## The Annals of Statistics

- Ann. Statist.
- Volume 25, Number 5 (1997), 2200-2209.

### On the relationship between two asymptotic expansions for the distribution of sample mean and its applications

#### Abstract

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 Lugannani and Rice's asymptotic expansion for the cumulative distribution function of the mean of a sample of i.i.d. observations may be differentiated to obtain Daniels's 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.

#### Article information

**Source**

Ann. Statist., Volume 25, Number 5 (1997), 2200-2209.

**Dates**

First available in Project Euclid: 20 November 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1069362394

**Digital Object Identifier**

doi:10.1214/aos/1069362394

**Mathematical Reviews number (MathSciNet)**

MR1474090

**Zentralblatt MATH identifier**

0942.62022

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]

**Keywords**

Saddlepoint approximation Daniels's series Lugannani and Rice's series asymptotic expansion uniform validity

#### Citation

Routledge, Rick; Tsao, Min. On the relationship between two asymptotic expansions for the distribution of sample mean and its applications. Ann. Statist. 25 (1997), no. 5, 2200--2209. doi:10.1214/aos/1069362394. https://projecteuclid.org/euclid.aos/1069362394