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

Covariance Autocorrelation## Preview

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