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Power transformations to induce normality and their applications
Author(s) -
Chen Willa W.,
Deo Rohit S.
Publication year - 2004
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/j.1467-9868.2004.00435.x
Subject(s) - mathematics , estimator , goodness of fit , statistics , normality , sample size determination , transformation (genetics) , cube root , power (physics) , sample (material) , biochemistry , chemistry , geometry , physics , chromatography , quantum mechanics , gene
Summary. Random variables which are positive linear combinations of positive independent random variables can have heavily right‐skewed finite sample distributions even though they might be asymptotically normally distributed. We provide a simple method of determining an appropriate power transformation to improve the normal approximation in small samples. Our method contains the Wilson–Hilferty cube root transformation for χ 2 random variables as a special case. We also provide some important examples, including test statistics of goodness‐of‐fit and tail index estimators, where such power transformations can be applied. In particular, we study the small sample behaviour of two goodness‐of‐fit tests for time series models which have been proposed recently in the literature. Both tests are generalizations of the popular Box–Ljung–Pierce portmanteau test, one in the time domain and the other in the frequency domain. A power transformation with a finite sample mean and variance correction is proposed, which ameliorates the small sample effect. It is found that the corrected versions of the tests have markedly better size properties. The correction is also found to result in an overall increase in power which can be significant under certain alternatives. Furthermore, the corrected tests also have better power than the Box–Ljung–Pierce portmanteau test, unlike the uncorrected versions.