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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 (Xᵢ_, Yᵢ_) consistent with the hypothesis H₀_ that x and y are statistically independent? This paper presents an easily applied permutation test for H₀_, 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 H₀_ amounts to testing validity of the cosmological model. Segal's chronomatic cosmological model is rejected under H₀_. On the other hand, for the quasar sample H₀_ 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.
Efron et al. (Sun,) studied this question.