Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid with the distribution \(N\left(\theta, \sigma^{2}\right),-\infty<\theta<\infty\). Prove that a necessary and sufficient condition that the statistics \(Z=\sum_{1}^{n} a_{i} X_{i}\) and \(Y=\sum_{1}^{n} X_{i}\), a complete sufficient statistic for \(\theta\), are independent is that \(\sum_{1}^{n} a_{i}=0\)

Short answer

The necessary and sufficient condition for the independence of \(Z=\sum_{1}^{n} a_{i} X_{i}\) and \(Y=\sum_{1}^{n} X_{i}\), a complete sufficient statistic for \(\theta\), is \(\sum_{i=1}^{n} a_{i} = 0\).

Step-by-step solution

Definition of Complete Sufficient Statistic

A statistic is said to be a complete sufficient statistic for a parameter, if it captures all the information about the parameter in the sample and no other statistic which can be a function of this statistic can provide additional information about the parameter. In our case, \(Y=\sum_{1}^{n} X_{i}\) is a complete sufficient statistic for \(\theta\).

Express Z in terms of Y

Given statistics, \(Z=\sum_{1}^{n} a_{i} X_{i}\) and \(Y=\sum_{1}^{n} X_{i}\). We know that the two statistics are independent if and only if \(E[Z|Y]=E[Z]\). To show this, express Z as \(\sum_{1}^{n} a_{i} X_{i} = \left(\sum_{1}^{n} a_{i}\right)\left(\frac{1}{n} \sum_{1}^{n} X_{i}\right) = bY\), where \(b = \sum_{1}^{n} a_{i}\). Substituting Z, we get \(E[bY|Y]=bE[Y|Y]\). This equals \(bY\) as \(E[Y|Y]\) equals Y.

Evaluate Expectation of Z

Next, calculate \(E[Z]\) where we substitute Z as \(bY\). This gives \(E[Z] = E[bY] = bE[Y]\) as b is a constant. But since Y is a complete sufficient statistic for \(\theta\) with a normal distribution, \(E[Y] = n\theta\).

Comparing the two expectations

Now equate the two expectations. We have \(E[bY|Y] = bY\) and \(E[Z] = bn\theta\). If the two expectations are equal, Z and Y are independent. So, \(bY = bn\theta\). This equation holds true for all Y if and only if \(b=0\), implying \(\sum_{i=1}^{n} a_{i} = 0\). Therefore \(\sum_{i=1}^{n} a_{i} = 0\) is a necessary and sufficient condition for the independence of Z and Y.