Let X1, X2, …, Xn be iid with the distribution N(θ, σ^2),-∞<θ<∞. Prove that a necessary and suff

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Let X1, X2, …, Xn be iid with the distribution N(θ,...

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 $\sum_{1}^{n} a_{i}=0$

Key Concepts

Linear Combinations of Normal Random Variables

Any linear combination of normally distributed random variables is itself normally distributed. This property greatly simplifies analysis because it allows the transformation of the original set of variables into new variables whose joint distribution is also multivariate normal. In the context of the problem, the statistic Z is a linear combination of the variables X?, X?, ..., X?, and its behavior, particularly in relation to the sum Y, is fully described by the means, variances, and covariances of the original variables.

Independence in Multivariate Normal Distributions

In a multivariate normal setting, the concepts of uncorrelatedness and independence coincide. This means that when two variables have a covariance of zero, they are not only uncorrelated but also independent. This property is crucial in the problem, as the independence of Z and Y is established by verifying that their covariance is zero.

Sufficient and Complete Statistics

A sufficient statistic is one that captures all the information about a parameter contained in the data, and a statistic is complete if no non-trivial function of it has an expected value of zero for all parameter values. In this problem, Y, being the sum of the iid observations, is a complete sufficient statistic for ?. This concept is important because it implies that any other statistic that is independent of Y does not contain additional information about ?, thus emphasizing the role of Y in parameter estimation.

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