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Student-t degrees of freedom

Degrees of freedom of a Student-t fitted to the z of H3: how heavy the tail is

Takeaway

A Student-t fitted to the z of Huchra's list has ν = 2.30 (1.91–2.98) against Planck and 1.97 (1.62–2.51) against Riess: a far heavier tail than a normal distribution (ν → ∞).
Table A2 gives 2.30 and 1.69, the same order as the edition-pair ν of the physical-constants chapter (1.00) and Bailey's (2017) 1.81.

Definition: The degrees of freedom ν and scale s of a zero-centered Student-t fitted by maximum likelihood to the z of H3. A smaller ν means a heavier tail (a normal distribution has ν → ∞). Fitted separately for each reference.
Formula: z ~ s · t(ν), ν and s by maximum likelihood. Rows with z = 0 are left out (the likelihood diverges at scale 0). 95% intervals: year-cluster bootstrap, 2,000 resamples.
Denominator: The same rows as H3: 548 in Huchra's list, 163 in Table A2 (162 against Planck).
How to read it: For Huchra's list ν = 2.30 against Planck (95% interval 1.91–2.98) and 1.97 against Riess (1.62–2.51). Table A2: 2.30 (1.53–12.21) and 1.69 (1.26–2.60). The edition-pair ν of the physical-constants chapter (M4) is 1.00; Bailey (2017) found ν = 1.81 ± 0.15 for pairs of individual measurements of fundamental constants.
Caveats: A ν near 2 is a tail whose variance is undefined. The Planck interval of Table A2 is open upward (1.53–12.21): 163 rows do not pin the tail down. The limits of H3 (choice of reference, dependence, unstated error levels) apply unchanged.

Observed (Table A2)NormalStudent-t

Table A2 against Planck 2018: share of rows whose |z| exceeds the value on the horizontal axis (log vertical axis) and the fitted Student-t (ν = 2.30, s = 1.22). Huchra’s list is shown only in aggregate, so it has no such plot.

Observed (Table A2)NormalStudent-t

Table A2 against Riess et al. 2022: share of rows whose |z| exceeds the value on the horizontal axis (log vertical axis) and the fitted Student-t (ν = 1.69, s = 1.13). Huchra’s list is shown only in aggregate, so it has no such plot.

Series	Start	Reference	n	Left out	ν	95% interval	Scale s	95% interval
Huchra's list	from 1929	Planck 2018	548	0	2.30	1.91–2.98	1.32	1.12–1.61
Huchra's list	from 1929	Riess et al. 2022	548	0	1.97	1.62–2.51	1.41	1.17–1.71
Huchra's list	from 1950	Planck 2018	543	0	2.47	2.04–3.19	1.32	1.14–1.61
Huchra's list	from 1950	Riess et al. 2022	543	0	2.06	1.70–2.65	1.41	1.18–1.70
Table A2	1976–2019	Planck 2018	162	1	2.30	1.53–12.21	1.22	0.89–1.85
Table A2	1976–2019	Riess et al. 2022	163	0	1.69	1.26–2.60	1.13	0.83–1.53
Merged (sensitivity)	from 1929	Planck 2018	641	1	2.24	1.87–2.82	1.30	1.10–1.57
Merged (sensitivity)	from 1929	Riess et al. 2022	642	0	1.87	1.56–2.33	1.34	1.12–1.62
Merged (sensitivity)	from 1950	Planck 2018	636	1	2.37	1.99–3.07	1.30	1.11–1.56
Merged (sensitivity)	from 1950	Riess et al. 2022	637	0	1.94	1.61–2.41	1.34	1.11–1.60

Download CSV: h0_h4_tail.csv

Sources: J. Huchra, Estimates of the Hubble Constant (hubble.plot.dat, last updated 2010-10-07), Harvard-Smithsonian Center for Astrophysics; assembled as part of the NASA/HST Key Project on the Extragalactic Distance Scale. Aggregates only. · Faerber, T.; López-Corredoira, M. A Chi-Squared Analysis of the Measurements of Two Cosmological Parameters over Time. Universe 2020, 6, 114, doi:10.3390/universe6080114, Appendix Table A2 (read from arXiv:2008.04035v1). CC BY 4.0.