Composite Distribution Variance Stable Transformation And Symmetry Transformation

Variance is the numeric feature which measures the deviation between variable and its mean, but fact the variance of many distributions are not stable, which means the variance is the function of the parameter of the distribution. When constructing the confidence intervals, the intervals are the complicated function of the parameter; Skewness coefficient is the numeric feature that describes the symmetry of the distribution, the distribution of the statistic related to the sample is asymptotically normal for large sample, which will be asymptotical symmetrizing if the skewness coefficient is asymptotical zero after being symmetrically transformed.In a variety of theories, the distribution of many random variables is not unitary, but compound. The compound distribution is applied in a wide range, especially in the actuarial,such as in the collective risk model, the distribution of random sum of claims variables is compound distribution.Compound Poisson distribution and compound negative binomial distribution are the two common compound distributions in the risk theory. This paper mainly studies some properties of the two compound distributions, and discusses the definition of variance stabilizing transformation and symmetrizing transformation and the method constructing the variance stabilizing transformation and symmetrizing transformation of the two compound distributions. Some corrections of the two transformations are made just for being applied easily, the confidence interval of specific parameter in the two different distributions are given by making use of the correction, in the Gamma distribution case for parameters given, the symmetrizing transformation of compound Poisson distribution can make more accurate inference than bias-correction transformation in the interval estimate, but the bias-correction transformation of compound negative binomial distribution makes more accurate inference than symmetrizing transformation. In the general Pareto distribution case, the symmetrizing transformation of the two compound distributions can make more accurate inference than bias-correction transformation.