DIRECT EXPANSIONS FOR THE DISTRIBUTIONFUNCTIONS AND DENSITY FUNCTIONS OF χ~2-TYPEAND t-TYPE DISTRIBUTED RANDOM VARIABLES

摘要

Suppose that Z1 , Z2,’’’ Zn are independent normal random variables with common meanμ and variance σ2. Then S2 and have distribution andtn-1 distribution respectively. If the normal assumption fails, there will be the remaindersof the distribution functions and density functions. This paper gives the direct expansions ofdistribution functions and density functions of S2 alld T up to o(n-1). They are more intuitiveand convenient than usual Edgeworth expansions.