DIRECT EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS AND DENSITY FUNCTIONS OF χ2-TYPE AND t-TYPE DISTRIBUTED RANDOM VARIABLES

Suppose that Z1,Z2…,Zn are independent normal random variables with common mean μ and variance σ^2. Then S^2=∑n n=1 (zi-z)^2/σ^2 and T =（n-1的平方根）-Z/（S^2/n的平方根） have x2n-1 distribution and tn-1 distribution respectively. If the normal assumption fails, there will be the remainders of the distribution functions and density functions. This paper gives the direct expansions of distribution functions and density functions of S^2 and T up to o(n^-1). They are more intuitive and convenient than usual Edgeworth expansions.

DIRECT EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS AND DENSITY FUNCTIONS OF x^2-TYPE AND t-TYPE DISTRIBUTED RANDOM VARIABLES

Suppose that $Z_1, Z_2, \cdots, Z_n$ are independent normal random variables with common mean $\mu$ and variance $\si^2.$ Then $S^2=\f{\sum_{i=1}^n \limits (Z_i-\ov Z)^2 }{\si^2}$ and $T=\f{\sqrt{n-1}\cdot \ov Z}{\sqrt{S^2/n}}$ have $\chi^2_{n-1}$ distribution and $t_{n-1}$ distribution respectively. If the normal assumption fails, there will be the remainders of the distribution functions and density functions. This paper gives the direct expansions of distribution functions and density functions of $S^2$ and $T$ up to $o(n^{-1})$. They are more intuitive and convenient than usual Edgeworth expansions.