Variance stabilizing transformations of Poisson, binomial and negative binomial distributions

Consider variance stabilizing transformations of Poisson distribution π(λ)π(λ), binomial distribution B(n,p)B(n,p) and negative binomial distribution NB(r,p)NB(r,p), with square root transformations for π(λ)π(λ), arcsin transformations for B(n,p)B(n,p) and inverse hyperbolic sine transformations for NB(r,p)NB(r,p). We will introduce three terms: critical point, domain of dependence and relative error. By comparing the relative errors of the transformed variances of π(λ)π(λ), B(n,p)B(n,p) and NB(r,p)NB(r,p), and comparing the skewness and kurtosis of π(λ)π(λ), B(n,p)B(n,p) and NB(r,p)NB(r,p) and their transformed variables, we obtain some better transformations with domains of dependence of the parameters. A new kind of transformation (n+12)1/2sin−1(2Y−nn+2a) for B(n,p)B(n,p) is suggested.