On the relationship between two asymptotic expansions for the distribution of sample mean and its applications. (English) Zbl 0942.62022

Summary: Although the cumulative distribution function may be differentiated to obtain the corresponding density function, whether or not a similar relationship exists between their asymptotic expansions remains a question. We provide a rigorous argument to prove that R. Lugannani and S. Rice’s [Adv. Appl. Probab. 12, 475-490 (1980; Zbl 0425.60042)] asymptotic expansion for the cumulative distribution function of the mean of a sample of i.i.d. observations may be differentiated to obtain H.E. Daniels’ [Ann. Math. Statistics 25, 631-650 (1954; Zbl 0058.35404); Biometrika 70, 89-96 (1983; Zbl 0539.62028)] asymptotic expansion for the corresponding density function. We then apply this result to study the relationship between the truncated versions of the two series, which establishes the derivative of a truncated Lugannani and Rice series as an alternative asymptotic approximation for the density function. This alternative approximation in general does not need to be renormalized. Numerical examples demonstrating its accuracy are included.


62E20 Asymptotic distribution theory in statistics
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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