Variance stabilizing transformations of Poisson, binomial and negative binomial distributions

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Consider variance stabilizing transformations of Poisson distribution π ( λ ) , binomial distribution B ( n , p ) and negative binomial distribution N B ( r , p ) , with square root transformations for π ( λ ) , arcsin transformations for B ( n , p ) and inverse hyperbolic sine transformations for N B ( 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 ) , B ( n , p ) and N B ( r , p ) , and comparing the skewness and kurtosis of π ( λ ) , B ( n , p ) and N B ( r , p ) and their transformed variables, we obtain some better transformations with domains of dependence of the parameters. A new kind of transformation ( n + 1 2 ) 1 / 2 sin − 1 ( 2 Y − n n + 2 a ) for B ( n , p ) is suggested.