Median Statistics, H₀, and the Accelerating Universe

We develop median statistics that provide powerful alternatives to χ² likelihood methods and require fewer assumptions about the data. Application to astronomical data demonstrates that median statistics lead to results that are quite similar and almost as constraining as χ² likelihood methods but with somewhat more confidence since they do not assume Gaussianity of the errors or that their magnitudes are known.

Applying median statistics to Huchra's compilation of nearly all estimates of the Hubble constant, we find a median value H₀ = 67 km s^-1 Mpc^-1. Median statistics assume only that the measurements are independent and free of systematic errors. This estimate is arguably the best summary of current knowledge because it uses all available data and, unlike other estimates, makes no assumption about the distribution of measurement errors. The 95% range of purely statistical errors is ±2 km s^-1 Mpc^-1. The high degree of statistical accuracy of this result demonstrates the power of using only these two assumptions and leads us to analyze the range of possible systematic errors in the median, which we estimate to be roughly ±5 km s^-1 Mpc^-1 (95% limits), dominating over the statistical errors.

Using a Bayesian median statistics treatment of high-redshift Type Ia supernovae (SNe Ia) apparent magnitude versus redshift data from Riess et al., we find the posterior probability that the cosmological constant Λ > 0 is 70% or 89%, depending on the prior information we include. We find the posterior probability of an open universe is about 47%, and the probability of a spatially flat universe is 51% or 38%. Our results generally support the observers' conclusions but indicate weaker evidence for Λ > 0 (less than 2 σ). Median statistics analysis of the Perlmutter et al. high-redshift SNe Ia data shows that the best-fit flat-Λ model is favored over the best-fit Λ = 0 open model by odds of 366 : 1; the corresponding Riess et al. odds are 3 : 1 (assuming in each case prior odds of 1 : 1).

A scalar field with a potential energy with a "tail" behaves like a time-variable Λ. Median statistics analyses of the SNe Ia data do not rule out such a time-variable Λ and may even favor it over a time-independent Λ and a Λ = 0 open model.