Prescriptions for Working Statisticians pp 92-119 | Cite as

# Testing for Independence of Observations

Chapter

## Abstract

The problem addressed in this chapter is one of testing whether a sequence of random variables and the

**x**_{1},…,**x**_{ n }are independent based on a set of observations*x*_{1}, …,*x*_{ n }of these random variables. One approach to this problem is to make an assumption about the distribution of the**x**_{ i }, both when they are independent and under the alternative that they are not independent, and test one hypothesis against the other. The simplest such pair of assumptions is that the**x**_{ i }are identically distributed as*N*(0, σ^{2}) random variables and, in the case of dependence, that the dependence can be modeled as a first-order auto-regressive series wherein$$\begin{gathered} {{x}_{1}} = {{u}_{1}}, \hfill \\ {{x}_{i}} = \rho {{x}_{{i - 1}}} + {{u}_{i}},i = 2, \ldots ,n, \hfill \\ \end{gathered}$$

**u**_{ i }are independent*N*(0, σ^{2}) random variables. The case*ρ*= 0 corresponds to the case in which the**x**_{ i }are independent*N*(0, σ^{2}) random variables. Under the alternative hypothesis,*C*(**x**_{ i },**x**_{ i }) =*ρ*^{|i − j|}and*V***x**_{ i }= σ^{2}(1 +*ρ*+ … +*ρ*^{ i − 1})= σ^{2}(1 –*ρ*^{ i })/(1 –*ρ*), so that the**x**_{ i }are not even identically distributed. Section 1 studies tests of this and more complex autoregressive models as alternatives to independence, both for a sequence of observations and for regression residuals.## Keywords

Serial Correlation American Statistical Association Regression Residual Parametric Procedure Nonparametric Procedure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York Inc. 1988