Statistics | Part IB, 2001

What is meant by a generalized likelihood ratio test? Explain in detail how to perform such a test

Let X1,,XnX_{1}, \ldots, X_{n} be independent random variables, and let XiX_{i} have a Poisson distribution with unknown mean λi,i=1,,n\lambda_{i}, i=1, \ldots, n.

Find the form of the generalized likelihood ratio statistic for testing H0:λ1==λnH_{0}: \lambda_{1}=\ldots=\lambda_{n}, and show that it may be approximated by

1Xˉi=1n(XiXˉ)2,\frac{1}{\bar{X}} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2},

where Xˉ=n1i=1nXi\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}.

If, for n=7n=7, you found that the value of this statistic was 27.327.3, would you accept H0H_{0} ? Justify your answer.

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