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Asymptotic Properties of the Square Root Transformation of the Gamma Distribution

Abstract Number:

3774 

Submission Type:

Contributed Abstract 

Contributed Abstract Type:

Paper 

Participants:

Mayla Ward (1), Kimihiro Noguchi (1)

Institutions:

(1) Western Washington University, Bellingham, WA

Co-Author:

Kimihiro Noguchi  
Western Washington University

First Author:

Mayla Ward  
Western Washington University

Presenting Author:

Mayla Ward  
N/A

Abstract Text:

Power transformations of the gamma distribution to approximate normality have been a topic of research for the past 100 years. Fisher (1925) proposed the square-root transformation of the chi-square distribution, while Wilson & Hilferty (1931) and Hernandez & Johnson (1980) proved the asymptotic optimality of the cube-root transformation. We employ the Kullback-Leibler information number criterion of Hernandez & Johnson (1980) to prove that the square-root transformation of the gamma distribution is asymptotically optimal when the normal distribution with a fixed variance is set as the target distribution. In particular, a stronger mode of convergence than the convergence in distribution is achieved in the normal case, implying that the square-root transformation is an asymptotically optimal variance-stabilizing power transformation. Additionally, by utilizing the asymptotic expansion of the normalized upper incomplete gamma function at the transition point, we show that the Kullback-Leibler information number is also minimized with the square-root transformation when the target distribution is set to be the Laplace distribution with a fixed scale parameter.

Keywords:

Box-Cox transformation|Incomplete gamma function|Kullback-Leibler divergence|Laplace distribution|Normality|Variance-stabilizing transformation

Sponsors:

IMS

Tracks:

Statistical Theory

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