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If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a beta distribution with parameters \(\alpha=\beta=\theta>0\), find a best critical region for testing \(H_{0}: \theta=1\) against \(H_{1}: \theta=2\)

Short Answer

Expert verified
The best critical region \(C = \{ x: \lambda \le k \}\) is determined by calculating the likelihood ratio and setting up criteria based on the level of significance \(\alpha\). Here, \(k\) is a constant determined to ensure that the size of the critical region is equal to the given level of significance \(\alpha\).

Step by step solution

01

Definition of Beta Distribution

The probability density function of the Beta distribution is given by: \[f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\] for \(0 \le x \le 1\) and \(\alpha > 0\), \(\beta > 0\). Where \(B(\alpha,\beta)\) stands for the beta function.
02

Calculate the Likelihood ratio

The likelihood ratio for two simple hypotheses \(H_{0}: \theta=\theta_{0}\) and \(H_{1}: \theta=\theta_{1}\) is computed as: \[ \lambda = \frac{L(\theta_{0})}{L(\theta_{1})}\] where \(L(\theta)\) is the likelihood function. In this case, \(\theta_{0} = 1\), \(\theta_{1} = 2\), hence \[ \lambda = \frac{L(1)}{L(2)} \] Calculate the likelihood ratios by replacing all \(\theta\) with 1 in \(H_{0}\) and 2 in \(H_{1}\) in the likelihood function derived from the Beta distribution.
03

Define the Best Critical Region

Define the best critical region (rejection region) as: \[C = \{ x: \lambda \le k \}\] for some constant \(k > 0\). In this step, find the value of \(k\) such that the size of the critical region is equal to the given level of significance \(\alpha\). Typically, \(\alpha\) is set to be 0.05.

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Most popular questions from this chapter

Show that the likelihood ratio principle leads to the same test when testing a simple hypothesis \(H_{0}\) against an alternative simple hypothesis \(H_{1}\), as that given by the Neyman-Pearson theorem. Note that there are only two points in \(\Omega\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid with pmf \(f(x ; p)=p^{x}(1-p)^{1-x}, x=0,1\), zero elsewhere. Show that \(C=\left\\{\left(x_{1}, \ldots, x_{n}\right): \sum_{1}^{n} x_{i} \leq c\right\\}\) is a best critical region for testing \(H_{0}: p=\frac{1}{2}\) against \(H_{1}: p=\frac{1}{3} .\) Use the Central Limit Theorem to find \(n\) and \(c\) so that approximately \(P_{H_{0}}\left(\sum_{1}^{n} X_{i} \leq c\right)=0.10\) and \(P_{H_{1}}\left(\sum_{1}^{n} X_{i} \leq c\right)=0.80\).

Let \(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample of size 10 from a normal distribution \(N\left(0, \sigma^{2}\right) .\) Find a best critical region of size \(\alpha=0.05\) for testing \(H_{0}: \sigma^{2}=1\) against \(H_{1}: \sigma^{2}=2 .\) Is this a best critical region of size \(0.05\) for testing \(H_{0}: \sigma^{2}=1\) against \(H_{1}: \sigma^{2}=4 ?\) Against \(H_{1}: \sigma^{2}=\sigma_{1}^{2}>1 ?\)

Let \(Y_{1}

Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) be a random sample from a bivariate normal distribution with \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}, \rho=\frac{1}{2}\), where \(\mu_{1}, \mu_{2}\), and \(\sigma^{2}>0\) are unknown real numbers. Find the likelihood ratio \(\Lambda\) for testing \(H_{0}: \mu_{1}=\mu_{2}=0, \sigma^{2}\) unknown against all alternatives. The likelihood ratio \(\Lambda\) is a function of what statistic that has a well- known distribution?

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