, with applications to quasars

We derive methods for applying the V/V/sub max/ variable when several complete samples are analyzed simultaneously. A new, related, variable, V/sub e//V/sub a/, which is superior to V/V/sub max/ when sufficient observational data are available, is introduced. We show how statistically independent complete samples can be generated from partially overlapping samples such that each observed object appears only once in the analysis. These methods make it possible to combine the information from different complete samples, and to make an efficient use of all available data. Illustrative examples for quasars are given. We show how the generalized V/V/sub max/ and V/sub e//V/sub a/ can be used to find the amount of cosmological evolution of quasars and its associated range of uncertainty. This range of uncertainty is of fundamental importance for theories of quasars and for their cosmological implications. We also give, for the first time, the formal justification of the V/V/sub max/ (and V/sub e//V/sub a/) test, using a variable-sample-size formalism that allows for the total number of objects in the sample to be a random variable