A Bayesian Estimate of the Skewness of the Cosmic Microwave Background

We propose a formalism for estimating the skewness and angular power spectrum of a general cosmic microwave background data set. We use the Edgeworth expansion to define a non-Gaussian likelihood function that takes into account the anisotropic nature of the noise and the incompleteness of the sky coverage. The formalism is then applied to estimate the skewness of the publicly available 4 yr COBE Differential Microwave Radiometer (DMR) data. We find that the data is consistent with a Gaussian skewness, and with isotropy. Inclusion of non-Gaussian degrees of freedom has essentially no effect on estimates of the power spectrum, if each Cl is regarded as a separate parameter or if the angular power spectrum is parametrized in terms of an amplitude (Q) and spectral index (n). Fixing the value of the angular power spectrum at its maximum-likelihood estimate, the best-fit skewness is S = 6.5 ± 6.0 × 104 μK3; marginalizing over Q, the estimate of the skewness is S = 6.5 ± 8.4 × 104 μK3, and marginalizing over n one has S = 6.5 ± 8.5 × 104 μK3.