Asymptotic Properties of the Square Root Transformation of the Gamma Distribution

Oregon Convention Center 

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.

Box-Cox transformation

Incomplete gamma function

Kullback-Leibler divergence

Laplace distribution

Normality

Variance-stabilizing transformation 

IMS