A Simple Test of Independence for Truncated Data with Applications to Redshift Surveys

In analysis of astronomical data, one is often faced with determination of bivariate distributions from truncated data. This leads to the following statistical question: Is a truncated sample of observed points (X_i_, Y_i_) consistent with the hypothesis H_0_ that x and y are statistically independent? This paper presents an easily applied permutation test for H_0_, closely related to Lynden-Bell's estimate of the marginal distribution of truncated data. The test is applied to two redshift-magnitude surveys, one of galaxies and one of quasars. Analysis of the galaxy survey by Loh & Spillar shows that in the framework of a simple Hubble Law model, that is, distance proportional to redshift, or most conventional models with zero cosmological constant and density parameters {OMEGA}~0(1), the absolute magnitude or luminosity and redshift are statistically independent. Therefore, assuming statistical independence, testing H_0_ amounts to testing validity of the cosmological model. Segal's chronomatic cosmological model is rejected under H_0_. On the other hand, for the quasar sample H_0_ is rejected strongly in a conventional cosmological model (and in a chronomatic model as well) indicating either incorrectness of the models or, as is more commonly assumed, indicating strong luminosity evolution.