Biased Estimations of Variance and Skewness

Nonlinear combinations of direct observables are often used to estimatequantities of theoretical interest. Without sufficient caution, this could leadto biased estimations. An example of great interest is the skewness $S_3$ ofthe galaxy distribution, defined as the ratio of the third moment $\xibar_3$and the variance squared $\xibar_2^2$. Suppose one is given unbiased estimatorsfor $\xibar_3$ and $\xibar_2^2$ respectively, taking a ratio of the two doesnot necessarily result in an unbiased estimator of $S_3$. Exactly such anestimation-bias affects most existing measurements of $S_3$. Furthermore,common estimators for $\xibar_3$ and $\xibar_2$ suffer also from this kind ofestimation-bias themselves: for $\xibar_2$, it is equivalent to what iscommonly known as the integral constraint. We present a unifying treatmentallowing all these estimation-biases to be calculated analytically. They are ingeneral negative, and decrease in significance as the survey volume increases,for a given smoothing scale. We present a re-analysis of some existingmeasurements of the variance and skewness and show that most of the well-knownsystematic discrepancies between surveys with similar selection criteria, butdifferent sizes, can be attributed to the volume-dependent estimation-biases.This affects the inference of the galaxy-bias(es) from these surveys. Ourmethodology can be adapted to measurements of analogous quantities in quasarspectra and weak-lensing maps. We suggest methods to reduce the aboveestimation-biases, and point out other examples in LSS studies which mightsuffer from the same type of a nonlinear-estimation-bias.Comment: 28 pages of text, 9 ps figures, submitted to Ap