Free Access
Volume 526, February 2011
Article Number A81
Number of page(s) 24
Section Cosmology (including clusters of galaxies)
Published online 24 December 2010

Online material

Appendix A: Sampling variance of weighted mean square error

If we assume the prediction errors are distributed normally with total variance, Eq. (5): , then the sampling variance of the weighted mean square error is (A.1)The standard error on WRMS is then .

Appendix B: Maximum likelihood estimators for intrinsic prediction error and covariance

Let the intrinsic variance of predictions () from method P,Q be . Denote the intrinsic covariance between distance prediction errors from P and Q as cPQ. The intrinsic correlation is ρPQ = cPQ/(σPσQ). Let θ = (σP,σQ,ρPQ) be the vector of this triplet of quantities. These parameters can be arranged in an intrinsic covariance matrix: (B.1)We wish to estimate them from the cross-validated predictions. We derive estimators for this intrinsic covariance using maximum likelihood, and also derive its uncertainty.

We assume that the pair of prediction errors are jointly distributed normally around zero with total covariance (B.2)The measurement error covariance matrix, , contains the measurement variances, , for model P and model Q, on the diagonal, and any covariance due to observational error in the off-diagonal. Since supernova s is subject to the same random peculiar velocity under both models P and Q, the covariance from peculiar velocity dispersion is (B.3)The negative log likelihood for the unknown (σP,σQ,ρPQ), given the set of distance predictions is (B.4)We numerically maximize the likelihood with the constraints σP,σQ > 0 and |ρPQ| < 1. Once we have found the maximum likelihood estimate (MLE) , we can compute its error by

numerically evaluating the Hessian of the negative log likelihood, ). The sampling covariance (error) of the MLE is estimated from the Fisher information: . The standard errors in each of (σP,σQ,ρPQ) are the square roots of the diagonal elements of . The off-diagonal elements contain the estimation covariance between the three parameters. If the difference in intrinsic prediction error between the two models is Δ = σP − σQ, the sampling variance of Δ is (B.5)where is the (i,j) element of the error covariance matrix of the MLE. This error estimate accounts for covariance from random peculiar velocities and the intrinsic correlation between two models. Notably, a large |ρPQ| will affect the significance of the difference, Δ.

From the prediction errors of a single method,  {Δμs} , we can estimate the rms intrinsic prediction error σpred. The negative log likelihood simplifies to (B.6)The maximum likelihood estimate is found by minimizing this or finding the zero of the score function . If N is large enough, the standard error on can be estimated using the Fisher information at the MLE: (B.7)An estimate of the sampling variance of the maximum likelihood estimate of the intrinsic variance is the inverse of the Fisher information . The standard error of itself is the square root of . This estimate of the intrinsic dispersion “subtracts” out the contribution of random peculiar velocities and measurement error to the total dispersion.

Appendix C: Results for other spectroscopic indicators at maximum light

We present our results using the absorption velocity (vabs), the full-width at half-maximum (FWHM), the relative absorption depth (dabs), the pseudo-equivalent width (pEW), and the various spectroscopic ratios ℛ(X) [Eqs. (16)–(22)] in Tables C.1C.5.

Table C.1

vabs (units of 104    km   s-1) at maximum light from 10-fold CV.

Table C.2

FWHM (units of 102 Å) at maximum light from 10-fold CV.

Table C.3

dabs at maximum light from 10-fold CV.

Table C.4

pEW (units of 102 Å) at maximum light from 10-fold CV.

Table C.5

ℛ(Ca) and ℛ(Si) at maximum light from 10-fold CV.

© ESO, 2010

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