Problem 8 If \(X_{1}, X_{2}, \ldots, X_{n}... [FREE SOLUTION]

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Introduction to Mathematical Statistics

Robert V. Hogg, Allen Craig, Joseph W. McKean

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 8: Problem 8

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

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.

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.

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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91影视

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

Step by step solution

Definition of Beta Distribution

Calculate the Likelihood ratio

Define the Best Critical Region

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