Question: Let \(X_{1}, \ldots, X_{n}\) be a random sample from a Pareto distribution, denoted by Pareto \((\gamma, \beta)\), of which the pdf is given by

and \(f(x \mid \gamma, \beta)=0\) if \(x<\gamma\), where \(\gamma\) and \(\beta\) are two positive parameters. Here, we assme \(\gamma\) known and \(\beta\) is the only unknown parameter.

1. Consider testing \(H_{0}: \beta=\beta_{0}\) versus \(H_{1}: \beta=\beta_{1}\), where \(\beta_{0}\) and \(\beta_{1}\) are two known constants with \(\beta_{1}>\beta_{0} .\) Derive the likelihood ratio statistic for this test.
2. Define \(W=\log X-\log \gamma\), where \(X \sim \operatorname{Pareto}(\gamma, \beta) .\) Show that \(W \sim \operatorname{exponential}(\beta)\).
3. With \(W_{i}=\log X_{i}-\log \gamma\), for \(i=1, \ldots, n\), denote by \(\bar{W}\) the sample mean of the random sample \(\left(W_{1}, \ldots, W_{n}\right) .\) Show that \(2 \beta n \bar{W} \sim \chi_{2 n}^{2}\)
4. Use the results from parts (a) and (c) to formulate a decision rule associated with a size- \(\alpha\) test for testing the hypotheses stated in part (a).
5. Is the test you provide in part (d) a uniformly most powerful test for testing \(H_{0}: \beta=\beta_{0}\) versus \(H_{1}: \beta>\beta_{0} ?\) Explain.
6. Now we observe a random sample of size 5 from Pareto \((2, \beta)\) given below,

2.5443,5.2518,2.9216,2.6347,2.3001

Consider testing \(H_{0}: \beta=0.5\) versus \(H_{1}: \beta=1\) using the size- 0.05 test developed in part

(d). What is the power of this test? What is the \(p\) -value of this test?

Price: $2.99

Solution: The downloadable solution consists of 4 pages
Deliverable: Word Document