Complete asymptotic expansions for the density function of t-distribution

Publication date: October 2018Source: Statistics & Probability Letters, Volume 141Author(s): Chao-Ping ChenAbstractThe density function of t-distribution with n degrees of freedom is given by the following formula: fn(x)=Γ(n+12)πnΓ(n2)1+x2n−(n+1)∕2,x>0.Ding (1988) obtained the following asymptotic formula in terms of 1∕n: fn(x)=φ(x){1+x4−2x2−14n+3x8−28x6+30x4+12x2+396n2+O1n3},n→∞,which derives the known result limn→∞fn(x)=φ(x), where φ(x)=12πe−x2∕2 is the probability density function of the standard normal distribution. In this paper, we develop Ding’s result to produce a complete asymptotic expansion.

Source: Statistics and Probability Letters - July 10, 2018 Category: Statistics Source Type: research