Measuring High-Order Moments of the Galaxy Distribution from Counts in Cells -- The Edgeworth Approximation

To probe the weakly non-linear regime, past the point where simple linear theory is sufficient to describe the statistics of the density distribution, we measure the skewness (S_3) and kurtosis (S_4) of the Count Probability Distribution Function (CPDF) of the IRAS 1.2 Jy sample obtained from counts in cells. These quantities are free parameters in a maximum likelihood fit of an Edgeworth expansion convolved with a Poissonian to the observed CPDF. This method, applicable on scales greater than 5 Mpc, is appreciably less sensitive to the tail of the distribution than are measurements of S_3 and S_4 from moments of the CPDF. We measure S_3 and S_4 to l~50 h^{-1} Mpc; the data are consistent with scale invariance, yielding averages of <S_3> = 2.83 +/- 0.09, and <S_4> = 6.89 +/- 0.68. These values are higher than those found using the moments method on the same data set, <S_3> = 1.5 +/- 0.5 and <S_4> = 4.4 +/- 3.7, due to lack of correction in the latter work for finite-volume effects. Unlike the moments method, our results are quite robust to the fact that IRAS galaxies are under-represented in cluster cores. We use N-body simulations to show that our method yields unbiased results.