Biased Estimations of Variance and Skewness

Abstract

Nonlinear combinations of direct observables are often used to estimate quantities of theoretical interest. Without sufficient caution, this could lead to biased estimations. An example of great interest is the skewness S3 of the galaxy distribution, defined as the ratio of the third moment ξ¯3 and the variance squared ξ¯22 smoothed at some scale R. Suppose one is given unbiased estimators for ξ¯3 and ξ¯22, respectively; taking a ratio of the two does not necessarily result in an unbiased estimator of S3. Exactly such an estimation bias (distinguished from the galaxy bias) affects most existing measurements of S3 from galaxy surveys. Furthermore, common estimators for ξ¯3 and ξ¯2 suffer also from this kind of estimation bias themselves because of a division by the estimated mean counts in cell. In the case of ξ¯2, the bias is equivalent to what is commonly known as the integral constraint. We present a unifying treatment allowing all these estimation biases to be calculated analytically. These estimation biases are in general negative, and decrease in significance as the survey volume increases, for a given smoothing scale. We present a preliminary reanalysis of some existing measurements of the variance and skewness (from the APM, CfA, SSRS, and IRAS) and show that most of the well-known systematic discrepancies between surveys with similar selection criteria, but different sizes, can be attributed to the volume-dependent estimation biases. This affects the inference of the galaxy bias(es) from these surveys. Our methodology can be adapted to measurements of the variance and skewness of, for instance, the transmission distribution in quasar spectra and the convergence distribution in weak-lensing maps. We discuss generalizations to N>3, suggest methods to reduce the estimation bias, and point out other examples in large-scale structure studies that might suffer from this type of a nonlinear estimation bias.