A few properties of sample variance
Collection titlePreprint Lamsade
Link to item filehttps://hal.archives-ouvertes.fr/hal-02012458
DeweyProbabilités et mathématiques appliquées
Sujetindependence between sample mean and variance; sample variance; variance of sample variance
MetadataShow full item record
989 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Item number of pages15
Abstract (EN)A basic result is that the sample variance for i.i.d. observations is an unbiased esti-mator of the variance of the underlying distribution (see for instance Casella and Berger(2002)). But what happens if the observations are neither independent nor identically distributed. What can we say? Can we in particular compute explicitly the firsttwo moments of the sample mean and hence generalize formulae provided in Tukey(1957a), Tukey (1957b) for the first two moments of the sample variance? We also know that the sample mean and variance are independent if they are computed onan i.i.d. normal distribution. This is one of the underlying assumption to derive theStudent distribution Student alias W. S. Gosset (1908). But does this result hold forany other underlying distribution? Can we still have independent sample mean andvariance if the distribution is not normal? This paper precisely answers these questions and extends previous work of Cho, Cho, and Eltinge (2004). We are able to derive ageneral formula for the first two moments and variance of the sample variance under nospecific assumptions. We also provide a faster proof of a seminal result of Lukacs (1942)by using the log characteristic function of the unbiased sample variance estimator.
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