2.II.12D

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

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

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

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

where $\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}$.

If, for $n=7$, you found that the value of this statistic was $27.3$, would you accept $H_{0}$ ? Justify your answer.

*Typos? Please submit corrections to this page on GitHub.*