Suppose that Y follows a χ^(p)-distribution with n degrees of freedom, and Z is the standardized form of Y. Let f(z,n,p) and F(z,n,p) denote the density function and the distribution function of Z, respectively. In t...Suppose that Y follows a χ^(p)-distribution with n degrees of freedom, and Z is the standardized form of Y. Let f(z,n,p) and F(z,n,p) denote the density function and the distribution function of Z, respectively. In this paper, we obtain the asymptotic expansion for f(z,n,p) and F(z,n,p). The validity of these results is illuminated by some numerical examples. We also investigate the power function of χ^(p)-test by the asymptotic expansion.展开更多
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 meth...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 the p _norm distribution law.The p _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 when p is larger than 1.Using p _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 the p _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 the p _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 parameter p is larger than 1 and less than 2 or p 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 w展开更多
基金Joint supported by Hubei Provincial Natural Science Foundation and Huangshi of China (2022CFD042)。
文摘Suppose that Y follows a χ^(p)-distribution with n degrees of freedom, and Z is the standardized form of Y. Let f(z,n,p) and F(z,n,p) denote the density function and the distribution function of Z, respectively. In this paper, we obtain the asymptotic expansion for f(z,n,p) and F(z,n,p). The validity of these results is illuminated by some numerical examples. We also investigate the power function of χ^(p)-test by the asymptotic expansion.
文摘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 the p _norm distribution law.The p _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 when p is larger than 1.Using p _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 the p _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 the p _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 parameter p is larger than 1 and less than 2 or p 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 w
