Approximate representation of thep-norm distribution

In surveying data processing, we generally suppose that the observational errors distribute normally. In this case the method of least squares can give the minimum variance unbiased estimation of the parameters. The method of least squares does not have the character of robustness, so the use of it will become unsuitable when a few measurements inheriting gross error mix with others. We can use the robust estimating methods that can avoid the influence of gross errors. With this kind of method there is no need to'know the exact distribution of the observations. But it will cause other difficulties such as the hypothesis testing for estimated parameters when the sample size is not so big. For non-normally distributed measurements we can suppose they obey thep-norm distribution law. Thep-norm distribution is a distributional class, which includes the most frequently used distributions such as the Laplace, Normal and Rectangular ones. This distribution is symmetric and has a kurtosis between 3 and −6/5 whenp is larger than 1 Usingp-norm distribution to describe the statistical character of the, errors, the only assumption is that the error distribution is a symmetric and unimodal curve. This method possesses the property of a kind of self-adapting. But the density function of thep-norm distribution is so complex that it makes the theoretical analysis more difficult. And the troublesome calculation also makes this method not suitable for practice. The research of this paper indicates that thep-norm distribution can be represented by the linear combination of Laplace distribution and normal distribution or by the linear combination of normal distribution and rectangular distribution approximately. Which kind of representation will be taken is according to whether the parameterp is larger than 1 and less than 2 orp is larger than 2. The approximate distribution have the same first four order moments, with the exact one. It means that approximate distribution has the same mathematical expectation, variance, skewness and kurtosis withp-norm distribution. Because every density function used in the approximate formulae has a simple form, using the approximate density function to replace thep-norm ones will simplify the problems ofp-norm distributed data processing obviously. Project supported by the Sustentation Plan for Outstanding Teachers of Advanced Colleges by Ministry of Education and the Scientific Research Find of Wuhan University.