## The Annals of Statistics

- Ann. Statist.
- Volume 15, Number 4 (1987), 1683-1698.

### Testing That a Stationary Time Series is Gaussian

#### Abstract

A class of procedures is proposed for testing the composite hypothesis that a stationary stochastic process is Gaussian. Requiring very limited prior knowledge about the structure of the process, the tests rely on quadratic forms in deviations of certain sample statistics from their population counterparts, minimized with respect to the unknown parameters. A specific test is developed, which employs differences between components of the sample and Gaussian characteristic functions, evaluated at certain points on the real line. By demonstrating that, under $H_0$, the normalized empirical characteristic function converges weakly to a continuous Gaussian process, it is shown that the test remains valid when arguments of the characteristic functions are in certain ways data dependent.

#### Article information

**Source**

Ann. Statist., Volume 15, Number 4 (1987), 1683-1698.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350618

**Digital Object Identifier**

doi:10.1214/aos/1176350618

**Mathematical Reviews number (MathSciNet)**

MR913582

**Zentralblatt MATH identifier**

0644.62093

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 60G15: Gaussian processes

**Keywords**

Chi-squared test empirical characteristic function Gaussian process goodness-of-fit test normal distribution spectral density stochastic process weak convergence

#### Citation

Epps, T. W. Testing That a Stationary Time Series is Gaussian. Ann. Statist. 15 (1987), no. 4, 1683--1698. doi:10.1214/aos/1176350618. https://projecteuclid.org/euclid.aos/1176350618