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| DIRECT EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS AND DENSITY FUNCTIONS OF χ2-TYPE AND t-TYPE DISTRIBUTED RANDOM VARIABLES |
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Citation： |
Zheng Zukang.DIRECT EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS AND DENSITY FUNCTIONS OF χ2-TYPE AND t-TYPE DISTRIBUTED RANDOM VARIABLES[J].Chinese Annals of Mathematics B,1996,17(3):289~300 |
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Net amount： 842 |
Authors： |
Zheng Zukang; |
Foundation： |
Project supported by the Doctoral Programme Foundation
and the National Natural Science Foundation of China |
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| Abstract： |
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. |
Keywords： |
Edgeworth expansion, Distribution function,
Density function |
Classification： |
62E20, 60F05 |
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