Biased Estimations of Variance and Skewness

Journal Article · Thu Jul 01 00:00:00 EDT 1999 · Astrophysical Journal

DOI:https://doi.org/10.1086/307401· OSTI ID:6372910

Hui, L ^[1]; Gaztanaga, E ^[2]

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1. NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, IL 60510 (United States)
2. Consejo Superior de Investigaciones Cientificas (CSIC), Institut d'Estudis Espacials de Catalunya, Edf. Nexus-104-c/ Gran Capita 2-4, 08034, Barcelona (Spain)

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 S[sub 3] of the galaxy distribution, defined as the ratio of the third moment [bar [xi]][sub 3] and the variance squared [bar [xi]][sup 2] [sub 2] smoothed at some scale [ital R]. Suppose one is given unbiased estimators for [bar [xi]][sub 3] and [bar [xi]][sup 2] [sub 2], respectively; taking a ratio of the two does not necessarily result in an unbiased estimator of S[sub 3]. Exactly such an estimation bias (distinguished from the [ital galaxy] bias) affects most existing measurements of S[sub 3] from galaxy surveys. Furthermore, common estimators for [bar [xi]][sub 3] and [bar [xi]][sub 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 [bar [xi]][sub 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 [ital 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[gt]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. [copyright] [ital [copyright] 1999.] [ital The American Astronomical Society]

• OSTI ID: 6372910
• Journal Information: Astrophysical Journal, Vol. 519:2; ISSN 0004-637X
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ACCRETION DISKS
FLUCTUATIONS
GALAXIES
GRAVITATIONAL INSTABILITY
QUASARS
RED SHIFT
STATISTICAL MODELS
STATISTICS
UNIVERSE
COSMIC RADIO SOURCES
INSTABILITY
MATHEMATICAL MODELS
MATHEMATICS
PLASMA INSTABILITY
VARIATIONS
661300* - Other Aspects of Physical Science- (1992-)